ContentsArticles

User:Rajah2770 1Operations research 5Mathematical optimization 14Mathematical model 25Game theory 30Simulation 47Artificial intelligence 67Probability theory 94Linear programming 99Nonlinear programming 114Integer programming 116Dynamic programming 117Goal programming 130Information theory 132Cybernetics 143Systems theory 151Queueing theory 162Markov chain 166Graph theory 179

ReferencesArticle Sources and Contributors 187Image Sources, Licenses and Contributors 191

Article LicensesLicense 192

User:Rajah2770 1

User:Rajah2770

Dr.A.B.Rajib Hazarika

Dr.A.B.Rajib Hazarika [[File:File:Dr.A.B.Rajib Hazarika & his two kids.jpg||alt=]]

Dr.A.B.Rajib Hazarika with Laquit(son) and Danisha(daughter)

Born Azad Bin Rajib HazarikaJuly 2, 1970Jammu, Jammu and Kashmir, India

Residence Nagaon, Assam, India

Nationality Indian

Ethnicity AssameseMuslim

Citizenship India

Education PhD, PDF, FRAS

Alma mater University of JodhpurJai Narayan Vyas UniversityInstitute of Advanced Study in Science & Technology[1]

Kendriya Vidyalaya[2]

Poona College of Arts, Science &Commerce[3]

Occupation Assistant Professor(Lecturer), Diphu Govt. College , Diphu,Assam,India

Years active 2004- onwards

Employer Diphu Government CollegeGovernment of Assam,Assam Education Service

Known for Lecturer ,Assistant Professor,Mathematician,Academician,Fusion,Astronomy

Home town Nagaon, Assam, India

Salary Rs 40000 per month

Height 6 feet and 2 inches

Weight 100 kg

Title Doctorate, Dr., FRAS (London), Assam Education Service, AES

Boardmember of

Member of Scientific and Technical committee & Editorial review board of Natuaral and Applied sciences World Academy ofScience ,Engineering & Technology[4]

Religion Sunni Islam,

Spouse Helmin Begum Hazarika

Children Laquit Ali Hazarika(son), Danisha Begum Hazarika(daughter)

Parents Rosmat Ali Hazarika@Rostam Ali Hazarika@Roufat Ali Hazarika and Anjena Begum Hazarika

Call-sign Drabrh or Raja

Website

[5][6] [7] [8] [9]

User:Rajah2770 2

Dr.A.B.Rajib Hazarika with Laquit (son) and Danisha(daughter)

Dr.A.B.Rajib Hazarika (born July 02, 1970, in Jammu, Jammu and Kashmir, India) is AssistantProfessor(Lecturer) Diphu Government College ,Diphu in Karbi Anglong district , Government of Assam [10] , [11] ,Karbi Anglong,Assam's largest conglomerate by Government of Assam . He is also the Fellow of RoyalAstronomical Society[12] ,London ,Member of International Association of Mathematical Physics, World Academyof Science ,Engineering & Technology, Focus Fusion Society, Dense Plasma Focus, Plasma Science Society ofIndia, Assam Science Society, Assam academy of mathematics,International Atomic Energy Agency,Nuclear andPlasma Sciences Society,Society of Industrial and Applied Mathematics,German Academy of Mathematics andMechanics,Fusion Science & Technology Society,Indian National Science Academy,Indian Science CongressAssociation,Advisory Committee of Mathematical Education,Royal Society,International Biographical Centre.

Early lifeDr.A.B.Rajib Hazarika was born into the famous Hazarika family, a prominent family belonging to Dhing's wealthyMuslim Assamese community of Nagaon district. He was born to Anjena Begum Hazarika and Rusmat AliHazarika. He is eldest of two childrens of his parents younger one is a Shamim Ara Rahman(nee Hazarika)daughter .

Early careerDr.A.B.Rajib Hazarika completed his PhD degree in Mathematics from J N Vyas University of Jodhpur in 1995 withspecialization in Plasma instability, the thesis was awarded “best thesis” by Association of Indian Universities in1998 and the Post-Doctoral Fellow Program from Institute of Advanced Study in Science & Technology [13] inGuwahati Assam in 1998 as Research Associate in Plasma Physics Division in theory group studying the Sheathphenomenon. As a Part-time Lecturer in Nowgong college, Assam before joining the present position in DiphuGovernment College ,Diphu in Karbi Anglong district[14] ,[15] He is a member of the wikipedia[16] , [17] . He isFellow of Royal Astronomical Society[18] ,member of International Association Mathematical Physics[19] , memberof World Academy of Science,Engineering & Technology [20] ,[21] , member of Plasma science Society of India [22] ,[23] ,member of Focus Fusion Society forum [24] ,member of Dense Plasma Focus [25] , Member of Assam ScienceSociety [26] , Member of Assam Academy of Mathematics [27]

User:Rajah2770 3

He joined the Diphu Government College in July2004 as Lecturer in Mathematics (Gazetted officer), through AssamPublic Service commission [28] in Assam Education Service [29] , AES-I. [30] now redesignated as AssistantProfessor.

CareerIn May 1993, Dr.A.B.Rajib Hazarika was awarded Junior Research Fellowship,University Grants Commission,National Eligibility Test and eligibility for Lecturership ,Govt. of India and worked as JRF(UGC,NET) inDepartment of Mathematics and Statistics of J N Vyas University in Jodhpur. Later on in May 1995 got SeniorResearch Fellowship(UGC,NET) and continued research for completion of PhD on 27th Dec 1995 .From 1993onwards taught in Kamala Nehru College for women, Jodhpur and in Faculty of Science in J N Vyas University inJodhpur up to the completion of PhD .In 1998 May joined Plasma Physics Division of Institute of Advanced Studyin Science & Technology in Guwahati as Research Associate for PDF in theory group to study the sheathphenomena of National Fusion Programme [31] of Govt. of India . Then joined Nowgong College as a part-timeLecturer after which in 2004, July joined the present position of Lecturer in Diphu Government College which isredesignated as Assistant Professor.

ResearchDuring PhD [32] [33] [34] [35] [36]

The research was based on Astronomy,Astrophysics, Geophysics , for plasma instability with the title of thesis as“Some Problems of instabilities in partially ionized and fully ionized plasmas” which later on in 1998 was assessedas best thesis of the year by Association of Indian Universities in New Delhi. He is known for Bhatia-HazarikalimitResearch at Diphu Govt. College [37] , [38] [39] [40] [41] [42] [43] [44] Applied for patent in US patent andtrademarks office [45] [46]

Research guidance is given to students in Mathematics for MPhil. He has written six books entitled Inventions ofDr.A.B.Rajib Hazarika on future devices and Dr.A.B.Rajib Hazarika's Pattern recognition on fusion,Application of Dr.A.B.Rajib Hazarika's conceptual devices , Green tecnology for next genration , Invention ofDr.A.B.Rajib Hazarika's devices ,VASIMR DANISHA:A Hall Thruster Space Odyssey ,[47] , [48] , [49]

He has derived a formula Hazarika's constant for VASIMR DANISHA as Hazarika constant Ch=1+4sin3φ sin θ-2sinφ-2sin θ the value is 2.646

Personal lifeDr.A.B.Rajib Hazarika has a metallic Scarlet red Tata Indigo CS of Tata motors make and loves to drive himself.Heis married to Helmin Begum Hazarika and have two chidrens Laquit(son) and Danisha(daughter).

Quotes• "Fakir(saint) and lakir(line) stops at nothing but at destination"• "Expert criticizes the wrong but demonstrates the right thing"• “Intellectuals are measured by their brain not by their age and experience”• “Two type of persons are happy in life one who knows everything another who doesn’t know anything”• “Implosion in device to prove every notion wrong for fusion”• “Meditation gives fakir(saint) long life and fusion devices the long lasting confinement”

User:Rajah2770 4

Awards and recognitionDr.A.B.Rajib Hazarika got Junior Research Fellowship,Government of IndiaSenior Research Fellowship,Government of IndiaResearch AssociateshipDSTGovernment of IndiaFellowof Royal Astronomical Society [50]

Member of Advisory committee of Mathematical Education Royal Society LondonMember of Scientific and Technical committee & editorial review board on Natural and applied sciences of WorldAcademy of Science ,Engineering &Technology [51]

Leading professional of the world-2010 as noted and eminent professional from International Biographical CentreCambridge

References[1] http:/ / www. iasst. in[2] http:/ / www. kvafsdigaru. org[3] http:/ / www. akipoonacollege. com[4] http:/ / www. waset. org/ NaturalandAppliedSciences. php?page=45[5] http:/ / www. facebook. com/ Drabrajib[6] http:/ / in. linkedin. com/ pub/ dr-a-b-rajib-hazarika/ 25/ 506/ 549[7] http:/ / en. wikipedia. org/ wiki/ Special:Contributions/ Drabrh[8] http:/ / www. diphugovtcollege. org[9] http:/ / www. karbianglong. nic. in/ diphugovtcollege. org/ teaching. html[10] http:/ / www. karbianglong. nic. in/ diphugovtcollege/ teaching. html[11] http:/ / www. diphugovtcollege. org/ DGC%20prospectus%2008-09. pdf[12] http:/ / www. ras. org. uk/ member?recid==5531[13] http:/ / www. iasst. in[14] {{cite web|url=http:/ / www. diphugovtcollege. org/ DGC%20prospectus%2008-09. pdf[15] http:/ / karbianglong. nic. in/ diphugovtcollege/ teaching. html[16] http:/ / en. wikipedia. org/ wiki/ User:Drabrh[17] http:/ / en. wikipedia. org/ wiki/ Special:Contributions/ Drabrh[18] http:/ / www. ras. org. uk/ member?recid=5531,[19] http:/ / www. iamp. org/ bulletins/ old-bulletins/ 201001. pdf[20] http:/ / www. waset. org/ NaturalandAppliedSciences. php?page=45[21] http:/ / www. waset. org/ Search. php?page=68& search=[22] http:/ / www. plasma. ernet. in/ ~pssi/ member/ pssi_new04. doc[23] http:/ / www. ipr. res. in/ ~pssi/ member/ pssidir_new-04. doc[24] http:/ / www. focusfusion. org/ index. php/ forums/ member/ 4165[25] http:/ / www. denseplasmafocus. org/ index. php/ forum/ member/ 4165[26] http:/ / www. assamsciencesociety. org/ member[27] http:/ / www. aam. org. in/ member/ 982004[28] http:/ / apsc. nic. in[29] http:/ / aasc. nic. in/ . . . / Education%20Department/ The%20Assam%20Education%20Service%20Rules%201982. pdf[30] (http:/ / www. diphugovtcollege. org/ DGC prospests 08-09. pdf)[31] http:/ / nfp. pssi. in[32] http:/ / www. iopscience. iop. org/ 1402-4896/ 51/ 6/ 012/ pdf/ physcr_51_6_012. pdf[33] http:/ / www. iopsciences. iop. org/ 1402-4896/ 53/ 1/ 011/ pdf/ 1402-4896_53_1_011. pdf,[34] http:/ / www. niscair. res. in/ sciencecommunication/ abstractingjournals/ isa_1jul08. asp[35] http:/ / en. wiktionary. org/ wiki/ Wikitionary%3ASandbox[36] http:/ / adsabs. harvard. edu/ abs/ 1996PhyS. . 53. . . 578[37] http:/ / en. wikipedia. org/ wiki/ Special:Contributions/ Drabrh/ File:Drabrhdouble_trios_saiph_star01. pdf[38] http:/ / en. wikipedia. org/ wiki/ File:Drabrh_bayer_rti. pdf[39] http:/ / en. wikipedia. org/ wiki/ File:Columb_drabrh. pdf[40] http:/ / en. wikipedia. org/ wiki/ File:Drabrh_double_trios. pdf[41] http:/ / en. wikipedia. org/ wiki/ File:Drabrhiterparabolic2007. pdf[42] http:/ / en. wikipedia. org/ wiki/ File:Drabrh_mctc_feedbackloop. pdf[43] http:/ / en. wikipedia. org/ wiki/ File:Drabrh_tasso_07. pdf

User:Rajah2770 5

[44] http:/ / en. wikipedia. org/ wiki/ File:Abstracts. pdf?page=2[45] http:/ / upload. wikimedia. org/ wikipedia/ en/ 5/ 50/ EfilingAck5530228. pdf[46] http:/ / upload. wikimedia. org/ wikipedia/ en/ c/ c4/ EfilingAck3442787. pdf[47] http:/ / www. pothi. com[48] http:/ / i-proclaimbookstore. com[49] http:/ / ipppserver. homelinux. org:8080/ view/ creators/ Hazarika=3ADr=2EA=2EB=2ERajib=3A=3A. html[50] http:/ / www. ras. org. uk/ members?recid=5531[51] http:/ / www. waset. org/ NaturalandAppliedSciences. php?page=46

External links• (http:/ / www. diphugovtcollege. org/ )• Dr.A.B.Rajib Hazarika's profile on the Linkedin Website (http:/ / in. linkedin. com/ pub/ dr-a-b-rajib-hazarika/ 25/

506/ 549=)• (http:/ / www. facebook. com/ Drabrajib)Rajah2770 (talk) 18:12, 7 February 2011 (UTC)

Operations researchOperations research (also referred to as decision science, or management science) is an interdisciplinarymathematical science that focuses on the effective use of technology by organizations. In contrast, many otherscience & engineering disciplines focus on technology giving secondary considerations to its use.Employing techniques from other mathematical sciences — such as mathematical modeling, statistical analysis, andmathematical optimization — operations research arrives at optimal or near-optimal solutions to complexdecision-making problems. Because of its emphasis on human-technology interaction and because of its focus onpractical applications, operations research has overlap with other disciplines, notably industrial engineering andmanagement science, and draws on psychology and organization science. Operations Research is often concernedwith determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of somereal-world objective. Originating in military efforts before World War II, its techniques have grown to concernproblems in a variety of industries.[1]

OverviewOperational research (OR) encompasses a wide range of problem-solving techniques and methods applied in thepursuit of improved decision-making and efficiency.[2] Some of the tools used by operational researchers arestatistics, optimization, probability theory, queuing theory, game theory, graph theory, decision analysis,mathematical modeling and simulation. Because of the computational nature of these fields, OR also has strong tiesto computer science and analytics. Operational researchers faced with a new problem must determine which of thesetechniques are most appropriate given the nature of the system, the goals for improvement, and constraints on timeand computing power.Work in operational research and management science may be characterized as one of three categories:[3]

• Fundamental or foundational work takes place in three mathematical disciplines: probability theory, mathematicaloptimization, and dynamical systems theory.

• Modeling work is concerned with the construction of models, analyzing them mathematically, implementing themon computers, solving them using software tools, and assessing their effectiveness with data. This level is mainlyinstrumental, and driven mainly by statistics and econometrics.

• Application work in operational research, like other engineering and economics' disciplines, attempts to usemodels to make a practical impact on real-world problems.

Operations research 6

The major subdisciplines in modern operational research, as identified by the journal Operations Research,[4] are:• Computing and information technologies• Decision analysis• Environment, energy, and natural resources• Financial engineering• Manufacturing, service sciences, and supply chain management• Marketing Engineering [5]

• Policy modeling and public sector work• Revenue management• Simulation• Stochastic models• Transportation

HistoryAs a formal discipline, operational research originated in the efforts of military planners during World War II. In thedecades after the war, the techniques began to be applied more widely to problems in business, industry and society.Since that time, operational research has expanded into a field widely used in industries ranging from petrochemicalsto airlines, finance, logistics, and government, moving to a focus on the development of mathematical models thatcan be used to analyze and optimize complex systems, and has become an area of active academic and industrialresearch.[1]

Historical originsIn the World War II era, operational research was defined as "a scientific method of providing executive departmentswith a quantitative basis for decisions regarding the operations under their control."[6] Other names for it includedoperational analysis (UK Ministry of Defence from 1962)[7] and quantitative management.[8]

Prior to the formal start of the field, early work in operational research was carried out by individuals such as CharlesBabbage. His research into the cost of transportation and sorting of mail led to England's universal "Penny Post" in1840, and studies into the dynamical behaviour of railway vehicles in defence of the GWR's broad gauge.[9] PercyBridgman brought operational research to bear on problems in physics in the 1920s and would later attempt toextend these to the social sciences.[10] The modern field of operational research arose during World War II.Modern operational research originated at the Bawdsey Research Station in the UK in 1937 and was the result of aninitiative of the station's superintendent, A. P. Rowe. Rowe conceived the idea as a means to analyse and improve theworking of the UK's early warning radar system, Chain Home (CH). Initially, he analyzed the operating of the radarequipment and its communication networks, expanding later to include the operating personnel's behaviour. Thisrevealed unappreciated limitations of the CH network and allowed remedial action to be taken.[11]

Scientists in the United Kingdom including Patrick Blackett (later Lord Blackett OM PRS), Cecil Gordon, C. H.Waddington, Owen Wansbrough-Jones, Frank Yates, Jacob Bronowski and Freeman Dyson, and in the United Stateswith George Dantzig looked for ways to make better decisions in such areas as logistics and training schedules. Afterthe war it began to be applied to similar problems in industry.

Operations research 7

Second World WarDuring the Second World War close to 1,000 men and women in Britain were engaged in operational research.About 200 operational research scientists worked for the British Army.[12]

Patrick Blackett worked for several different organizations during the war. Early in the war while working for theRoyal Aircraft Establishment (RAE) he set up a team known as the "Circus" which helped to reduce the number ofanti-aircraft artillery rounds needed to shoot down an enemy aircraft from an average of over 20,000 at the start ofthe Battle of Britain to 4,000 in 1941.[13]

In 1941 Blackett moved from the RAE to the Navy, first to the Royal Navy's Coastal Command, in 1941 and thenearly in 1942 to the Admiralty.[14] Blackett's team at Coastal Command's Operational Research Section (CC-ORS)included two future Nobel prize winners and many other people who went on to be preeminent in their fields.[15]

They undertook a number of crucial analyses that aided the war effort. Britain introduced the convoy system toreduce shipping losses, but while the principle of using warships to accompany merchant ships was generallyaccepted, it was unclear whether it was better for convoys to be small or large. Convoys travel at the speed of theslowest member, so small convoys can travel faster. It was also argued that small convoys would be harder forGerman U-boats to detect. On the other hand, large convoys could deploy more warships against an attacker.Blackett's staff showed that the losses suffered by convoys depended largely on the number of escort vessels present,rather than on the overall size of the convoy. Their conclusion, therefore, was that a few large convoys are moredefensible than many small ones.[16]

While performing an analysis of the methods used by RAF Coastal Command to hunt and destroy submarines, oneof the analysts asked what colour the aircraft were. As most of them were from Bomber Command they were paintedblack for nighttime operations. At the suggestion of CC-ORS a test was run to see if that was the best colour tocamouflage the aircraft for daytime operations in the grey North Atlantic skies. Tests showed that aircraft paintedwhite were on average not spotted until they were 20% closer than those painted black. This change indicated that30% more submarines would be attacked and sunk for the same number of sightings.[17]

Other work by the CC-ORS indicated that on average if the trigger depth of aerial delivered depth charges (DCs) waschanged from 100 feet to 25 feet, the kill ratios would go up. The reason was that if a U-boat saw an aircraft onlyshortly before it arrived over the target then at 100 feet the charges would do no damage (because the U-boatwouldn't have time to descend as far as 100 feet), and if it saw the aircraft a long way from the target it had time toalter course under water so the chances of it being within the 20 feet kill zone of the charges was small. It was moreefficient to attack those submarines close to the surface when these targets' locations were better known than toattempt their destruction at greater depths when their positions could only be guessed. Before the change of settingsfrom 100 feet to 25 feet, 1% of submerged U-boats were sunk and 14% damaged. After the change, 7% were sunkand 11% damaged. (If submarines were caught on the surface, even if attacked shortly after submerging, the numbersrose to 11% sunk and 15% damaged). Blackett observed "there can be few cases where such a great operational gainhad been obtained by such a small and simple change of tactics".[18]

Bomber Command's Operational Research Section (BC-ORS), analysed a report of a survey carried out by RAFBomber Command. For the survey, Bomber Command inspected all bombers returning from bombing raids overGermany over a particular period. All damage inflicted by German air defences was noted and the recommendationwas given that armour be added in the most heavily damaged areas. Their suggestion to remove some of the crew sothat an aircraft loss would result in fewer personnel loss was rejected by RAF command. Blackett's team insteadmade the surprising and counter-intuitive recommendation that the armour be placed in the areas which werecompletely untouched by damage in the bombers which returned. They reasoned that the survey was biased, since itonly included aircraft that returned to Britain. The untouched areas of returning aircraft were probably vital areas,which, if hit, would result in the loss of the aircraft.(Info to help verify some/most of the above paragraph is found in "Dirty Little Secrets of the Twentieth Century"authored by James F. Dunnigan on pages 215-217

Operations research 8

Map of Kammhuber Line

When Germany organised its air defences into the Kammhuber Line, it wasrealised that if the RAF bombers were to fly in a bomber stream they couldoverwhelm the night fighters who flew in individual cells directed to theirtargets by ground controllers. It was then a matter of calculating the statisticalloss from collisions against the statistical loss from night fighters to calculatehow close the bombers should fly to minimise RAF losses.[19]

The "exchange rate" ratio of output to input was a characteristic feature ofoperational research. By comparing the number of flying hours put in byAllied aircraft to the number of U-boat sightings in a given area, it waspossible to redistribute aircraft to more productive patrol areas. Comparisonof exchange rates established "effectiveness ratios" useful in planning. Theratio of 60 mines laid per ship sunk was common to several campaigns:German mines in British ports, British mines on German routes, and UnitedStates mines in Japanese routes.[20]

Operational research doubled the on-target bomb rate of B-29s bombingJapan from the Marianas Islands by increasing the training ratio from 4 to 10 percent of flying hours; revealed thatwolf-packs of three United States submarines were the most effective number to enable all members of the pack toengage targets discovered on their individual patrol stations; revealed that glossy enamel paint was more effectivecamouflage for night fighters than traditional dull camouflage paint finish, and the smooth paint finish increasedairspeed by reducing skin friction.[20]

On land, the operational research sections of the Army Operational Research Group (AORG) of the Ministry ofSupply (MoS) were landed in Normandy in 1944, and they followed British forces in the advance across Europe.They analysed, among other topics, the effectiveness of artillery, aerial bombing, and anti-tank shooting.

After World War IIWith expanded techniques and growing awareness of the field at the close of the war, operational research was nolonger limited to only operational, but was extended to encompass equipment procurement, training, logistics andinfrastructure.Individuals such as Stafford Beer, George Dantzig and András Prékopa pioneered early academic efforts inoperational research.

Problems addressed with operational research• critical path analysis or project planning: identifying those processes in a complex project which affect the overall

duration of the project• floorplanning: designing the layout of equipment in a factory or components on a computer chip to reduce

manufacturing time (therefore reducing cost)• network optimization: for instance, setup of telecommunications networks to maintain quality of service during

outages• Allocation problems• Facility location• Assignment Problems:

• Assignment problem• Generalized assignment problem• Quadratic assignment problem• Weapon target assignment problem

Operations research 9

• Bayesian search theory : looking for a target• optimal search• routing, such as determining the routes of buses so that as few buses are needed as possible• supply chain management: managing the flow of raw materials and products based on uncertain demand for the

finished products• efficient messaging and customer response tactics• automation: automating or integrating robotic systems in human-driven operations processes• globalization: globalizing operations processes in order to take advantage of cheaper materials, labor, land or

other productivity inputs• transportation: managing freight transportation and delivery systems (Examples: LTL Shipping, intermodal

freight transport)• scheduling:

• personnel staffing• manufacturing steps• project tasks• network data traffic: these are known as queueing models or queueing systems.• sports events and their television coverage

• blending of raw materials in oil refineries• determining optimal prices, in many retail and B2B settings, within the disciplines of pricing scienceOperational research is also used extensively in government where evidence-based policy is used.

Management scienceIn 1967 Stafford Beer characterized the field of management science as "the business use of operations research".[21]

However, in modern times the term management science may also be used to refer to the separate fields oforganizational studies or corporate strategy. Like operational research itself, management science (MS), is aninterdisciplinary branch of applied mathematics devoted to optimal decision planning, with strong links witheconomics, business, engineering, and other sciences. It uses various scientific research-based principles, strategies,and analytical methods including mathematical modeling, statistics and numerical algorithms to improve anorganization's ability to enact rational and meaningful management decisions by arriving at optimal or near optimalsolutions to complex decision problems. In short, management sciences help businesses to achieve their goals usingthe scientific methods of operational research.The management scientist's mandate is to use rational, systematic, science-based techniques to inform and improvedecisions of all kinds. Of course, the techniques of management science are not restricted to business applicationsbut may be applied to military, medical, public administration, charitable groups, political groups or communitygroups.Management science is concerned with developing and applying models and concepts that may prove useful inhelping to illuminate management issues and solve managerial problems, as well as designing and developing newand better models of organizational excellence.[22]

The application of these models within the corporate sector became known as Management science.[23]

Operations research 10

TechniquesSome of the fields that have considerable overlap with Management Science include:

• Data mining • Mathematical optimization• Decision analysis • Probability and statistics• Engineering • Project management• Forecasting • Simulation• Game theory • Social network/Transportation forecasting models• Industrial engineering • Supply chain management• Logistics • Financial engineering• Mathematical modeling

Applications of management scienceApplications of management science are abundant in industry as airlines, manufacturing companies, serviceorganizations, military branches, and in government. The range of problems and issues to which managementscience has contributed insights and solutions is vast. It includes:[22]

• scheduling airlines, including both planes and crew,• deciding the appropriate place to site new facilities such as a warehouse, factory or fire station,• managing the flow of water from reservoirs,• identifying possible future development paths for parts of the telecommunications industry,• establishing the information needs and appropriate systems to supply them within the health service, and• identifying and understanding the strategies adopted by companies for their information systemsManagement science is also concerned with so-called ”soft-operational analysis”, which concerns methods forstrategic planning, strategic decision support, and Problem Structuring Methods (PSM). In dealing with these sorts ofchallenges mathematical modeling and simulation are not appropriate or will not suffice. Therefore, during the past30 years, a number of non-quantified modeling methods have been developed. These include:• stakeholder based approaches including metagame analysis and drama theory• morphological analysis and various forms of influence diagrams.• approaches using cognitive mapping• the Strategic Choice Approach• robustness analysis

Societies and journalsSocietiesThe International Federation of Operational Research Societies (IFORS)[24] is an umbrella organization foroperational research societies worldwide, representing approximately 50 national societies including those in theUS,[25] UK,[26] France,[27] Germany, Canada,[28] Australia,[29] New Zealand,[30] Philippines,[31] India,[32] Japan andSouth Africa (ORSSA).[33] The constituent members of IFORS form regional groups, such as that in Europe.[34]

Other important operational research organizations are Simulation Interoperability Standards Organization(SISO)[35] and Interservice/Industry Training, Simulation and Education Conference (I/ITSEC)[36]

In 2004 the US-based organization INFORMS began an initiative to market the OR profession better, including awebsite entitled The Science of Better[37] which provides an introduction to OR and examples of successfulapplications of OR to industrial problems. This initiative has been adopted by the Operational Research Society inthe UK, including a website entitled Learn about OR,.[38]

Journals

Operations research 11

INFORMS publishes twelve scholarly journals about operations research, including the top two journals in theirclass, according to 2005 Journal Citation Reports.[39] They are:• Decision Analysis[40]

• Information Systems Research• INFORMS Journal on Computing• INFORMS Transactions on Education[41] (an open access journal)• Interfaces: An International Journal of the Institute for Operations Research and the Management Sciences• Management Science: A Journal of the Institute for Operations Research and the Management Sciences• Manufacturing & Service Operations Management• Marketing Science• Mathematics of Operations Research• Operations Research: A Journal of the Institute for Operations Research and the Management Sciences• Organization Science• Transportation Science.Other journals• 4OR-A Quarterly Journal of Operations Research: jointly published the Belgian, French and Italian Operations

Research Societies (Springer);• Decision Sciences published by Wiley-Blackwell on behalf of the Decision Sciences Institute• European Journal of Operational Research (EJOR): Founded in 1975 and is presently by far the largest

operational research journal in the world, with its around 9,000 pages of published papers per year. In 2004, itstotal number of citations was the second largest amongst Operational Research and Management Sciencejournals;

• INFOR Journal: published and sponsored by the Canadian Operational Research Society;• International Journal of Operations Research and Information Systems (IJORIS)": an official publication of the

Information Resources Management Association, published quarterly by IGI Global;[42]

• Journal of Defense Modeling and Simulation (JDMS): Applications, Methodology, Technology: a quarterlyjournal devoted to advancing the science of modeling and simulation as it relates to the military and defense.[43]

• Journal of the Operational Research Society (JORS): an official journal of The OR Society; this is the oldestcontinuously published journal of OR in the world, published by Palgrave;[44]

• Journal of Simulation (JOS): an official journal of The OR Society, published by Palgrave;[44]

• Mathematical Methods of Operations Research (MMOR): the journal of the German and Dutch OR Societies,published by Springer;[45]

• Military Operations Research (MOR): published by the Military Operations Research Society;• Opsearch: official journal of the Operational Research Society of India;• OR Insight: a quarterly journal of The OR Society, published by Palgrave;[44]

• Production and Operations Management, the official journal of the Production and Operations ManagementSociety

• TOP: the official journal of the Spanish Society of Statistics and Operations Research.[46]

Operations research 12

Notes[1] http:/ / www. hsor. org/ what_is_or. cfm[2] http:/ / www. bls. gov/ oco/ ocos044. htm[3] What is Management Science Research? (http:/ / www. jbs. cam. ac. uk/ programmes/ mphil_mgtscience/ mgtresearch. html) University of

Cambridge 2008. Retrieved 5 June 2008.[4] http:/ / www3. informs. org/ site/ OperationsResearch/ index. php?c=10& kat=Forthcoming+ Papers[5] http:/ / www. decisionpro. biz[6] "Operational Research in the British Army 1939–1945, October 1947, Report C67/3/4/48, UK National Archives file WO291/1301

Quoted on the dust-jacket of: Morse, Philip M, and Kimball, George E, Methods of Operations Research, 1st Edition Revised, pub MIT Press& J Wiley, 5th printing, 1954.

[7] UK National Archives Catalogue for WO291 (http:/ / www. nationalarchives. gov. uk/ catalogue/ displaycataloguedetails. asp?CATID=109&CATLN=2& Highlight=& FullDetails=True) lists a War Office organisation called Army Operational Research Group (AORG) that existedfrom 1946 to 1962. "In January 1962 the name was changed to Army Operational Research Establishment (AORE). Following the creation ofa unified Ministry of Defence, a tri-service operational research organisation was established: the Defence Operational ResearchEstablishment (DOAE) which was formed in 1965, and it absorbed the Army Operational Research Establishment based at West Byfleet."

[8] http:/ / brochure. unisa. ac. za/ myunisa/ data/ subjects/ Quantitative%20Management. pdf[9] M.S. Sodhi, "What about the 'O' in O.R.?" OR/MS Today, December, 2007, p. 12, http:/ / www. lionhrtpub. com/ orms/ orms-12-07/ frqed.

html[10] P. W. Bridgman, The Logic of Modern Physics, The MacMillan Company, New York, 1927[11] http:/ / www. britannica. com/ EBchecked/ topic/ 682073/ operations-research/ 68171/ History#ref22348[12] Kirby, p. 117 (http:/ / books. google. co. uk/ books?id=DWITTpkFPEAC& lpg=PA141& pg=PA117)[13] Kirby, pp. 91–94 (http:/ / books. google. co. uk/ books?id=DWITTpkFPEAC& lpg=PA141& pg=PA94)[14] Kirby, p. 96,109 (http:/ / books. google. co. uk/ books?id=DWITTpkFPEAC& lpg=PA141& pg=PA109)[15] Kirby, p. 96 (http:/ / books. google. co. uk/ books?id=DWITTpkFPEAC& lpg=PA141& pg=PA96)[16] "Numbers are Essential": Victory in the North Atlantic Reconsidered, March–May 1943 (http:/ / www. familyheritage. ca/ Articles/

victory1943. html)[17] Kirby, p. 101 (http:/ / books. google. co. uk/ books?id=DWITTpkFPEAC& lpg=PA141& pg=PA101)[18] (Kirby, pp. 102,103 (http:/ / books. google. co. uk/ books?id=DWITTpkFPEAC& lpg=PA141& pg=PA103))[19] (http:/ / www. raf. mod. uk/ bombercommand/ thousands. html)[20] Milkman, Raymond H. (May 1968). Operations Research in World War II. United States Naval Institute Proceedings.[21] Stafford Beer (1967) Management Science: The Business Use of Operations Research[22] What is Management Science? (http:/ / www. lums. lancs. ac. uk/ departments/ ManSci/ DeptProfile/ WhatisManSci/ ) Lancaster University,

2008. Retrieved 5 June 2008.[23] What is Management Science? (http:/ / bus. utk. edu/ soms/ information/ whatis_msci. html) The University of Tennessee, 2006. Retrieved 5

June 2008.[24] IFORS (http:/ / www. ifors. org/ )[25] INFORMS (http:/ / www. informs. org/ )[26] The OR Society (http:/ / www. orsoc. org. uk)[27] http:/ / www. roadef. org/ content/ index. htm[28] CORS (http:/ / www. cors. ca)[29] ASOR (http:/ / www. asor. org. au)[30] ORSNZ (http:/ / www. orsnz. org. nz/ )[31] ORSP (http:/ / www. orsp. org. ph/ )[32] ORSI (http:/ / www. orsi. in/ )[33] ORSSA (http:/ / www. orssa. org. za/ )[34] EURO (http:/ / www. euro-online. org/ )[35] SISO (http:/ / www. sisostds. org/ )[36] I/ITSEC (http:/ / www. iitsec. org/ )[37] The Science of Better (http:/ / www. scienceofbetter. org/ )[38] Learn about OR (http:/ / www. learnaboutor. co. uk/ )[39] INFORMS Journals (http:/ / www. informs. org/ index. php?c=31& kat=-+ INFORMS+ Journals)[40] Decision Analysis (http:/ / www. informs. org/ site/ DA/ )[41] INFORMS Transactions on Education (http:/ / www. informs. org/ site/ ITE/ )[42] (http:/ / www. igi-global. com/ Bookstore/ TitleDetails. aspx?TitleId=1141)[43] JDMS (http:/ / www. scs. org/ pubs/ jdms/ jdms. html)[44] The OR Society (http:/ / www. orsoc. org. uk);[45] Mathematical Methods of Operations Research website (http:/ / www. springer. com/ mathematics/ journal/ 186)

Operations research 13

[46] TOP (http:/ / www. springer. com/ east/ home/ business/ operations+research?SGWID=5-40521-70-173677307-detailsPage=journal|description)

References• Kirby, M. W. (Operational Research Society (Great Britain)). Operational Research in War and Peace: The

British Experience from the 1930s to 1970, Imperial College Press, 2003. ISBN 1860943667, 9781860943669

Further reading• C. West Churchman, Russell L. Ackoff & E. L. Arnoff, Introduction to Operations Research, New York: J.

Wiley and Sons, 1957• Joseph G. Ecker & Michael Kupferschmid, Introduction to Operations Research, Krieger Publishing Co.• Frederick S. Hillier & Gerald J. Lieberman, Introduction to Operations Research, McGraw-Hill: Boston MA; 8th.

(International) Edition, 2005• Michael Pidd, Tools for Thinking: Modelling in Management Science, J. Wiley & Sons Ltd., Chichester; 2nd.

Edition, 2003• Hamdy A. Taha, Operations Research: An Introduction, Prentice Hall; 9th. Edition, 2011• Wayne Winston, Operations Research: Applications and Algorithms, Duxbury Press; 4th. Edition, 2003• Kenneth R. Baker, Dean H. Kropp (1985). Management Science: An Introduction to the Use of Decision Models• David Charles Heinze (1982). Management Science: Introductory Concepts and Applications• Lee J. Krajewski, Howard E. Thompson (1981). "Management Science: Quantitative Methods in Context"• Thomas W. Knowles (1989). Management science: Building and Using Models• Kamlesh Mathur, Daniel Solow (1994). Management Science: The Art of Decision Making• Laurence J. Moore, Sang M. Lee, Bernard W. Taylor (1993). Management Science• William Thomas Morris (1968). Management Science: A Bayesian Introduction.• William E. Pinney, Donald B. McWilliams (1987). Management Science: An Introduction to Quantitative

Analysis for Management• Shrader, Charles R. (2006). History of Operations Research in the United States Army, Volume 1:1942-1962

(http:/ / www. history. army. mil/ html/ books/ hist_op_research/ index. html). Washington, D.C.: United StatesArmy Center of Military History. CMH Pub 70-102-1.

• Gerald E. Thompson (1982). Management Science: An Introduction to Modern Quantitative Analysis andDecision Making. New York : McGraw-Hill Publishing Co.

• Saul I. Gass & Arjang A. Assad (2005). An Annotated Timeline of Operations Research: An Informal History.New York : Kluwer Academic Publishers.

External links• INFORMS OR/MS Resource Collection (http:/ / www. informs. org/ Resources/ ): a comprehensive set of OR

links.• International Federation of Operational Research Societies (http:/ / www. ifors. org/ )• Occupational Outlook Handbook, U.S. Department of Labor Bureau of Labor Statistics (http:/ / stats. bls. gov/

oco/ ocos044. htm/ )• "Operation Everything: It Stocks Your Grocery Store, Schedules Your Favorite Team's Games, and Helps Plan

Your Vacation. The Most Influential Academic Discipline You've Never Heard Of." Boston Globe, June 27, 2004(http:/ / www. boston. com/ news/ globe/ reprints/ 062704_postrel/ )

• "Optimal Results: IT-powered advances in operations research can enhance business processes and boost thecorporate bottom line." Computerworld, November 20, 2000 (http:/ / www. computerworld. com/ s/ article/54157/ Optimal_Results?taxonomyId=063/ )

Mathematical optimization 14

Mathematical optimization

The maximum of a paraboloid

In mathematics and computational science, mathematicaloptimization (alternatively, optimization or mathematicalprogramming) refers to the selection of a best element from some setof available alternatives.

In the simplest case, this means solving problems in which one seeks tominimize or maximize a real function by systematically choosing thevalues of real or integer variables from within an allowed set. Thisformulation, using a real-valued objective function, is probably thesimplest example; the generalization of optimization theory andtechniques to other formulations comprises a large area of appliedmathematics. More generally, it means finding "best available" valuesof some objective function given a defined domain, including a variety of different types of objective functions anddifferent types of domains.

In applications, optimization is used in engineering and economics.

OverviewAn optimization problem can be represented in the following way

Given: a function f : A R from some set A to the real numbersSought: an element x0 in A such that f(x0) ≤ f(x) for all x in A ("minimization") or such that f(x0) ≥ f(x) for all xin A ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term notdirectly related to computer programming, but still in use for example in linear programming - see History below).Many real-world and theoretical problems may be modeled in this general framework. Problems formulated usingthis technique in the fields of physics and computer vision may refer to the technique as energy minimization,speaking of the value of the function f as representing the energy of the system being modeled.Typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities orinequalities that the members of A have to satisfy. The domain A of f is called the search space or the choice set,while the elements of A are called candidate solutions or feasible solutions.The function f is called, variously, an objective function, cost function, energy function, or energy functional. Afeasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimalsolution.Generally, when the feasible region or the objective function of the problem does not present convexity, there maybe several local minima and maxima, where a local minimum x* is defined as a point for which there exists some δ >0 so that for all x such that

the expression

holds; that is to say, on some region around x* all of the function values are greater than or equal to the value at thatpoint. Local maxima are defined similarly.A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal

Mathematical optimization 15

solutions, and will treat the former as actual solutions to the original problem. The branch of applied mathematicsand numerical analysis that is concerned with the development of deterministic algorithms that are capable ofguaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called globaloptimization.

NotationOptimization problems are often expressed with special notation. Here are some examples.

Minimum and maximum value of a functionConsider the following notation:

This denotes the minimum value for the objective function x2 , when choosing x from the set of real numbers. The minimum value in this case is , occurring at .

Similarly, the notation

asks for the maximum value for the objective function 2x, where x may be any real number. In this case, there is nosuch maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

Optimal input argumentsConsider the following notation:

or equivalently

This represents the value (or values) of x in the interval that minimizes (or minimize) the objectivefunction x2 + 1 (the actual minimum value of that function is not what the problem asks for). In this case, the answeris x = -1, since x = 0 is infeasible, i.e. does not belong to the feasible set.Similarly,

or equivalently

represents the pair (or pairs) that maximizes (or maximize) the value of the objective function ,with the added constraint that x lie in the interval (again, the actual maximum value of the expression doesnot matter). In this case, the solutions are the pairs of the form (5, 2kπ) and (−5,(2k+1)π), where k ranges over allintegers.Arg min and arg max are sometimes also written argmin and argmax, and stand for argument of the minimumand argument of the maximum.

Mathematical optimization 16

HistoryFermat and Lagrange found calculus-based formulas for identifying optima, while Newton and Gauss proposediterative methods for moving towards an optimum. Historically, the first term for optimization was "linearprogramming", which was due to George B. Dantzig, although much of the theory had been introduced by LeonidKantorovich in 1939. Dantzig published the Simplex algorithm in 1947, and John von Neumann developed thetheory of duality in the same year.The term programming in this context does not refer to computer programming. Rather, the term comes from the useof program by the United States military to refer to proposed training and logistics schedules, which were theproblems Dantzig studied at that time.Later important researchers in mathematical optimization include the following:

• Richard Bellman • Arkadii Nemirovskii• Ronald A. Howard • Yurii Nesterov• Narendra Karmarkar • Boris Polyak• William Karush • Lev Pontryagin• Leonid Khachiyan • James Renegar• Bernard Koopman • R. Tyrrell Rockafellar• Harold Kuhn • Cornelis Roos• Joseph Louis Lagrange • Naum Z. Shor• László Lovász • Michael J. Todd

• Albert Tucker

Major subfields• Convex programming studies the case when the objective function is convex and the constraints, if any, form a

convex set. This can be viewed as a particular case of nonlinear programming or as generalization of linear orconvex quadratic programming.• Linear programming (LP), a type of convex programming, studies the case in which the objective function f is

linear and the set of constraints is specified using only linear equalities and inequalities. Such a set is called apolyhedron or a polytope if it is bounded.

• Second order cone programming (SOCP) is a convex program, and includes certain types of quadraticprograms.

• Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables aresemidefinite matrices. It is generalization of linear and convex quadratic programming.

• Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conicprograms with the appropriate type of cone.

• Geometric programming is a technique whereby objective and inequality constraints expressed as posynomialsand equality constraints as monomials can be transformed into a convex program.

• Integer programming studies linear programs in which some or all variables are constrained to take on integervalues. This is not convex, and in general much more difficult than regular linear programming.

• Quadratic programming allows the objective function to have quadratic terms, while the feasible set must bespecified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convexprogramming.

• Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concavefractional programs can be transformed to a convex optimization problem.

• Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the

Mathematical optimization 17

difficulty of solving it.• Stochastic programming studies the case in which some of the constraints or parameters depend on random

variables.• Robust programming is, like stochastic programming, an attempt to capture uncertainty in the data underlying the

optimization problem. This is not done through the use of random variables, but instead, the problem is solvedtaking into account inaccuracies in the input data.

• Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can bereduced to a discrete one.

• Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of aninfinite-dimensional space, such as a space of functions.

• Metaheuristics make few or no assumptions about the problem being optimized and can search very large spacesof candidate solutions. However, metaheuristics do not guarantee an optimal solution is ever found.

• Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificialintelligence, particularly in automated reasoning).• Constraint programming.

• Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular usein scheduling.

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is,decision making over time):• Calculus of variations seeks to optimize an objective defined over many points in time, by considering how the

objective function changes if there is a small change in the choice path.• Optimal control theory is a generalization of the calculus of variations.• Dynamic programming studies the case in which the optimization strategy is based on splitting the problem into

smaller subproblems. The equation that describes the relationship between these subproblems is called theBellman equation.

• Mathematical programming with equilibrium constraints is where the constraints include variational inequalitiesor complementarities.

Multi-objective optimizationAdding more than one objective to an optimization problem adds complexity. For example, to optimize a structuraldesign, one would want a design that is both light and rigid. Because these two objectives conflict, a trade-off exists.There will be one lightest design, one stiffest design, and an infinite number of designs that are some compromise ofweight and stiffness. The set of trade-off designs that cannot be improved upon according to one criterion withouthurting another criterion is known as the Pareto set. The curve created plotting weight against stiffness of the bestdesigns is known as the Pareto frontier.A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominatedby any other design: If it is worse than another design in some respects and no better in any respect, then it isdominated and is not Pareto optimal.

Mathematical optimization 18

Multi-modal optimizationOptimization problems are often multi-modal; that is they possess multiple good solutions. They could all beglobally good (same cost function value) or there could be a mix of globally good and locally good solutions.Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used toobtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with differentstarting points in multiple runs of the algorithm. Evolutionary Algorithms are however a very popular approach toobtain multiple solutions in a multi-modal optimization task. See Evolutionary multi-modal optimization.

Classification of critical points and extrema

Feasibility problemThe satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solutionat all without regard to objective value. This can be regarded as the special case of mathematical optimization wherethe objective value is the same for every solution, and thus any solution is optimal.Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to relax thefeasibility conditions using a slack variable; with enough slack, any starting point is feasible. Then, minimize thatslack variable until slack is null or negative.

ExistenceThe extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attainsits maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains itsminimum; an upper semi-continuous function on a compact set attains its maximum.

Necessary conditions for optimalityOne of Fermat's theorems states that optima of unconstrained problems are found at stationary points, where the firstderivative or the gradient of the objective function is zero (see First derivative test). More generally, they may befound at critical points, where the first derivative or gradient of the objective function is zero or is undefined, or onthe boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at aninterior optimum is called a 'first-order condition' or a set of first-order conditions.Optima of inequality-constrained problems are instead found by the Lagrange multiplier method. This methodcalculates a system of inequalities called the 'Karush–Kuhn–Tucker conditions' or 'complementary slacknessconditions', which may then be used to calculate the optimum.

Sufficient conditions for optimalityWhile the first derivative test identifies points that might be optima, this test does not distinguish a point which is aminimum from one that is a maximum or one that is neither. When the objective function is twice differentiable,these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called theHessian matrix) in unconstrained problems, or the matrix of second derivatives of the objective function and theconstraints called the bordered Hessian in constrained problems. The conditions that distinguish maxima, or minima,from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solutionsatisfies the first-order conditions, then satisfaction of the second-order conditions as well is sufficient to establish atleast local optimality.

Mathematical optimization 19

Sensitivity and continuity of optimaThe envelope theorem describes how the value of an optimal solution changes when an underlying parameterchanges. The process of computing this change is called comparative statics.The maximum theorem of Claude Berge (1963) describes the continuity of an optimal solution as a function ofunderlying parameters.

Calculus of optimizationFor unconstrained problems with twice-differentiable functions, some critical points can be found by finding thepoints where the gradient of the objective function is zero (that is, the stationary points). More generally, a zerosubgradient certifies that a local minimum has been found for minimization problems with convex functions andother locally Lipschitz functions.Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positivedefinite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the pointis a local maximum; finally, if indefinite, then the point is some kind of saddle point.Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers.Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.When the objective function is convex, then any local minimum will also be a global minimum. There exist efficientnumerical techniques for minimizing convex functions, such as interior-point methods.

Computational optimization techniquesTo solve problems, researchers may use algorithms that terminate in a finite number of steps, or iterative methodsthat converge to a solution (on some specified class of problems), or heuristics that may provide approximatesolutions to some problems (although their iterates need not converge).

Optimization algorithms• Simplex algorithm of George Dantzig, designed for linear programming.• Extensions of the simplex algorithm, designed for quadratic programming and for linear-fractional programming.• Variants of the simplex algorithm that are especially suited for network optimization.• Combinatorial algorithms

Iterative methodsThe iterative methods used to solve problems of nonlinear programming differ according to whether they evaluateHessians, gradients, or only function values. While evaluating Hessians and gradients improves the rate ofconvergence, such evaluations increase the computational complexity (or computational cost) of each iteration. Insome cases, the computational complexity may be excessively high.• Methods that evaluate Hessians (or approximate Hessians, using finite differences):

• Newton's method• Sequential quadratic programming: A method for small-medium scale constrained problems. Some versions

can handle large-dimensional problems.• Methods that evaluate gradients or approximate gradients using finite differences (or even subgradients):

• Quasi-Newton methods: Iterative methods for medium-large problems.• Conjugate gradient methods: Iterative methods for large problems. (In theory, these methods terminate in a

finite number of steps with quadratic objective functions, but this finite termination is not observed in practiceon finite–precision computers.)

Mathematical optimization 20

• Interior point methods: This is a large class of methods for constrained optimization. Some interior-pointmethods use only (sub)gradient information, and others of which require the evaluation of Hessians.

• Gradient descent (alternatively, "steepest descent" or "steepest ascent"): A method of historical and theoreticalinterest, which has had renewed interest for find approximate solutions of enormous problems.

• Subgradient methods - An iterative method for large locally Lipschitz functions using generalized gradients.Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.

• Bundle method of descent: An iterative method for small–medium sized problems with locally Lipschitzfunctions, particularly for convex minimization problems. (Similar to conjugate gradient methods)

• Ellipsoid method: An iterative method for small problems with quasiconvex objective functions and of greattheoretical interest, particularly in establishing the polynomial time complexity of some combinatorialoptimization problems. It has similarities with Quasi-Newton methods.

• Reduced gradient method (Frank–Wolfe) for approximate minimization of specially structured problems withlinear constraints, especially with traffic networks. For general unconstrained problems, this method reduces tothe gradient method, which is regarded as obsolete (for almost all problems).

• Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can beapproximated using finite differences, in which case a gradient-based method can be used.• Interpolation methods (Michael J. D. Powell)• Pattern search methods, which have better convergence properties than the Nelder–Mead simplicial heuristic

(listed below, under heuristics).

Global convergence

More generally, if the objective function is not a quadratic function, then many optimization methods use othermethods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popularmethod for ensuring convergence relies on-line searches, which optimize a function along one dimension. A secondand increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions areused in modern methods of non-differentiable optimization.

HeuristicsBesides (finitely terminating) algorithms and (convergent) iterative methods, there are heuristics that can provideapproximate solutions to some optimization problems:• Memetic algorithm• Differential evolution• Dynamic relaxation• Genetic algorithms• Hill climbing• Nelder-Mead simplicial heuristic: A popular heuristic for approximate minimization (without calling gradients)• Particle swarm optimization• Simulated annealing• Tabu search

Mathematical optimization 21

Applications

Mechanics and engineeringProblems in rigid body dynamics (in particular articulated rigid body dynamics) often require mathematicalprogramming techniques, since you can view rigid body dynamics as attempting to solve an ordinary differentialequation on a constraint manifold; the constraints are various nonlinear geometric constraints such as "these twopoints must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhereon this curve". Also, the problem of computing contact forces can be done by solving a linear complementarityproblem, which can also be viewed as a QP (quadratic programming) problem.Many design problems can also be expressed as optimization programs. This application is called designoptimization. One subset is the engineering optimization, and another recent and growing subset of this field ismultidisciplinary design optimization, which, while useful in many problems, has in particular been applied toaerospace engineering problems.

EconomicsEconomics relies so heavily on optimization that some economists define their field as the study of optimizationunder constraints or optimization under scarcity. Modern optimization theory overlaps with game theory and thestudy of (economic) equilibria, as well as traditional optimization theory. The Journal of Economic Literatureclassifies optimization theory and related topics as mathematical and quantitative methods in classes C61–C63.In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem,are economic optimization problems. Insofar as they behave consistently, consumers are assumed to maximize theirutility, while firms are usually assumed to maximize their profit. Also, agents are often modeled as being risk-averse,thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlyingmathematics relies on optimizing stochastic processes rather than on static optimization. Trade theory also usesoptimization to explain trade patterns between nations.Since the 1970s, economists have modeled dynamic decisions over time using control theory. For example,microeconomists use dynamic search models to study labor-market behavior. Macroeconomists build dynamicstochastic general equilibrium (DSGE) models that describe the dynamics of the whole economy as the result of theinterdependent optimizing decisions of workers, consumers, investors, and governments.[1]

Operations researchAnother field that uses optimization techniques extensively is operations research. Operations research also usesstochastic modeling and simulation to support improved decision-making. Increasingly, operations research usesstochastic programming to model dynamic decisions that adapt to events; such problems can be solved withlarge-scale optimization and stochastic optimization methods.

Notes[1] Julio Rotemberg and Michael Woodford (1997), 'An optimization-based econometric framework for the evaluation of monetary policy.'

NBER Macroeconomics Annual 12, pp. 297-346.

Mathematical optimization 22

Further reading

Comprehensive

Undergraduate level

• Bradley, S.; Hax, A.; Magnanti, T. (1977). Applied mathematical programming. Addison Wesley.• Rardin, Ronald L. (1997). Optimization in operations research. Prentice Hall. pp. 919. ISBN 0-02-398415-5.• Strang, Gilbert (1986). Introduction to applied mathematics (http:/ / www. wellesleycambridge. com/ tocs/

toc-appl). Wellesley, MA: Wellesley-Cambridge Press (Strang's publishing company). pp. xii+758.ISBN 0-9614088-0-4. MR870634.

Graduate level

• Magnanti, Thomas L. (1989). "Twenty years of mathematical programming". In Cornet, Bernard; Tulkens, Henry.Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from thesymposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. pp. 163–227.ISBN 0-262-03149-3. MR1104662.

• Minoux, M. (1986). Mathematical programming: Theory and algorithms (Translated by Steven Vajda from the(1983 Paris: Dunod) French ed.). Chichester: A Wiley-Interscience Publication. John Wiley & Sons, Ltd..pp. xxviii+489. ISBN 0-471-90170-9. MR2571910. (2008 Second ed., in French: Programmation mathématique:Théorie et algorithmes. Editions Tec & Doc, Paris, 2008. xxx+711 pp. ISBN 978-2-7430-1000-3..

• Nemhauser, G. L.; Rinnooy Kan, A. H. G.; Todd, M. J., eds (1989). Optimization. Handbooks in OperationsResearch and Management Science. 1. Amsterdam: North-Holland Publishing Co.. pp. xiv+709.ISBN 0-444-87284-1. MR1105099.• J. E. Dennis, Jr. and Robert B. Schnabel, A view of unconstrained optimization (pp. 1–72);• Donald Goldfarb and Michael J. Todd, Linear programming (pp. 73–170);• Philip E. Gill, Walter Murray, Michael A. Saunders, and Margaret H. Wright, Constrained nonlinear

programming (pp. 171–210);• Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin, Network flows (pp. 211–369);• W. R. Pulleyblank, Polyhedral combinatorics (pp. 371–446);• George L. Nemhauser and Laurence A. Wolsey, Integer programming (pp. 447–527);• Claude Lemaréchal, Nondifferentiable optimization (pp. 529–572);• Roger J-B Wets, Stochastic programming (pp. 573–629);• A. H. G. Rinnooy Kan and G. T. Timmer, Global optimization (pp. 631–662);• P. L. Yu, Multiple criteria decision making: five basic concepts (pp. 663–699).

• Shapiro, Jeremy F. (1979). Mathematical programming: Structures and algorithms. New York:Wiley-Interscience [John Wiley & Sons]. pp. xvi+388. ISBN 0-471-77886-9. MR544669.

Continuous optimization• Mordecai Avriel (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing.

ISBN 0-486-43227-0.• Bonnans, J. Frédéric; Gilbert, J. Charles; Lemaréchal, Claude; Sagastizábal, Claudia A. (2006). Numerical

optimization: Theoretical and practical aspects (http:/ / www. springer. com/ mathematics/ applications/ book/978-3-540-35445-1). Universitext (Second revised ed. of translation of 1997 French ed.). Berlin: Springer-Verlag.pp. xiv+490. doi:10.1007/978-3-540-35447-5. ISBN 3-540-35445-X. MR2265882.

• Bonnans, J. Frédéric; Shapiro, Alexander (2000). Perturbation analysis of optimization problems. Springer Seriesin Operations Research. New York: Springer-Verlag. pp. xviii+601. ISBN 0-387-98705-3. MR1756264.

Mathematical optimization 23

• Stephen Boyd and Lieven Vandenberghe (2004). Convex Optimization (http:/ / www. stanford. edu/ ~boyd/cvxbook/ ). Cambridge University Press. ISBN 0-521-83378-7.

• Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization (http:/ / www. ece. northwestern. edu/~nocedal/ book/ num-opt. html). Springer.. ISBN 0-387-30303-0.

Applications

• Dixit, Avinash (1990). Optimization in economic theory (2nd ed.). Oxford University Press. pp. 206.ISBN 0198772106.

Combinatorial optimization• R. K. Ahuja, Thomas L. Magnanti, and James B. Orlin (1993). Network Flows: Theory, Algorithms, and

Applications. Prentice-Hall, Inc. ISBN 0-13-617549-X.• William J. Cook, William H. Cunningham, William R. Pulleyblank, Alexander Schrijver; Combinatorial

Optimization; John Wiley & Sons; 1 edition (November 12, 1997); ISBN 0-471-55894-X.• Gondran, Michel; Minoux, Michel (1984). Graphs and algorithms. Wiley-Interscience Series in Discrete

Mathematics (Translated by Steven Vajda from the second (Collection de la Direction des Études et Recherchesd'Électricité de France [Collection of the Department of Studies and Research of Électricité de France], v. 37.Paris: Éditions Eyrolles 1985. xxviii+545 pp. MR868083) French ed.). Chichester: John Wiley & Sons, Ltd..pp. xix+650. ISBN 0-471-10374-8. MR2552933. (Fourth ed. Collection EDF R&D. Paris: Editions Tec & Doc2009. xxxii+784 pp..

• Eugene Lawler (2001). Combinatorial Optimization: Networks and Matroids. Dover. ISBN 0486414531.• Lawler, E. L.; Lenstra, J. K.; Rinnooy Kan, A. H. G.; Shmoys, D. B. (1985), The traveling salesman problem: A

guided tour of combinatorial optimization, John Wiley & Sons, ISBN 0-471-90413-9.• Jon Lee; A First Course in Combinatorial Optimization (http:/ / books. google. com/

books?id=3pL1B7WVYnAC& printsec=frontcover& source=gbs_ge_summary_r& cad=0#v=onepage& q&f=false); Cambridge University Press; 2004; ISBN 0-521-01012-8.

• Christos H. Papadimitriou and Kenneth Steiglitz Combinatorial Optimization : Algorithms and Complexity;Dover Pubns; (paperback, Unabridged edition, July 1998) ISBN 0-486-40258-4.

External links• COIN-OR (http:/ / www. coin-or. org/ )—Computational Infrastructure for Operations Research• Decision Tree for Optimization Software (http:/ / plato. asu. edu/ guide. html) Links to optimization source codes• Global optimization (http:/ / www. mat. univie. ac. at/ ~neum/ glopt. html)• Mathematical Programming Glossary (http:/ / glossary. computing. society. informs. org/ )• Mathematical Programming Society (http:/ / www. mathprog. org/ )• NEOS Guide (http:/ / www-fp. mcs. anl. gov/ otc/ Guide/ index. html) currently being replaced by the NEOS

Wiki (http:/ / wiki. mcs. anl. gov/ neos)• Optimization Online (http:/ / www. optimization-online. org) A repository for optimization e-prints• Optimization Related Links (http:/ / www2. arnes. si/ ~ljc3m2/ igor/ links. html)• Convex Optimization I (http:/ / see. stanford. edu/ see/ courseinfo.

aspx?coll=2db7ced4-39d1-4fdb-90e8-364129597c87) EE364a: Course from Stanford University• Convex Optimization – Boyd and Vandenberghe (http:/ / www. stanford. edu/ ~boyd/ cvxbook) Book on Convex

Optimization• Simplemax Online Optimization Services (http:/ / simplemax. net) Web applications to access nonlinear

optimization servicesSolvers:

Mathematical optimization 24

• APOPT (http:/ / wiki. mcs. anl. gov/ NEOS/ index. php/ APOPT) - large-scale nonlinear programming• Free Optimization Software by Systems Optimization Laboratory, Stanford University (http:/ / www. stanford.

edu/ group/ SOL/ software. html)• MIDACO-Solver (http:/ / www. midaco-solver. com/ ) General purpose (MINLP) optimization software based on

Ant colony optimization algorithms (Matlab, Excel, C/C++, Fortran)• Moocho (http:/ / trilinos. sandia. gov/ packages/ moocho/ ) - a very flexible open-source NLP solver• TANGO Project (http:/ / www. ime. usp. br/ ~egbirgin/ tango/ ) - Trustable Algorithms for Nonlinear General

Optimization - FortranLibraries:

• NAG Libraries (http:/ / www. nag. co. uk/ numeric/ numerical_libraries. asp)- optimization routines available formultiple programming languages (C, C++, Fortran, Python, Java, .NET, GPUs), packages (MATLAB, Maple,Excel) and for SMP and multicore

• IMSL Numerical Libraries Mathematical and statistical algorithms available in C/C++, Fortran, Java andC#/.NET. Optimization routines in the IMSL Libraries include unconstrained, linearly and nonlinearlyconstrained minimizations, and linear programming algorithms.

• ALGLIB (http:/ / www. alglib. net/ optimization/ ) Open-source optimization routines (unconstrained andbound-constrained optimization). C++, C#, Delphi, Visual Basic.

• IOptLib (Investigative Optimization Library) (http:/ / www2. arnes. si/ ~ljc3m2/ igor/ ioptlib/ ) - a free,open-source library for optimization algorithms (ANSI C).

• OAT (Optimization Algorithm Toolkit) (http:/ / optalgtoolkit. sourceforge. net/ ) - a set of standard optimizationalgorithms and problems in Java.

• Java Parallel Optimization Package (JPOP) (http:/ / www5. informatik. uni-erlangen. de/ research/ software/java-parallel-optimization-package/ ) An open-source java package which allows the parallel evaluation offunctions, gradients, and hessians.

• OOL (Open Optimization library) (http:/ / ool. sourceforge. net/ )-optimization routines in C.• FuncLib (http:/ / funclib. codeplex. com/ ) Open source non-linear optimization library in C# with support for

non-linear constraints and automatic differentiation.

Mathematical model 25

Mathematical modelNote: The term model has a different meaning in model theory, a branch of mathematical logic. An artifactwhich is used to illustrate a mathematical idea may also be called a mathematical model, and this usage is thereverse of the sense explained below.

A mathematical model is a description of a system using mathematical concepts and language. The process ofdeveloping a mathematical model is termed mathematical modelling. Mathematical models are used not only in thenatural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computerscience, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology andpolitical science); physicists, engineers, statisticians, operations research analysts and economists use mathematicalmodels most extensively.Mathematical models can take many forms, including but not limited to dynamical systems, statistical models,differential equations, or game theoretic models. These and other types of models can overlap, with a given modelinvolving a variety of abstract structures. In general, mathematical models may include logical models, as far aslogic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well themathematical models developed on the theoretical side agree with results of repeatable experiments. Lack ofagreement between theoretical mathematical models and experimental measurements often leads to importantadvances as better theories are developed.

Examples of mathematical models• Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model.

A slightly more realistic and largely used population growth model is the logistic function, and its extensions.• Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which

describes a trajectory in space which is modeled by a function giving its coordinates in space as a function oftime. The potential field is given by a function V : R3 → R and the trajectory is a solution of the differentialequation

Note this model assumes the particle is a point mass, which is certainly known to be false in many cases inwhich we use this model; for example, as a model of planetary motion.

• Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commoditieslabeled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have a cardinal utility functionU (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1,x2,..., xn consumed. The model further assumes that the consumer has a budget M which is used to purchase avector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this modelthen becomes an optimization problem, that is:

subject to:

This model has been used in general equilibrium theory, particularly to show existence and Pareto efficiency of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the

Mathematical model 26

theory and again this is an idealization.• Neighbour-sensing model explains the mushroom formation from the initially chaotic fungal network.• Computer Science: models in Computer Networks, data models, surface model,...• Mechanics: movement of rocket model,...Modelling requires selecting and identifying relevant aspects of a situation in the real world.

BackgroundOften when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis,engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try toestimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try outdifferent control approaches in simulations.A mathematical model usually describes a system by a set of variables and a set of equations that establishrelationships between the variables. The values of the variables can be practically anything; real or integer numbers,boolean values or strings, for example. The variables represent some properties of the system, for example, measuredsystem outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model isthe set of functions that describe the relations between the different variables.

Building blocksThere are six basic groups of variables namely: decision variables, input variables, state variables, exogenousvariables, random variables, and output variables. Since there can be many variables of each type, the variables aregenerally represented by vectors.Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known asparameters or constants. The variables are not independent of each other as the state variables are dependent on thedecision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of thesystem (represented by the state variables).Objectives and constraints of the system and its users can be represented as functions of the output variables or statevariables. The objective functions will depend on the perspective of the model's user. Depending on the context, anobjective function is also known as an index of performance, as it is some measure of interest to the user. Althoughthere is no limit to the number of objective functions and constraints a model can have, using or optimizing themodel becomes more involved (computationally) as the number increases.

Classifying mathematical modelsMany mathematical models can be classified in some of the following ways:1. Linear vs. nonlinear: Mathematical models are usually composed by variables, which are abstractions of

quantities of interest in the described systems, and operators that act on these variables, which can be algebraic operators, functions, differential operators, etc. If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The question of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.

Mathematical model 27

Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility.Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. Acommon approach to nonlinear problems is linearization, but this can be problematic if one is trying to studyaspects such as irreversibility, which are strongly tied to nonlinearity.

2. Deterministic vs. probabilistic (stochastic): A deterministic model is one in which every set of variable states isuniquely determined by parameters in the model and by sets of previous states of these variables. Therefore,deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model,randomness is present, and variable states are not described by unique values, but rather by probabilitydistributions.

3. Static vs. dynamic: A static model does not account for the element of time, while a dynamic model does.Dynamic models typically are represented with difference equations or differential equations.

4. Discrete vs. Continuous: A discrete model does not take into account the function of time and usually usestime-advance methods, while a Continuous model does. Continuous models typically are represented with f(t) andthe changes are reflected over continuous time intervals.

A priori informationMathematical modelling problems are used often classified into black box or white box models, according to howmuch a priori information is available of the system. A black-box model is a system of which there is no a prioriinformation available. A white-box model (also called glass box or clear box) is a system where all necessaryinformation is available. Practically all systems are somewhere between the black-box and white-box models, so thisconcept is useful only as an intuitive guide for deciding which approach to take.Usually it is preferable to use as much a priori information as possible to make the model more accurate. Thereforethe white-box models are usually considered easier, because if you have used the information correctly, then themodel will behave correctly. Often the a priori information comes in forms of knowing the type of functions relatingdifferent variables. For example, if we make a model of how a medicine works in a human system, we know thatusually the amount of medicine in the blood is an exponentially decaying function. But we are still left with severalunknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine inblood? This example is therefore not a completely white-box model. These parameters have to be estimated throughsome means before one can use the model.In black-box models one tries to estimate both the functional form of relations between variables and the numericalparameters in those functions. Using a priori information we could end up, for example, with a set of functions thatprobably could describe the system adequately. If there is no a priori information we would try to use functions asgeneral as possible to cover all different models. An often used approach for black-box models are neural networkswhich usually do not make assumptions about incoming data. The problem with using a large set of functions todescribe a system is that estimating the parameters becomes increasingly difficult when the amount of parameters(and different types of functions) increases.

Subjective informationSometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: one specifies a prior probability distribution (which can be subjective) and then updates this distribution based on empirical data. An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown, so the experimenter would need to make an arbitrary decision (perhaps by looking at the shape of the coin) about what

Mathematical model 28

prior distribution to use. Incorporation of the subjective information is necessary in this case to get an accurateprediction of the probability, since otherwise one would guess 1 or 0 as the probability of the next flip being heads,which would be almost certainly wrong.[1]

ComplexityIn general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is aprinciple particularly relevant to modelling; the essential idea being that among models with roughly equal predictivepower, the simplest one is the most desirable. While added complexity usually improves the realism of a model, itcan make the model difficult to understand and analyze, and can also pose computational problems, includingnumerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complexbefore a Paradigm shift offers radical simplification.For example, when modelling the flight of an aircraft, we could embed each mechanical part of the aircraft into ourmodel and would thus acquire an almost white-box model of the system. However, the computational cost of addingsuch a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertaintywould increase due to an overly complex system, because each separate part induces some amount of variance intothe model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size.Engineers often can accept some approximations in order to get a more robust and simple model. For exampleNewton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficientfor most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we studymacro-particles only.

TrainingAny model which is not pure white-box contains some parameters that can be used to fit the model to the system it isintended to describe. If the modelling is done by a neural network, the optimization of parameters is called training.In more conventional modelling through explicitly given mathematical functions, parameters are determined bycurve fitting.

Model evaluationA crucial part of the modelling process is the evaluation of whether or not a given mathematical model describes asystem accurately. This question can be difficult to answer as it involves several different types of evaluation.

Fit to empirical dataUsually the easiest part of model evaluation is checking whether a model fits experimental measurements or otherempirical data. In models with parameters, a common approach to test this fit is to split the data into two disjointsubsets: training data and verification data. The training data are used to estimate the model parameters. An accuratemodel will closely match the verification data even though these data were not used to set the model's parameters.This practice is referred to as cross-validation in statistics.Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit.In statistics, decision theory, and some economic models, a loss function plays a similar role.While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validityof the general mathematical form of a model. In general, more mathematical tools have been developed to test the fitof statistical models than models involving differential equations. Tools from non-parametric statistics cansometimes be used to evaluate how well the data fit a known distribution or to come up with a general model thatmakes only minimal assumptions about the model's mathematical form.

Mathematical model 29

Scope of the modelAssessing the scope of a model, that is, determining what situations the model is applicable to, can be lessstraightforward. If the model was constructed based on a set of data, one must determine for which systems orsituations the known data is a "typical" set of data.The question of whether the model describes well the properties of the system between data points is calledinterpolation, and the same question for events or data points outside the observed data is called extrapolation.As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, wecan note that Newton made his measurements without advanced equipment, so he could not measure properties ofparticles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of moleculesand other small particles, but macro particles only. It is then not surprising that his model does not extrapolate wellinto these domains, even though his model is quite sufficient for ordinary life physics.

Philosophical considerationsMany types of modelling implicitly involve claims about causality. This is usually (but not always) true of modelsinvolving differential equations. As the purpose of modelling is to increase our understanding of the world, thevalidity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situationsor data beyond those originally described in the model. One can argue that a model is worthless unless it providessome insight which goes beyond what is already known from direct investigation of the phenomenon being studied.An example of such criticism is the argument that the mathematical models of Optimal foraging theory do not offerinsight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.[2]

References[1] MacKay, D.J. Information Theory, Inference, and Learning Algorithms, Cambridge, (2003-2004). ISBN 0521642981[2] "Optimal Foraging Theory: A Critical Review - Annual Review of Ecology and Systematics, 15(1):523 - First Page Image" (http:/ /

arjournals. annualreviews. org/ doi/ abs/ 10. 1146/ annurev. es. 15. 110184. 002515?journalCode=ecolsys). Arjournals.annualreviews.org.2003-11-28. . Retrieved 2011-03-27.

Further readingBooks• Aris, Rutherford [ 1978 ] ( 1994 ). Mathematical Modelling Techniques, New York : Dover. ISBN 0-486-68131-9• Bender, E.A. [ 1978 ] ( 2000 ). An Introduction to Mathematical Modelling, New York : Dover. ISBN

0-486-41180-X• Lin, C.C. & Segel, L.A. ( 1988 ). Mathematics Applied to Deterministic Problems in the Natural Sciences,

Philadelphia : SIAM. ISBN 0-89871-229-7• Gershenfeld, N., The Nature of Mathematical Modelling, Cambridge University Press, (1998).ISBN 0521570956• Yang, X.-S., Mathematical Modelling for Earth Sciences, Dudedin Academic, (2008). ISBN 1903765927Specific applications• Peierls, Rudolf. Model-making in physics (http:/ / www. informaworld. com/ smpp/

content~content=a752582770~db=all~order=page), Contemporary Physics, Volume 21 (1), January 1980, 3-17• Korotayev A., Malkov A., Khaltourina D. ( 2006 ). Introduction to Social Macrodynamics: Compact

Macromodels of the World System Growth (http:/ / cliodynamics. ru/ index. php?option=com_content&task=view& id=124& Itemid=70). Moscow : Editorial URSS (http:/ / urss. ru/ cgi-bin/ db. pl?cp=& lang=en&blang=en& list=14& page=Book& id=34250). ISBN 5-484-00414-4

Mathematical model 30

External linksGeneral reference material• McLaughlin, Michael P. ( 1999 ) 'A Tutorial on Mathematical Modelling' (http:/ / www. causascientia. org/

math_stat/ Tutorial. pdf) PDF (264 KiB)• Patrone, F. Introduction to modeling via differential equations (http:/ / www. diptem. unige. it/ patrone/

differential_equations_intro. htm), with critical remarks.• Plus teacher and student package: Mathematical Modelling. (http:/ / plus. maths. org/ issue44/ package/ index.

html) Brings together all articles on mathematical modelling from Plus, the online mathematics magazineproduced by the Millennium Mathematics Project at the University of Cambridge.

Philosophical background• Frigg, R. and S. Hartmann, Models in Science (http:/ / plato. stanford. edu/ entries/ models-science/ ), in: The

Stanford Encyclopedia of Philosophy, (Spring 2006 Edition)

Game theoryIn mathematics, game theory models strategic situations, or games, in which an individual's success in makingchoices depends on the choices of others (Myerson, 1991). It is used in the social sciences (most notably ineconomics, management, operations research, political science, and social psychology) as well as in other formalsciences (logic, computer science, and statistics) and biology (particularly evolutionary biology and ecology). Whileinitially developed to analyze competitions in which one individual does better at another's expense (zero sumgames), it has been expanded to treat a wide class of interactions, which are classified according to several criteria.Today, "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social'is interpreted broadly, to include human as well as non-human players (computers, animals, plants)." (Aumann1987).Traditional applications of game theory define and study equilibria in these games. In an equilibrium, each player ofthe game has adopted a strategy that cannot improve his outcome, given the others' strategy. Many equilibriumconcepts have been developed (most famously the Nash equilibrium) to describe aspects of strategic equilibria.These equilibrium concepts are motivated differently depending on the area of application, although they oftenoverlap or coincide. This methodology has received criticism, and debates continue over the appropriateness ofparticular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematicalmodels in the social sciences.Mathematical game theory had beginnings with some publications by Émile Borel, which led to his 1938 bookApplications aux Jeux de Hasard. However, Borel's results were limited, and his conjecture about the non-existenceof mixed-strategy equilibria in two-person zero-sum games was wrong. The modern epoch of game theory beganwith the statement of the theorem on the existence of mixed-strategy equilibria in two-person zero-sum games and itsproof by John von Neumann. Von Neumann's original proof used Brouwer's fixed-point theorem on continuousmappings into compact convex sets, which became a standard method in game theory and mathematical economics.His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, whichconsidered cooperative games of several players. The second edition of this book provided an axiomatic theory ofexpected utility, which allowed mathematical statisticians and economists to treat decision-making underuncertainty.This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game-theorists have won the Nobel Memorial Prize in Economic Sciences, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to

Game theory 31

biology.

HistoryEarly discussions of examples of two-person games occurred long before the rise of modern, mathematical gametheory. The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In thisletter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her.James Madison made what we now recognize as a game-theoretic analysis of the ways states can be expected tobehave under different systems of taxation.[1] [2] In his 1838 Recherches sur les principes mathématiques de lathéorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine AugustinCournot considered a duopoly and presents a solution that is a restricted version of the Nash equilibrium.The Danish mathematician Zeuthen proved that a mathematical model has a winning strategy by using Brouwer'sfixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel 1938 bookproved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric.Borel conjectured that non-existence of a mixed-strategy equilibria in two-person zero-sum games would occur, aconjecture that was proved false.Game theory did not really exist as a unique field until John von Neumann published a paper in 1928.[3] VonNeumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets,which became a standard method in game theory and mathematical economics. His paper was followed by his 1944book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games ofseveral players. The second edition of this book provided an axiomatic theory of expected utility, which allowedmathematical statisticians and economists to treat decision-making under uncertainty. Von Neumann's work in gametheory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and OskarMorgenstern. This foundational work contains the method for finding mutually consistent solutions for two-personzero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory,which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements betweenthem about proper strategies.[4]

In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game atthe RAND corporation. Around this same time, John Nash developed a criterion for mutual consistency of players'strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by vonNeumann and Morgenstern. This equilibrium is sufficiently general to allow for the analysis of non-cooperativegames in addition to cooperative ones.Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensiveform game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applicationsof Game theory to philosophy and political science occurred during this time.In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined theNash equilibrium (later he would introduce trembling hand perfection as well). In 1967, John Harsanyi developedthe concepts of complete information and Bayesian games. Nash, Selten and Harsanyi became Economics NobelLaureates in 1994 for their contributions to economic game theory.In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smithand his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection,and common knowledge[5] were introduced and analyzed.In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as NobelLaureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributedmore to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing anextensive formal analysis of the assumption of common knowledge and of its consequences.

Game theory 32

In 2007, Roger Myerson, together with Leonid Hurwicz and Eric Maskin, was awarded the Nobel Prize inEconomics "for having laid the foundations of mechanism design theory." Myerson's contributions include thenotion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict (Myerson 1997).

Representation of gamesThe games studied in game theory are well-defined mathematical objects. A game consists of a set of players, a setof moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies.Most cooperative games are presented in the characteristic function form, while the extensive and the normal formsare used to define noncooperative games.

Extensive form

An extensive form game

The extensive form can be used to formalize games with a timesequencing of moves. Games here are played on trees (as pictured tothe left). Here each vertex (or node) represents a point of choice for aplayer. The player is specified by a number listed by the vertex. Thelines out of the vertex represent a possible action for that player. Thepayoffs are specified at the bottom of the tree. The extensive form canbe viewed as a multi-player generalization of a decision tree.(Fudenberg & Tirole 1991, p. 67)

In the game pictured to the left, there are two players. Player 1 moves first and chooses either F or U. Player 2 seesPlayer 1's move and then chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player1 gets 8 and Player 2 gets 2.The extensive form can also capture simultaneous-move games and games with imperfect information. To representit, either a dotted line connects different vertices to represent them as being part of the same information set (i.e., theplayers do not know at which point they are), or a closed line is drawn around them. (See example in the imperfectinformation section.)

Normal form

Player 2chooses Left

Player 2chooses Right

Player 1chooses Up

4, 3 –1, –1

Player 1chooses Down

0, 0 3, 4

Normal form or payoff matrix of a 2-player,2-strategy game

The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, andpayoffs (see the example to the right). More generally it can be represented by any function that associates a payofffor each player with every possible combination of actions. In the accompanying example there are two players; onechooses the row and the other chooses the column. Each player has two strategies, which are specified by the numberof rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff receivedby the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in ourexample). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2gets 3.

Game theory 33

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, withoutknowing the actions of the other. If players have some information about the choices of other players, the game isusually presented in extensive form.Every extensive-form game has an equivalent normal-form game, however the transformation to normal form mayresult in an exponential blowup in the size of the representation, making it computationally impractical.(Leyton-Brown & Shoham 2008, p. 35)

Characteristic function formIn cooperative games with transferable utility no individual payoffs are given. Instead, the characteristic functiondetermines the payoff of each coalition. The standard assumption is that the empty coalition obtains a payoff of 0.The origin of this form is to be found in the seminal book of von Neumann and Morgenstern who, when studyingcoalitional normal form games, assumed that when a coalition forms, it plays against the complementarycoalition ( ) as if they were playing a 2-player game. The equilibrium payoff of is characteristic. Nowthere are different models to derive coalitional values from normal form games, but not all games in characteristicfunction form can be derived from normal form games.Formally, a characteristic function form game (also known as a TU-game) is given as a pair , where denotes a set of players and is a characteristic function.The characteristic function form has been generalised to games without the assumption of transferable utility.

Partition function formThe characteristic function form ignores the possible externalities of coalition formation. In the partition functionform the payoff of a coalition depends not only on its members, but also on the way the rest of the players arepartitioned (Thrall & Lucas 1963).

General and applied usesAs a method of applied mathematics, game theory has been used to study a wide variety of human and animalbehaviors. It was initially developed in economics to understand a large collection of economic behaviors, includingbehaviors of firms, markets, and consumers. The use of game theory in the social sciences has expanded, and gametheory has been applied to political, sociological, and psychological behaviors as well.Game-theoretic analysis was initially used to study animal behavior by Ronald Fisher in the 1930s (although evenCharles Darwin makes a few informal game-theoretic statements). This work predates the name "game theory", but itshares many important features with this field. The developments in economics were later applied to biology largelyby John Maynard Smith in his book Evolution and the Theory of Games.In addition to being used to predict and explain behavior, game theory has also been used to attempt to developtheories of ethical or normative behavior. In economics and philosophy, scholars have applied game theory to help inthe understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back asPlato.[6]

Game theory 34

Description and modeling

A three stage Centipede Game

The first known use is to describe and modelhow human populations behave. Somescholars believe that by finding theequilibria of games they can predict howactual human populations will behave whenconfronted with situations analogous to thegame being studied. This particular view ofgame theory has come under recentcriticism. First, it is criticized because theassumptions made by game theorists are often violated. Game theorists may assume players always act in a way todirectly maximize their wins (the Homo economicus model), but in practice, human behavior often deviates fromthis model. Explanations of this phenomenon are many; irrationality, new models of deliberation, or even differentmotives (like that of altruism). Game theorists respond by comparing their assumptions to those used in physics.Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin tothe models used by physicists. However, additional criticism of this use of game theory has been levied becausesome experiments have demonstrated that individuals do not play equilibrium strategies. For instance, in thecentipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria.There is an ongoing debate regarding the importance of these experiments.[7]

Alternatively, some authors claim that Nash equilibria do not provide predictions for human populations, but ratherprovide an explanation for why populations that play Nash equilibria remain in that state. However, the question ofhow populations reach those points remains open.Some game theorists have turned to evolutionary game theory in order to resolve these worries. These modelspresume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary gametheory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includesboth biological as well as cultural evolution and also models of individual learning (for example, fictitious playdynamics).

Prescriptive or normative analysis

Cooperate Defect

Cooperate -1, -1 -10,0

Defect 0, -10 -5,-5

The Prisoner's Dilemma

On the other hand, some scholars see game theory not as a predictive tool for the behavior of human beings, but as asuggestion for how people ought to behave. Since a Nash equilibrium of a game constitutes one's best response to theactions of the other players, playing a strategy that is part of a Nash equilibrium seems appropriate. However, thisuse for game theory has also come under criticism. First, in some cases it is appropriate to play a non-equilibriumstrategy if one expects others to play non-equilibrium strategies as well. For an example, see Guess 2/3 of theaverage.Second, the Prisoner's dilemma presents another potential counterexample. In the Prisoner's Dilemma, each playerpursuing his own self-interest leads both players to be worse off than had they not pursued their own self-interests.

Game theory 35

Economics and businessEconomists have long used game theory to analyze a wide array of economic phenomena and methods, includingauctions, bargaining, fair division, duopolies, oligopolies, social network formation, general equilibrium, mechanismdesign,[8] and voting systems and to model across such broad classifications as mathematical economics,[9]

behavioral economics,[10] political economy,[11] and industrial organization.[12]

This research usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts"are usually based on what is required by norms of rationality. In non-cooperative games, the most famous of these isthe Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the otherstrategies. So, if all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive todeviate, since their strategy is the best they can do given what others are doing.The payoffs of the game are generally taken to represent the utility of individual players. Often in modelingsituations the payoffs represent money, which presumably corresponds to an individual's utility. This assumption,however, can be faulty.A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of someparticular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategysets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use should thisinformation be put. Economists and business professors suggest two primary uses: descriptive and prescriptive.

Political scienceThe application of game theory to political science is focused in the overlapping areas of fair division, politicaleconomy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas,researchers have developed game-theoretic models in which the players are often voters, states, special interestgroups, and politicians.For early examples of game theory applied to political science, see the work of Anthony Downs. In his book AnEconomic Theory of Democracy (Downs 1957), he applies the Hotelling firm location model to the political process.In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. The theoristshows how the political candidates will converge to the ideology preferred by the median voter.A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear andreliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions ofnondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrustand unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy (Levy &Razin 2003).

Biology

Game theory 36

Hawk Dove

Hawk 20,20

80,40

Dove 40,80

60,60

The hawk-dovegame

Unlike economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, thefocus has been less on equilibria that correspond to a notion of rationality, but rather on ones that would bemaintained by evolutionary forces. The best known equilibrium in biology is known as the evolutionarily stablestrategy (or ESS), and was first introduced in (Smith & Price 1973). Although its initial motivation did not involveany of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.In biology, game theory has been used to understand many different phenomena. It was first used to explain theevolution (and stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a resultof evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animalcommunication (Harper & Maynard Smith 2003). The analysis of signaling games and other communication gameshas provided some insight into the evolution of communication among animals. For example, the mobbing behaviorof many species, in which a large number of prey animals attack a larger predator, seems to be an example ofspontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion, seeButterfly Economics.Biologists have used the game of chicken to analyze fighting behavior and territoriality.Maynard Smith, in the preface to Evolution and the Theory of Games, writes, "paradoxically, it has turned out thatgame theory is more readily applied to biology than to the field of economic behaviour for which it was originallydesigned". Evolutionary game theory has been used to explain many seemingly incongruous phenomena innature.[13]

One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in away that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruismbecause such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness.Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night'shunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for theirentire lives and never mate, to Vervet monkeys that warn group members of a predator's approach, even when itendangers that individual's chance of survival.[14] All of these actions increase the overall fitness of a group, butoccur at a cost to the individual.Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary reasoning behind this selection with the equation c<b*r where the cost ( c ) to the altruist must be less than the benefit ( b ) to the recipient multiplied by the coefficient of relatedness ( r ). The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on, (through survival of its offspring) can forgo the option of having offspring itself because the same number of alleles are passed on. Helping a sibling for example (in diploid animals), has a coefficient of ½, because (on average) an individual shares ½ of the alleles in its sibling's offspring. Ensuring that enough of a sibling’s offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.[14] The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between

Game theory 37

all humans only accounts for approximately 1% of the diversity in the playing field, a co-efficient that was ½ in thesmaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g.epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepanciessmaller.

Computer science and logicGame theory has come to play an increasingly important role in logic and in computer science. Several logicaltheories have a basis in game semantics. In addition, computer scientists have used games to model interactivecomputations. Also, game theory provides a theoretical basis to the field of multi-agent systems.Separately, game theory has played a role in online algorithms. In particular, the k-server problem, which has in thepast been referred to as games with moving costs and request-answer games (Ben David, Borodin & Karp etal. 1994). Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexityof randomized algorithms, and especially of online algorithms.The field of algorithmic game theory combines computer science concepts of complexity and algorithm design withgame theory and economic theory. The emergence of the internet has motivated the development of algorithms forfinding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and informationmarkets.[15]

Philosophy

Stag Hare

Stag 3,3

0,2

Hare 2,0

2,2

Stag hunt

Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967),Lewis (1969) used game theory to develop a philosophical account of convention. In so doing, he provided the firstanalysis of common knowledge and employed it in analyzing play in coordination games. In addition, he firstsuggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued byseveral philosophers since Lewis (Skyrms (1996), Grim, Kokalis, and Alai-Tafti et al. (2004)). Following Lewis(1969) game-theoretic account of conventions, Ullmann Margalit (1977) and Bicchieri (2006) have developedtheories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game intoa coordination game.[16]

Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for acollective to have common beliefs or knowledge, and what are the consequences of this knowledge for the socialoutcomes resulting from agents' interactions. Philosophers who have worked in this area include Bicchieri (1989,1993),[17] Skyrms (1990),[18] and Stalnaker (1999).[19]

In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality fromself-interest. Since games like the Prisoner's dilemma present an apparent conflict between morality and self-interest,explaining why cooperation is required by self-interest is an important component of this project. This generalstrategy is a component of the general social contract view in political philosophy (for examples, see Gauthier (1986)and Kavka (1986).[20]

Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the Prisoner's

Game theory 38

dilemma, Stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes aboutmorality (see, e.g., Skyrms (1996, 2004) and Sober and Wilson (1999)).Some assumptions used in some parts of game theory have been challenged in philosophy; psychological egoismstates that rationality reduces to self-interest—a claim debated among philosophers. (see Psychologicalegoism#Criticisms)

Types of games

Cooperative or non-cooperativeA game is cooperative if the players are able to form binding commitments. For instance the legal system requiresthem to adhere to their promises. In noncooperative games this is not possible.Often it is assumed that communication among players is allowed in cooperative games, but not in noncooperativeones. This classification on two binary criteria has been rejected (Harsanyi 1974).Of the two types of games, noncooperative games are able to model situations to the finest details, producingaccurate results. Cooperative games focus on the game at large. Considerable efforts have been made to link the twoapproaches. The so-called Nash-programme has already established many of the cooperative solutions asnoncooperative equilibria.Hybrid games contain cooperative and non-cooperative elements. For instance, coalitions of players are formed in acooperative game, but these play in a non-cooperative fashion.

Symmetric and asymmetric

E F

E 1,2

0,0

F 0,0

1,2

An asymmetricgame

A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategiesemployed, not on who is playing them. If the identities of the players can be changed without changing the payoff tothe strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standardrepresentations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars wouldconsider certain asymmetric games as examples of these games as well. However, the most common payoffs foreach of these games are symmetric.Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It ispossible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the gamepictured to the right is asymmetric despite having identical strategy sets for both players.

Game theory 39

Zero-sum and non-zero-sum

A B

A –1,1

3,–3

B 0,0

–2,2

A zero-sumgame

Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nordecrease the available resources. In zero-sum games the total benefit to all players in the game, for everycombination of strategies, always adds to zero (more informally, a player benefits only at the equal expense ofothers). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactlythe amount one's opponents lose. Other zero-sum games include matching pennies and most classical board gamesincluding Go and chess.Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, becausesome outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one playerdoes not necessarily correspond with a loss by another.Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situationin which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric)zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate theplayers' net winnings.

Simultaneous and sequentialSimultaneous games are games where both players move simultaneously, or if they do not move simultaneously, thelater players are unaware of the earlier players' actions (making them effectively simultaneous). Sequential games (ordynamic games) are games where later players have some knowledge about earlier actions. This need not be perfectinformation about every action of earlier players; it might be very little knowledge. For instance, a player may knowthat an earlier player did not perform one particular action, while he does not know which of the other availableactions the first player actually performed.The difference between simultaneous and sequential games is captured in the different representations discussedabove. Often, normal form is used to represent simultaneous games, and extensive form is used to representsequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive formgames correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games areinsufficient for reasoning about sequential games; see subgame perfection.

Game theory 40

Perfect information and imperfect information

A game of imperfect information (the dotted line representsignorance on the part of player 2, formally called an information set)

An important subset of sequential games consists ofgames of perfect information. A game is one of perfectinformation if all players know the moves previouslymade by all other players. Thus, only sequential gamescan be games of perfect information, since insimultaneous games not every player knows the actionsof the others. Most games studied in game theory areimperfect-information games, although there are someinteresting examples of perfect-information games,including the ultimatum game and centipede game.Recreational games of perfect information games include chess, go, and mancala. Many card games are games ofimperfect information, for instance poker or contract bridge.

Perfect information is often confused with complete information, which is a similar concept. Complete informationrequires that every player know the strategies and payoffs of the other players but not necessarily the actions. Gamesof incomplete information can be reduced, however, to games of imperfect information by introducing "moves bynature" (Leyton-Brown & Shoham 2008, p. 60).

Combinatorial gamesGames in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are calledcombinatorial games. Examples include chess and go. Games that involve imperfect or incomplete information mayalso have a strong combinatorial character, for instance backgammon. There is no unified theory addressingcombinatorial elements in games. There are, however, mathematical tools that can solve particular problems andanswer some general questions.[21]

Games of perfect information have been studied in combinatorial game theory, which has developed novelrepresentations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proofmethods to solve games of certain types, including some "loopy" games that may result in infinitely long sequencesof moves. These methods address games with higher combinatorial complexity than those usually considered intraditional (or "economic") game theory.[22] [23] A typical game that has been solved this way is hex. A related fieldof study, drawing from computational complexity theory, is game complexity, which is concerned with estimatingthe computational difficulty of finding optimal strategies.[24]

Research in artificial intelligence has addressed both perfect and imperfect (or incomplete) information games thathave very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimalstrategies have been found. The practical solutions involve computational heuristics, like alpha-beta pruning or useof artificial neural networks trained by reinforcement learning, which make games more tractable in computingpractice.[21] [25]

Game theory 41

Infinitely long gamesGames, as studied by economists and real-world game players, are generally finished in finitely many moves. Puremathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves,with the winner (or other payoff) not known until after all those moves are completed.The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether oneor the other player has a winning strategy. (It can be proven, using the axiom of choice, that there are games—evenwith perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winningstrategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptiveset theory.

Discrete and continuous gamesMuch of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events,outcomes, etc. Many concepts can be extended, however. Continuous games allow players to choose a strategy froma continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being anynon-negative quantities, including fractional quantities.Differential games such as the continuous pursuit and evasion game are continuous games.

Many-player and population gamesGames with an arbitrary, but finite number of players are often called n-person games (Luce & Raiffa 1957).Evolutionary game theory considers games involving a population of decision makers, where the frequency withwhich a particular decision is made can change over time in response to the decisions made by all individuals in thepopulation. In biology, this is intended to model (biological) evolution, where genetically programmed organismspass along some of their strategy programming to their offspring. In economics, the same theory is intended tocapture population changes because people play the game many times within their lifetime, and consciously (andperhaps rationally) switch strategies (Webb 2007).

Stochastic outcomes (and relation to other fields)Individual decision problems with stochastic outcomes are sometimes considered "one-player games". Thesesituations are not considered game theoretical by some authors. They may be modeled using similar tools within therelated disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AIplanning (with uncertainty) and multi-agent system. Although these fields may have different motivators, themathematics involved are substantially the same, e.g. using Markov decision processes (MDP).Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes"chance moves", also known as "moves by nature" (Osborne & Rubinstein 1994). This player is not typicallyconsidered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dicewhere required by the game.For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. Forexample, the difference in approach between MDPs and the minimax solution is that the latter considers theworst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixedprobability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are notavailable, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategyin such scenarios if it is assumed that an adversary can force such an event to happen.[26] (See black swan theory formore discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses ininvestment banking.)

Game theory 42

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (ofmoves by other players) have also been studied. The "gold standard" is considered to be partially observablestochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.[26]

MetagamesThese are games the play of which is the development of the rules for another game, the target or subject game.Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related tomechanism design theory.The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard (Howard 1971)whereby a situation is framed as a strategic game in which stakeholders try to realise their objectives by means of theoptions available to them. Subsequent developments have led to the formulation of drama theory.

Notes[1] James Madison, Vices of the Political System of the United States, April, 1787. Link (http:/ / www. constitution. org/ jm/ 17870400_vices.

htm)[2] Jack Rakove, "James Madison and the Constitution", History Now, Issue 13 September 2007. Link (http:/ / www. historynow. org/ 09_2007/

historian2. html)[3] J. v. Neumann (1928). "Zur Theorie der Gesellschaftsspiele," Mathematische Annalen, 100(1), p p. 295 (http:/ / www. springerlink. com/

content/ q07530916862223p/ )-320. English translation: "On the Theory of Games of Strategy," in A. W. Tucker and R. D. Luce, ed.(1959),Contributions to the Theory of Games, v. 4, p p. 13 (http:/ / books. google. com/ books?hl=en& lr=& id=9lSVFzsTGWsC& oi=fnd&pg=PA13& dq==P_RGaKOVtC& sig=J-QB_GglFSVWw9KfXjut62E6AmM#v=onepage& q& f=false)- 42. (http:/ / books. google. com/books?hl=en& lr=& id=9lSVFzsTGWsC& oi=fnd& pg=PA42& dq==P_RGaKOVtC&sig=J-QB_GglFSVWw9KfXjut62E6AmM#v=onepage& q& f=false)

[4] Leonard, Robert. Von Neumann, Morgenstern, and the Creation of Game Theory. Cambridge University Press, 2010[5] Although common knowledge was first discussed by the philosopher David Lewis in his dissertation (and later book) Convention in the late

1960s, it was not widely considered by economists until Robert Aumann's work in the 1970s.[6] Ross, Don. "Game Theory" (http:/ / plato. stanford. edu/ archives/ spr2008/ entries/ game-theory/ ). The Stanford Encyclopedia of Philosophy

(Spring 2008 Edition). Edward N. Zalta (ed.). . Retrieved 2008-08-21.[7] Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game theory are

several. For a recent discussion on this field see Camerer (2003).[8] From (2008), The New Palgrave Dictionary of Economics, 2nd Edition:

   • Roger B. Myerson (2008). "mechanism design." Abstract. (http:/ / www. dictionaryofeconomics. com/ article?id=pde2008_M000132&edition=current& q=mechanism design& topicid=& result_number=3)   • _____. "revelation principle." Abstract. (http:/ / www. dictionaryofeconomics. com/ article?id=pde2008_R000137& edition=current&q=moral& topicid=& result_number=1)

[9] • Kenneth J. Arrow and Michael D. Intriligator, ed. (1981), Handbook of Mathematical Economics (1981), v. 1. (http:/ / www. sciencedirect.com/ science?_ob=PublicationURL& _tockey=#TOC#24615#1981#999989999#565707#FLP#& _cdi=24615& _pubType=HS& _auth=y&_acct=C000050221& _version=1& _urlVersion=0& _userid=10& md5=01881ea3fe7d7990fed1c5b78d9f7be6)   • Martin Shubik (1978), "Game Theory: Economic Applications," International Encyclopedia of Statistics, v. 2, William Kruskal, Judith M.Tanur, ed., pp. 372-78.

[10] From (2008), The New Palgrave Dictionary of Economics, 2nd Edition:   • Faruk Gul. "behavioural economics and game theory." Abstract. (http:/ / www. dictionaryofeconomics. com/article?id=pde2008_G000210& q=Behavioral economics & topicid=& result_number=2)   • Colin F. Camerer "behavioral game theory." Abstract. (http:/ / www. dictionaryofeconomics. com/ article?id=pde2008_B000302&q=Behavioral economics & topicid=& result_number=13)

[11] Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in Handbook of Mathematical Economics, , v. 1, pp. 285(http:/ / www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B7P5Y-4FDF0FN-C& _user=10& _coverDate=01/ 01/ 1981&_rdoc=11& _fmt=high& _orig=browse& _origin=browse& _zone=rslt_list_item&_srch=doc-info(#toc#24615#1981#999989999#565707#FLP#display#Volume)& _cdi=24615& _sort=d& _docanchor=& _ct=14&_acct=C000050221& _version=1& _urlVersion=0& _userid=10& md5=cb34198ec88c9ab8fa59af6d5634e9cf& searchtype=a)-330.

[12] Jean Tirole, 1988. The Theory of Industrial Organization, MIT Press. Description (http:/ / mitpress. mit. edu/ catalog/ item/ default.asp?ttype=2& tid=8224) and chapter-preview links (http:/ / books. google. com/ books?id=HIjsF0XONF8C& dq=The+ theory+ of+industrial+ organization+ By+ Jean+ Tirole& printsec=frontcover& source=bn& hl=en& ei=e2vES-H-O8T68Abxp_GyDw& sa=X&oi=book_result& ct=result& resnum=4& ved=0CB8Q6AEwAw#v=onepage& q& f=false) via scroll down.

[13] Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) (http:/ / plato. stanford. edu/ entries/ game-evolutionary/ )

Game theory 43

[14] Biological Altruism (Stanford Encyclopedia of Philosophy) (http:/ / www. seop. leeds. ac. uk/ entries/ altruism-biological/ )[15] Algorithmic Game Theory (http:/ / www. cambridge. org/ journals/ nisan/ downloads/ Nisan_Non-printable. pdf), Cambridge University

Press,[16] E. Ullmann Margalit, The Emergence of Norms, Oxford University Press, 1977. C. Bicchieri, The Grammar of Society: the Nature and

Dynamics of Social Norms, Cambridge University Press, 2006.[17] "Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge ", Erkenntnis 30, 1989: 69-85. See also Rationality and

Coordination, Cambridge University Press, 1993.[18] The Dynamics of Rational Deliberation, Harvard University Press, 1990.[19] "Knowledge, Belief, and Counterfactual Reasoning in Games." In Cristina Bicchieri, Richard Jeffrey, and Brian Skyrms, eds., The Logic of

Strategy. New York: Oxford University Press, 1999.[20] For a more detailed discussion of the use of Game Theory in ethics see the Stanford Encyclopedia of Philosophy's entry game theory and

ethics (http:/ / plato. stanford. edu/ entries/ game-ethics/ ).[21] Jörg Bewersdorff (2005). Luck, logic, and white lies: the mathematics of games. A K Peters, Ltd.. pp. ix-xii and chapter 31.

ISBN 9781568812106.[22] Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: In Introduction to Combinatorial Game Theory. A K

Peters Ltd. pp. 3–4. ISBN 978-1-56881-277-9.[23] Beck, József (2008). Combinatorial games: tic-tac-toe theory. Cambridge University Press. pp. 1–3. ISBN 9780521461009.[24] Robert A. Hearn; Erik D. Demaine (2009). Games, Puzzles, and Computation. A K Peters, Ltd.. ISBN 9781568813226.[25] M. Tim Jones (2008). Artificial Intelligence: A Systems Approach. Jones & Bartlett Learning. pp. 106–118. ISBN 9780763773373.[26] Hugh Brendan McMahan (2006), Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability (http:/ /

www. cs. cmu. edu/ ~mcmahan/ research/ mcmahan_thesis. pdf), CMU-CS-06-166, pp. 3-4

References and further reading

Textbooks and general references• Aumann, Robert J. (1987), game theory,, The New Palgrave: A Dictionary of Economics, 2, pp. 460–82.• Aumann, Robert J., and Sergiu Hart, ed. (1992, 1994, 2002). Handbook of Game Theory with Economic

Applications, 3 v., Elsevier. Table of Contents and "Review Article" (Abstract) links (http:/ / www. sciencedirect.com/ science?_ob=PublicationURL& _tockey=#TOC#24608#2002#999969999#565225#FLP#& _cdi=24608&_pubType=HS& _auth=y& _acct=C000050221& _version=1& _urlVersion=0& _userid=10&md5=7eacc99525be5eae95d41390ba838af5).

• The New Palgrave Dictionary of Economics, (2008). 2nd Edition:"game theory" by Robert J. Aumann. Abstract. (http:/ / www. dictionaryofeconomics. com/article?id=pde2008_G000007& q=game theory& topicid=& result_number=3)"game theory in economics, origins of," by Robert Leonard. Abstract. (http:/ / www. dictionaryofeconomics.com/ article?id=pde2008_G000193& goto=a& topicid=B2& result_number=10)"behavioural economics and game theory" by Faruk Gul. Abstract. (http:/ / www. dictionaryofeconomics.com/ article?id=pde2008_G000210& q=Behavioral economics & topicid=& result_number=2)

• Camerer, Colin (2003), Behavioral Game Theory: Experiments in Strategic Interaction, Russell Sage Foundation,ISBN 978-0-691-09039-9 Description (http:/ / press. princeton. edu/ titles/ 7517. html) amd Introduction (http:/ /press. princeton. edu/ chapters/ i7517. html), pp. 1–25.

• Dutta, Prajit K. (1999), Strategies and games: theory and practice, MIT Press, ISBN 978-0-262-04169-0.Suitable for undergraduate and business students.

• Fernandez, L F.; Bierman, H S. (1998), Game theory with economic applications, Addison-Wesley,ISBN 978-0-201-84758-1. Suitable for upper-level undergraduates.

• Fudenberg, Drew; Tirole, Jean (1991), Game theory, MIT Press, ISBN 978-0-262-06141-4. Acclaimed referencetext. Description. (http:/ / mitpress. mit. edu/ catalog/ item/ default. asp?ttype=2& tid=8204)

• Gibbons, Robert D. (1992), Game theory for applied economists, Princeton University Press,ISBN 978-0-691-00395-5. Suitable for advanced undergraduates.

Game theory 44

• Published in Europe as Robert Gibbons (2001), A Primer in Game Theory, London: Harvester Wheatsheaf,ISBN 978-0-7450-1159-2.

• Gintis, Herbert (2000), Game theory evolving: a problem-centered introduction to modeling strategic behavior,Princeton University Press, ISBN 978-0-691-00943-8

• Green, Jerry R.; Mas-Colell, Andreu; Whinston, Michael D. (1995), Microeconomic theory, Oxford UniversityPress, ISBN 978-0-19-507340-9. Presents game theory in formal way suitable for graduate level.

• edited by Vincent F. Hendricks, Pelle G. Hansen. (2007), Hansen, Pelle G.; Hendricks, Vincent F., eds., GameTheory: 5 Questions, New York, London: Automatic Press / VIP, ISBN 9788799101344. Snippets frominterviews (http:/ / www. gametheorists. com).

• Howard, Nigel (1971), Paradoxes of Rationality: Games, Metagames, and Political Behavior, Cambridge,Massachusetts: The MIT Press, ISBN 978-0262582377

• Isaacs, Rufus (1999), Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit,Control and Optimization, New York: Dover Publications, ISBN 978-0-486-40682-4

• Leyton-Brown, Kevin; Shoham, Yoav (2008), Essentials of Game Theory: A Concise, MultidisciplinaryIntroduction (http:/ / www. gtessentials. org), San Rafael, CA: Morgan & Claypool Publishers,ISBN 978-1-598-29593-1. An 88-page mathematical introduction; free online (http:/ / www. morganclaypool.com/ doi/ abs/ 10. 2200/ S00108ED1V01Y200802AIM003) at many universities.

• Miller, James H. (2003), Game theory at work: how to use game theory to outthink and outmaneuver yourcompetition, New York: McGraw-Hill, ISBN 978-0-07-140020-6. Suitable for a general audience.

• Myerson, Roger B. (1991), Game theory: analysis of conflict, Harvard University Press,ISBN 978-0-674-34116-6

• Osborne, Martin J. (2004), An introduction to game theory, Oxford University Press, ISBN 978-0-19-512895-6.Undergraduate textbook.

• Papayoanou, Paul (2010), Game Theory for Business, Probabilistic Publishing, ISBN 978-09647938-7-3. Primerfor business men and women.

• Osborne, Martin J.; Rubinstein, Ariel (1994), A course in game theory, MIT Press, ISBN 978-0-262-65040-3. Amodern introduction at the graduate level.

• Poundstone, William (1992), Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb,Anchor, ISBN 978-0-385-41580-4. A general history of game theory and game theoreticians.

• Rasmusen, Eric (2006), Games and Information: An Introduction to Game Theory (http:/ / www. rasmusen. org/GI/ index. html) (4th ed.), Wiley-Blackwell, ISBN 978-1-4051-3666-2

• Shoham, Yoav; Leyton-Brown, Kevin (2009), Multiagent Systems: Algorithmic, Game-Theoretic, and LogicalFoundations (http:/ / www. masfoundations. org), New York: Cambridge University Press,ISBN 978-0-521-89943-7. A comprehensive reference from a computational perspective; downloadable freeonline (http:/ / www. masfoundations. org/ download. html).

• Williams, John Davis (1954) (PDF), The Compleat Strategyst: Being a Primer on the Theory of Games ofStrategy (http:/ / www. rand. org/ pubs/ commercial_books/ 2007/ RAND_CB113-1. pdf), Santa Monica: RANDCorp., ISBN 9780833042224 Praised primer and popular introduction for everybody, never out of print.

• Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy (http:/ / faculty. lebow.drexel. edu/ McCainR/ / top/ eco/ game/ game. html) (Revised Edition)

• Christopher Griffin (2010) Game Theory: Penn State Math 486 Lecture Notes (http:/ / www. personal. psu. edu/cxg286/ Math486. pdf), pp. 169, CC-BY-NC-SA license, suitable introduction for undergraduates

• Webb, James N. (2007), Game theory: decisions, interaction and evolution, Springer undergraduate mathematicsseries, Springer, ISBN 1846284236 Consistent treatment of game types usually claimed by different appliedfields, e.g. Markov decision processes.

• Joseph E. Harrington (2008) Games, strategies, and decision making, Worth, ISBN 0716766302. Textbooksuitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation.

Game theory 45

Historically important texts• Aumann, R.J. and Shapley, L.S. (1974), Values of Non-Atomic Games, Princeton University Press• Cournot, A. Augustin (1838), "Recherches sur les principles mathematiques de la théorie des richesses", Libraire

des sciences politiques et sociales (Paris: M. Rivière & C.ie)• Edgeworth, Francis Y. (1881), Mathematical Psychics, London: Kegan Paul• Farquharson, Robin (1969), Theory of Voting, Blackwell (Yale U.P. in the U.S.), ISBN 0631124608• Luce, R. Duncan; Raiffa, Howard (1957), Games and decisions: introduction and critical survey, New York:

Wiley• reprinted edition: R. Duncan Luce ; Howard Raiffa (1989), Games and decisions: introduction and critical

survey, New York: Dover Publications, ISBN 978-0-486-65943-5• Maynard Smith, John (1982), Evolution and the theory of games, Cambridge University Press,

ISBN 978-0-521-28884-2• Maynard Smith, John; Price, George R. (1973), "The logic of animal conflict", Nature 246 (5427): 15–18,

Bibcode 1973Natur.246...15S, doi:10.1038/246015a0• Nash, John (1950), "Equilibrium points in n-person games" (http:/ / www. pnas. org/ cgi/ search?sendit=Search&

pubdate_year=& volume=& firstpage=& DOI=& author1=nash& author2=& title=equilibrium&andorexacttitle=and& titleabstract=& andorexacttitleabs=and& fulltext=& andorexactfulltext=and&fmonth=Jan& fyear=1915& tmonth=Feb& tyear=2008& fdatedef=15+ January+ 1915& tdatedef=6+ February+2008& tocsectionid=all& RESULTFORMAT=1& hits=10& hitsbrief=25& sortspec=relevance&sortspecbrief=relevance), Proceedings of the National Academy of Sciences of the United States of America 36(1): 48–49, doi:10.1073/pnas.36.1.48, PMC 1063129, PMID 16588946

• Shapley, L. S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H. W.Kuhn and A. W. Tucker (eds.)

• Shapley, L. S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. 39, pp. 1095–1100.• von Neumann, John (1928), "Zur Theorie der Gesellschaftsspiele", Mathematische Annalen 100 (1): p. 295 (http:/

/ www. springerlink. com/ content/ q07530916862223p/ )–320. English translation: "On the Theory of Games ofStrategy," in A. W. Tucker and R. D. Luce, ed. (1959), Contributions to the Theory of Games, v. 4, p p. 13 (http:// books. google. com/ books?hl=en& lr=& id=9lSVFzsTGWsC& oi=fnd& pg=PA13& dq==P_RGaKOVtC&sig=J-QB_GglFSVWw9KfXjut62E6AmM#v=onepage& q& f=false)- 42. (http:/ / books. google. com/books?hl=en& lr=& id=9lSVFzsTGWsC& oi=fnd& pg=PA42& dq==P_RGaKOVtC&sig=J-QB_GglFSVWw9KfXjut62E6AmM#v=onepage& q& f=false) Princeton University Press.

• von Neumann, John; Morgenstern, Oskar (1944), Theory of games and economic behavior, Princeton UniversityPress

• Zermelo, Ernst (1913), "Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels", Proceedingsof the Fifth International Congress of Mathematicians 2: 501–4

Other print references• Ben David, S.; Borodin, Allan; Karp, Richard; Tardos, G.; Wigderson, A. (1994), "On the Power of

Randomization in On-line Algorithms" (http:/ / www. math. ias. edu/ ~avi/ PUBLICATIONS/ MYPAPERS/BORODIN/ paper. pdf) (PDF), Algorithmica 11 (1): 2–14, doi:10.1007/BF01294260

• Bicchieri, Cristina (1993, 2nd. edition, 1997), Rationality and Coordination, Cambridge University Press,ISBN 0-521-57444-7

• Downs, Anthony (1957), An Economic theory of Democracy, New York: Harper• Gauthier, David (1986), Morals by agreement, Oxford University Press, ISBN 978-0-19-824992-4• Allan Gibbard, "Manipulation of voting schemes: a general result", Econometrica, Vol. 41, No. 4 (1973),

pp. 587–601.

Game theory 46

• Grim, Patrick; Kokalis, Trina; Alai-Tafti, Ali; Kilb, Nicholas; St Denis, Paul (2004), "Making meaning happen",Journal of Experimental & Theoretical Artificial Intelligence 16 (4): 209–243,doi:10.1080/09528130412331294715

• Harper, David; Maynard Smith, John (2003), Animal signals, Oxford University Press, ISBN 978-0-19-852685-8• Harsanyi, John C. (1974), "An equilibrium point interpretation of stable sets", Management Science 20 (11):

1472–1495, doi:10.1287/mnsc.20.11.1472• Levy, Gilat; Razin, Ronny (2003), "It Takes Two: An Explanation of the Democratic Peace" (http:/ / papers. ssrn.

com/ sol3/ papers. cfm?abstract_id=433844), Working Paper• Lewis, David (1969), Convention: A Philosophical Study, ISBN 978-0-631-23257-5 (2002 edition)• McDonald, John (1950 - 1996), Strategy in Poker, Business & War, W. W. Norton, ISBN 0-393-31457-X. A

layman's introduction.• Quine, W.v.O (1967), "Truth by Convention", Philosophica Essays for A.N. Whitehead, Russel and Russel

Publishers, ISBN 978-0-8462-0970-6• Quine, W.v.O (1960), "Carnap and Logical Truth", Synthese 12 (4): 350–374, doi:10.1007/BF00485423• Mark A. Satterthwaite, "Strategy-proofness and Arrow's Conditions: Existence and Correspondence Theorems for

Voting Procedures and Social Welfare Functions", Journal of Economic Theory 10 (April 1975), 187–217.• Siegfried, Tom (2006), A Beautiful Math, Joseph Henry Press, ISBN 0-309-10192-1• Skyrms, Brian (1990), The Dynamics of Rational Deliberation, Harvard University Press, ISBN 0-674-21885-X• Skyrms, Brian (1996), Evolution of the social contract, Cambridge University Press, ISBN 978-0-521-55583-8• Skyrms, Brian (2004), The stag hunt and the evolution of social structure, Cambridge University Press,

ISBN 978-0-521-53392-8• Sober, Elliott; Wilson, David Sloan (1998), Unto others: the evolution and psychology of unselfish behavior,

Harvard University Press, ISBN 978-0-674-93047-6• Thrall, Robert M.; Lucas, William F. (1963), " -person games in partition function form", Naval Research

Logistics Quarterly 10 (4): 281–298, doi:10.1002/nav.3800100126

Websites• Paul Walker: History of Game Theory Page (http:/ / www. econ. canterbury. ac. nz/ personal_pages/ paul_walker/

gt/ hist. htm).• David Levine: Game Theory. Papers, Lecture Notes and much more stuff. (http:/ / dklevine. com)• Alvin Roth: Game Theory and Experimental Economics page (http:/ / www. economics. harvard. edu/ ~aroth/

alroth. html) - Comprehensive list of links to game theory information on the Web• Adam Kalai: Game Theory and Computer Science (http:/ / wiki. cc. gatech. edu/ theory/ index. php/

CS_8803_-_Game_Theory_and_Computer_Science. _Spring_2008) - Lecture notes on Game Theory andComputer Science

• Mike Shor: Game Theory .net (http:/ / www. gametheory. net) - Lecture notes, interactive illustrations and otherinformation.

• Jim Ratliff's Graduate Course in Game Theory (http:/ / virtualperfection. com/ gametheory/ ) (lecture notes).• Don Ross: Review Of Game Theory (http:/ / plato. stanford. edu/ entries/ game-theory/ ) in the Stanford

Encyclopedia of Philosophy.• Bruno Verbeek and Christopher Morris: Game Theory and Ethics (http:/ / plato. stanford. edu/ entries/

game-ethics/ )• Chris Yiu's Game Theory Lounge (http:/ / web. archive. org/ web/ 20090110073411/ http:/ / www. yiu. co. uk/

gametheory. php) Archived from the original (http:/ / www. yiu. co. uk/ gametheory. php) on 2009-01-17• Elmer G. Wiens: Game Theory (http:/ / www. egwald. ca/ operationsresearch/ gameintroduction. php) -

Introduction, worked examples, play online two-person zero-sum games.

Game theory 47

• Marek M. Kaminski: Game Theory and Politics (http:/ / webfiles. uci. edu/ mkaminsk/ www/ courses. html) -syllabuses and lecture notes for game theory and political science.

• Web sites on game theory and social interactions (http:/ / www. socialcapitalgateway. org/ eng-gametheory. htm)• Kesten Green's Conflict Forecasting (http:/ / conflictforecasting. com) - See Papers for evidence on the accuracy

of forecasts from game theory and other methods.• McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007) Gambit: Software Tools for

Game Theory (http:/ / gambit. sourceforge. net).• Benjamin Polak: Open Course on Game Theory at Yale (http:/ / oyc. yale. edu/ economics/ game-theory) videos

of the course (http:/ / www. youtube. com/ view_play_list?p=6EF60E1027E1A10B)• Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007) Spieltheorie-Software.de: An

application for Game Theory implemented in JAVA (http:/ / www. spieltheorie-software. de).

Simulation

Wooden mechanical horse simulator during WWI.

Simulation is the imitation of some real thing, state ofaffairs, or process. The act of simulating somethinggenerally entails representing certain key characteristicsor behaviours of a selected physical or abstract system.

Simulation is used in many contexts, such as simulationof technology for performance optimization, safetyengineering, testing, training, education, and videogames. Training simulators include flight simulators fortraining aircraft pilots in order to provide them with alifelike experience. Simulation is also used for scientificmodeling of natural systems or human systems in order togain insight into their functioning.[1] Simulation can beused to show the eventual real effects of alternativeconditions and courses of action. Simulation is also used when the real system cannot be engaged, because it may notbe accessible, or it may be dangerous or unacceptable to engage, or it is being designed but not yet built, or it maysimply not exist .[2]

Key issues in simulation include acquisition of valid source information about the relevant selection of keycharacteristics and behaviours, the use of simplifying approximations and assumptions within the simulation, andfidelity and validity of the simulation outcomes.

Simulation 48

Classification and terminology

Human-in-the-loop simulation of outer space.

Visualization of a direct numerical simulationmodel.

Historically, simulations used in different fields developed largelyindependently, but 20th century studies of Systems theory andCybernetics combined with spreading use of computers across all thosefields have led to some unification and a more systematic view of theconcept.

Physical simulation refers to simulation in which physical objects aresubstituted for the real thing (some circles[3] use the term for computersimulations modelling selected laws of physics, but this article doesn't).These physical objects are often chosen because they are smaller orcheaper than the actual object or system.

Interactive simulation is a special kind of physical simulation, oftenreferred to as a human in the loop simulation, in which physicalsimulations include human operators, such as in a flight simulator or adriving simulator.

Human in the loop simulations can include a computer simulation as aso-called synthetic environment.[4]

Computer simulation

A computer simulation (or "sim") is an attempt to model a real-life orhypothetical situation on a computer so that it can be studied to seehow the system works. By changing variables, predictions may bemade about the behaviour of the system.[1]

Computer simulation has become a useful part of modeling manynatural systems in physics, chemistry and biology,[5] and humansystems in economics and social science (the computational sociology)as well as in engineering to gain insight into the operation of thosesystems. A good example of the usefulness of using computers tosimulate can be found in the field of network traffic simulation. In such simulations, the model behaviour willchange each simulation according to the set of initial parameters assumed for the environment.

Traditionally, the formal modeling of systems has been via a mathematical model, which attempts to find analyticalsolutions enabling the prediction of the behaviour of the system from a set of parameters and initial conditions.Computer simulation is often used as an adjunct to, or substitution for, modeling systems for which simple closedform analytic solutions are not possible. There are many different types of computer simulation, the common featurethey all share is the attempt to generate a sample of representative scenarios for a model in which a completeenumeration of all possible states would be prohibitive or impossible.Several software packages exist for running computer-based simulation modeling (e.g. Monte Carlo simulation,stochastic modeling, multimethod modeling) that makes the modeling almost effortless.Modern usage of the term "computer simulation" may encompass virtually any computer-based representation.

Simulation 49

Computer scienceIn computer science, simulation has some specialized meanings: Alan Turing used the term "simulation" to refer towhat happens when a universal machine executes a state transition table (in modern terminology, a computer runs aprogram) that describes the state transitions, inputs and outputs of a subject discrete-state machine. The computersimulates the subject machine. Accordingly, in theoretical computer science the term simulation is a relation betweenstate transition systems, useful in the study of operational semantics.Less theoretically, an interesting application of computer simulation is to simulate computers using computers. Incomputer architecture, a type of simulator, typically called an emulator, is often used to execute a program that hasto run on some inconvenient type of computer (for example, a newly designed computer that has not yet been built oran obsolete computer that is no longer available), or in a tightly controlled testing environment (see Computerarchitecture simulator and Platform virtualization). For example, simulators have been used to debug amicroprogram or sometimes commercial application programs, before the program is downloaded to the targetmachine. Since the operation of the computer is simulated, all of the information about the computer's operation isdirectly available to the programmer, and the speed and execution of the simulation can be varied at will.Simulators may also be used to interpret fault trees, or test VLSI logic designs before they are constructed. Symbolicsimulation uses variables to stand for unknown values.In the field of optimization, simulations of physical processes are often used in conjunction with evolutionarycomputation to optimize control strategies...

Simulation in education and trainingSimulation is extensively used for educational purposes. It is frequently used by way of adaptive hypermedia.Simulation is often used in the training of civilian and military personnel.[6] This usually occurs when it isprohibitively expensive or simply too dangerous to allow trainees to use the real equipment in the real world. In suchsituations they will spend time learning valuable lessons in a "safe" virtual environment yet living a lifelikeexperience (or at least it is the goal).. Often the convenience is to permit mistakes during training for a safety-criticalsystem. For example, in simSchool [7] teachers practice classroom management and teaching techniques onsimulated students, which avoids "learning on the job" that can damage real students. There is a distinction, though,between simulations used for training and Instructional simulation.Training simulations typically come in one of three categories:[8]

• "live" simulation (where actual players use genuine systems in a real environment);• "virtual" simulation (where actual players use simulated systems in a synthetic environment [4] ), or• "constructive" simulation (where virtual players use simulated systems in a synthetic environment). Constructive

simulation is often referred to as "wargaming" since it bears some resemblance to table-top war games in whichplayers command armies of soldiers and equipment that move around a board.

In standardized tests, "live" simulations are sometimes called "high-fidelity", producing "samples of likelyperformance", as opposed to "low-fidelity", "pencil-and-paper" simulations producing only "signs of possibleperformance",[9] but the distinction between high, moderate and low fidelity remains relative, depending on thecontext of a particular comparison.Simulations in education are somewhat like training simulations. They focus on specific tasks. The term 'microworld' is used to refer to educational simulations which model some abstract concept rather than simulating a realistic object or environment, or in some cases model a real world environment in a simplistic way so as to help a learner develop an understanding of the key concepts. Normally, a user can create some sort of construction within the microworld that will behave in a way consistent with the concepts being modeled. Seymour Papert was one of the first to advocate the value of microworlds, and the Logo (programming language) programming environment developed by Papert is one of the most famous microworlds. As another example, the Global Challenge Award

Simulation 50

online STEM learning web site uses microworld simulations to teach science concepts related to global warming andthe future of energy. Other projects for simulations in educations are Open Source Physics, NetSim etc.Management games (or business simulations) have been finding favour in business education in recent years.[10]

Business simulations that incorporate a dynamic model enable experimentation with business strategies in a risk freeenvironment and provide a useful extension to case study discussions.Social simulations may be used in social science classrooms to illustrate social and political processes inanthropology, economics, history, political science, or sociology courses, typically at the high school or universitylevel. These may, for example, take the form of civics simulations, in which participants assume roles in a simulatedsociety, or international relations simulations in which participants engage in negotiations, alliance formation, trade,diplomacy, and the use of force. Such simulations might be based on fictitious political systems, or be based oncurrent or historical events. An example of the latter would be Barnard College's "Reacting to the Past" series ofeducational simulations.[11] The "Reacting to the Past" series also includes simulation games that address scienceeducation.In recent years, there has been increasing use of social simulations for staff training in aid and development agencies.The Carana simulation, for example, was first developed by the United Nations Development Programme, and isnow used in a very revised form by the World Bank for training staff to deal with fragile and conflict-affectedcountries.[12]

Common User Interaction Systems for Virtual SimulationsVirtual Simulations represent a specific category of simulation that utilizes simulation equipment to create asimulated world for the user. Virtual Simulations allow users to interact with a virtual world. Virtual worlds operateon platforms of integrated software and hardware components. In this manner, the system can accept input from theuser (e.g., body tracking, voice/sound recognition, physical controllers) and produce output to the user (e.g., visualdisplay, aural display, haptic display) .[13] Virtual Simulations use the aforementioned modes of interaction toproduce a sense of immersion for the user.

Virtual Simulation Input HardwareThere is a wide variety of input hardware available to accept user input for virtual simulations. The following listbriefly describes several of them:Body Tracking The motion capture method is often used to record the user’s movements and translate the captureddata into inputs for the virtual simulation. For example, if a user physically turns their head, the motion would becaptured by the simulation hardware in some way and translated to a corresponding shift in view within thesimulation.• Capture Suits and/or gloves may be used to capture movements of users body parts. The systems may have

sensors incorporated inside them to sense movements of different body parts (e.g., fingers). Alternatively, thesesystems may have exterior tracking devices or marks that can be detected by external ultrasound, optical receiversor electromagnetic sensors. Internal inertial sensors are also available on some systems. The units may transmitdata either wirelessly or through cables.

• Eye trackers can also be used to detect eye movements so that the system can determine precisely where a user islooking at any given instant.

Physical Controllers Physical controllers provide input to the simulation only through direct manipulation by theuser. In virtual simulations, tactile feedback from physical controllers is highly desirable in a number of simulationenvironments.• Omni directional treadmills can be used to capture the users locomotion as they walk or run.

Simulation 51

• High fidelity instrumentation such as instrument panels in virtual aircraft cockpits provides users with actualcontrols to raise the level of immersion. For example, pilots can use the actual global positioning system controlsfrom the real device in a simulated cockpit to help them practice procedures with the actual device in the contextof the integrated cockpit system.

Voice/Sound Recognition This form of interaction may be used either to interact with agents within the simulation(e.g., virtual people) or to manipulate objects in the simulation (e.g., information). Voice interaction presumablyincreases the level of immersion for the user.• Users may use headsets with boom microphones, lapel microphones or the room may be equipped with

strategically located microphones.Current Research into User Input Systems Research in future input systems hold a great deal of promise forvirtual simulations. Systems such as brain-computer interfaces (BCIs)Brain-computer interface offer the ability tofurther increase the level of immersion for virtual simulation users. Lee, Keinrath, Scherer, Bischof, Pfurtscheller [14]

proved that naïve subjects could be trained to use a BCI to navigate a virtual apartment with relative ease. Using theBCI, the authors found that subjects were able to freely navigate the virtual environment with relatively minimaleffort. It is possible that these types of systems will become standard input modalities in future virtual simulationsystems.

Virtual Simulation Output HardwareThere is a wide variety of output hardware available to deliver stimulus to users in virtual simulations. The followinglist briefly describes several of them:Visual Display Visual displays provide the visual stimulus to the user.• Stationary displays can vary from a conventional desktop display to 360-degree wrap around screens to stereo

three-dimensional screens. Conventional desktop displays can vary in size from 15 to 60+ inches. Wrap aroundscreens are typically utilized in what is known as a Cave Automatic Virtual Environment (CAVE) CaveAutomatic Virtual Environment. Stereo three-dimensional screens produce three-dimensional images either withor without special glasses—depending on the design.

• Head mounted displays (HMDs) have small displays that are mounted on headgear worn by the user. Thesesystems are connected directly into the virtual simulation to provide the user with a more immersive experience.Weight, update rates and field of view are some of the key variables that differentiate HMDs. Naturally, heavierHMDs are undesirable as they cause fatigue over time. If the update rate is too slow, the system is unable toupdate the displays fast enough to correspond with a quick head turn by the user. Slower update rates tend tocause simulation sickness and disrupt the sense of immersion. Field of view or the angular extent of the world thatis seen at a given moment Field of view can vary from system to system and has been found to affect the userssense of immersion.

Aural Display Several different types of audio systems exist to help the user hear and localize sounds spatially.Special software can be used to produce 3D audio effects 3D audio to create the illusion that sound sources areplaced within a defined three-dimensional space around the user.• Stationary conventional speaker systems may be used provide dual or multi-channel surround sound. However,

external speakers are not as effective as headphones in producing 3D audio effects.[13]

• Conventional headphones offer a portable alternative to stationary speakers. They also have the added advantagesof masking real world noise and facilitate more effective 3D audio sound effects.[13]

Haptic Display These displays provide sense of touch to the user Haptic technology. This type of output issometimes referred to as force feedback.• Tactile Tile Displays use different types of actuators such as inflatable bladders, vibrators, low frequency

sub-woofers, pin actuators and/or thermo-actuators to produce sensations for the user.

Simulation 52

• End Effector Displays can respond to users inputs with resistance and force.[13] These systems are often used inmedical applications for remote surgeries that employ robotic instruments.[15]

Vestibular Display These displays provide a sense of motion to the user Motion simulator. They often manifest asmotion bases for virtual vehicle simulation such as driving simulators or flight simulators. Motion bases are fixed inplace but use actuators to move the simulator in ways that can produce the sensations pitching, yawing or rolling.The simulators can also move in such a way as to produce a sense of acceleration on all axes (e.g., the motion basecan produce the sensation of falling).

Clinical healthcare simulatorsMedical simulators are increasingly being developed and deployed to teach therapeutic and diagnostic procedures aswell as medical concepts and decision making to personnel in the health professions. Simulators have beendeveloped for training procedures ranging from the basics such as blood draw, to laparoscopic surgery [16] andtrauma care. They are also important to help on prototyping new devices[17] for biomedical engineering problems.Currently, simulators are applied to research and development of tools for new therapies,[18] treatments[19] and earlydiagnosis[20] in medicine.Many medical simulators involve a computer connected to a plastic simulation of the relevant anatomy.Sophisticated simulators of this type employ a life size mannequin that responds to injected drugs and can beprogrammed to create simulations of life-threatening emergencies. In other simulations, visual components of theprocedure are reproduced by computer graphics techniques, while touch-based components are reproduced by hapticfeedback devices combined with physical simulation routines computed in response to the user's actions. Medicalsimulations of this sort will often use 3D CT or MRI scans of patient data to enhance realism. Some medicalsimulations are developed to be widely distributed (such as web-enabled simulations [21] that can be viewed viastandard web browsers) and can be interacted with using standard computer interfaces, such as the keyboard andmouse.Another important medical application of a simulator — although, perhaps, denoting a slightly different meaning ofsimulator — is the use of a placebo drug, a formulation that simulates the active drug in trials of drug efficacy (seePlacebo (origins of technical term)).

Improving Patient Safety through New InnovationsPatient safety is a concern in the medical industry. Patients have been known to suffer injuries and even death due tomanagement error, and lack of using best standards of care and training. According to Building a National Agendafor Simulation-Based Medical Education (Eder-Van Hook, Jackie, 2004) , “A health care provider’s ability to reactprudently in an unexpected situation is one of the most critical factors in creating a positive outcome in medicalemergency, regardless of whether it occurs on the battlefield, freeway, or hospital emergency room.” simulation.Eder-Van Hook (2004) also noted that medical errors kill up to 98,000 with an estimated cost between $37 and $50million and $17 to $29 billion for preventable adverse events dollars per year. “Deaths due to preventable adverseevents exceed deaths attributable to motor vehicle accidents, breast cancer, or AIDS” Eder-Van Hook (2004). Withthese types of statistics it is no wonder that improving patient safety is a prevalent concern in the industry.New innovative simulation training solutions are now being used to train medical professionals in an attempt to reduce the number of safety concerns that have adverse effects on the patients. However, according to the article Does Simulation Improve Patient Safety? Self-efficacy, Competence, Operational Performance, and Patient Safety (Nishisaki A., Keren R., and Nadkarni, V., 2007), the jury is still out. Nishisaki states that “There is good evidence that simulation training improves provider and team self-efficacy and competence on manikins. There is also good evidence that procedural simulation improves actual operational performance in clinical settings.[22] However, no evidence yet shows that crew resource management training through simulation, despite its promise, improves team operational performance at the bedside. Also, no evidence to date proves that simulation training actually improves

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patient outcome. Even so, confidence is growing in the validity of medical simulation as the training tool of thefuture.” This could be because there are not enough research studies yet conducted to effectively determine thesuccess of simulation initiatives to improve patient safety. Examples of [recently implemented] research simulationsused to improve patient care [and its funding] can be found at Improving Patient Safety through Simulation Research(US Department of Human Health Services) http:/ / www. ahrq. gov/ qual/ simulproj. htm.One such attempt to improve patient safety through the use of simulations training is pediatric care to deliverjust-in-time service or/and just-in-place. This training consists of 20 minutes of simulated training just beforeworkers report to shift. It is hoped that the recentness of the training will increase the positive and reduce thenegative results that have generally been associated with the procedure. The purpose of this study is to determine ifjust-in-time training improves patient safety and operational performance of orotracheal intubation and decreaseoccurrences of undesired associated events and “to test the hypothesis that high fidelity simulation may enhance thetraining efficacy and patient safety in simulation settings.” The conclusion as reported in Abstract P38: Just-In-TimeSimulation Training Improves ICU Physician Trainee Airway Resuscitation Participation without CompromisingProcedural Success or Safety (Nishisaki A., 2008), were that simulation training improved resident participation inreal cases; but did not sacrifice the quality of service. It could be therefore hypothesized that by increasing thenumber of highly trained residents through the use of simulation training, that the simulation training does in factincrease patient safety. This hypothesis would have to be researched for validation and the results may or may notgeneralize to other situations.

History of simulation in healthcareThe first medical simulators were simple models of human patients.[23]

Since antiquity, these representations in clay and stone were used to demonstrate clinical features of disease statesand their effects on humans. Models have been found from many cultures and continents. These models have beenused in some cultures (e.g., Chinese culture) as a "diagnostic" instrument, allowing women to consult malephysicians while maintaining social laws of modesty. Models are used today to help students learn the anatomy ofthe musculoskeletal system and organ systems.[23]

Type of modelsActive models

Active models that attempt to reproduce living anatomy or physiology are recent developments. The famous“Harvey” mannequin was developed at the University of Miami and is able to recreate many of the physicalfindings of the cardiology examination, including palpation, auscultation, and electrocardiography.[24]

Interactive modelsMore recently, interactive models have been developed that respond to actions taken by a student orphysician.[24] Until recently, these simulations were two dimensional computer programs that acted more likea textbook than a patient. Computer simulations have the advantage of allowing a student to make judgements,and also to make errors. The process of iterative learning through assessment, evaluation, decision making,and error correction creates a much stronger learning environment than passive instruction.

Computer simulators

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3DiTeams learner is percussing the patients chestin virtual field hospital

Simulators have been proposed as an ideal tool for assessment ofstudents for clinical skills.[25] For patients, "cybertherapy" can beused for sessions simulating traumatic expericences, from fear ofheights to social anxiety.[26]

Programmed patients and simulated clinical situations, includingmock disaster drills, have been used extensively for educationand evaluation. These “lifelike” simulations are expensive, andlack reproducibility. A fully functional "3Di" simulator would bethe most specific tool available for teaching and measurement ofclinical skills. Gaming platforms have been applied to createthese virtual medical environments to create an interactivemethod for learning and application of information in a clinicalcontext.[27] [28]

Immersive disease state simulations allow a doctor or HCP to experience what a disease actually feels like.Using sensors and transducers symptomatic effects can be delivered to a participant allowing them toexperience the patients disease state.Such a simulator meets the goals of an objective and standardized examination for clinical competence.[29]

This system is superior to examinations that use "standard patients" because it permits the quantitativemeasurement of competence, as well as reproducing the same objective findings.[30]

Simulation in entertainmentEntertainment simulation is a term that encompasses many large and popular industries such as film, television,video games (including serious games) and rides in theme parks. Although modern simulation is thought to have itsroots in training and the military, in the 20th century it also became a conduit for enterprises which were morehedonistic in nature. Advances in technology in the 1980s and 1990s caused simulation to become more widely usedand it began to appear in movies such as Jurassic Park (1993) and in computer-based games such as Atari’sBattlezone.

History

Early History (1940’s and 50’s)

The first simulation game may have been created as early as 1947 by Thomas T. Goldsmith Jr. and Estle Ray Mann.This was a straightforward game that simulated a missile being fired at a target. The curve of the missile and itsspeed could be adjusted using several knobs. In 1958 a computer game called “Tennis for Two” was created by WillyHigginbotham which simulated a tennis game between two players who could both play at the same time using handcontrols and was displayed on an oscilloscope.[31] This was one of the first electronic video games to use a graphicaldisplay.

Modern Simulation (1980’s-present)

Advances in technology in the 1980s made the computer more affordable and more capable than they were in previous decades [32] which facilitated the rise of computer gaming. The first video game consoles released in the 1970s and early '80s fell prey to the industry crash in 1983, but in 1985 Nintendo released the Nintendo Entertainment System (NES) which became the best selling console in video game history.[33] In the 1990s computer games became widely popular with the release of such game as The Sims and Command and Conquer and the still increasing power of desktop computers. Today, computer simulation games such as World of Warcraft are played by millions of people around the world.

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Computer-generated imagery was used in film to simulate objects as early as 1976, though in 1982 the movie Tronwas the first film to use computer-generated imagery for more than a couple of minutes. However, the commercialfailure of the movie may have caused the industry to step away from the technology.[34] In 1993, the movie JurassicPark became the first popular film to use computer-generated graphics extensively, integrating the simulateddinosaurs almost seamlessly into live action scenes. This event transformed the film industry; in 1995 the movie ToyStory was the first film to use only computer-generated images and by the new millennium computer generatedgraphics were the leading choice for special effects in movies.[35] Simulators have been used for entertainment since the Link Trainer in the 1930s.[36] The first modern simulator rideto open at a theme park was Disney’s Star Tours in 1987 soon followed by Universal’s The Funtastic World ofHanna-Barbera in 1990 which was the first ride to be done entirely with computer graphics.[37]

Examples of entertainment simulation

Computer and video games

Simulation games, as opposed to other genres of video and computer games, represent or simulate an environmentaccurately. Moreover, they represent the interactions between the playable characters and the environmentrealistically. These kinds of games are usually more complex in terms of game play.[38] Simulation games havebecome incredibly popular among people of all ages.[39] Popular simulation games include SimCity, Tiger WoodsPGA Tour and Virtonomics.

Film

Computer-generated imagery is “the application of the field of 3D computer graphics to special effects”. Thistechnology is used for visual effects because they are high in quality, controllable, and can create effects that wouldnot be feasible using any other technology either because of cost, resources or safety.[40] Computer-generatedgraphics can be seen in many live action movies today, especially those of the action genre. Further, computergenerated imagery has almost completely supplanted hand-drawn animation in children's movies which areincreasingly computer-generated only. Examples of movies that use computer-generated imagery include FindingNemo, 300 and Iron Man.

Theme park rides

Simulator rides are the progeny of military training simulators and commercial simulators, but they are different in afundamental way. While military training simulators react realistically to the input of the trainee in real time, ridesimulators only feel like they move realistically and move according to prerecorded motion scripts.[37] One of thefirst simulator rides, Star Tours, which cost $32 million, used a hydraulic motion based cabin. The movement wasprogrammed by a joystick. Today’s simulator rides, such as The Amazing Adventures of Spider-Man includeelements to increase the amount of immersion experienced by the riders such as: 3D imagery, physical effects(spraying water or producing scents), and movement through an environment.[41] Examples of simulation ridesinclude Mission Space and The Simpsons Ride.

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Simulation and ManufacturingManufacturing represents one of the most important applications of Simulation. This technique represents a valuabletool used by engineers when evaluating the effect of capital investment in equipments and physical facilities likefactory plants, warehouses, and distribution centers. Simulation can be used to predict the performance of an existingor planned system and to compare alternative solutions for a particular design problem.[42]

Another important goal of manufacturing-simulations is to quantify system performance. Common measures ofsystem performance include the following:[43]

• Throughput under average and peak loads;• System cycle time (how long it take to produce one part);• Utilization of resource, labor, and machines;• Bottlenecks and choke points;• Queuing at work locations;• Queuing and delays caused by material-handling devices and systems;• WIP storage needs;• Staffing requirements;• Effectiveness of scheduling systems;• Effectiveness of control systems.

More examples in various areas

Automobile simulator

A soldier tests out a heavy-wheeled-vehicledriver simulator.

An automobile simulator provides an opportunity to reproduce thecharacteristics of real vehicles in a virtual environment. It replicates theexternal factors and conditions with which a vehicle interacts enablinga driver to feel as if they are sitting in the cab of their own vehicle.Scenarios and events are replicated with sufficient reality to ensure thatdrivers become fully immersed in the experience rather than simplyviewing it as an educational experience.The simulator provides a constructive experience for the novice driverand enables more complex exercises to be undertaken by the moremature driver. For novice drivers, truck simulators provide anopportunity to begin their career by applying best practice. For maturedrivers, simulation provides the ability to enhance good driving or to detect poor practice and to suggest thenecessary steps for remedial action. For companies, it provides an opportunity to educate staff in the driving skillsthat achieve reduced maintenance costs, improved productivity and, most importantly, to ensure the safety of theiractions in all possible situations.

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Biomechanics simulatorsA biomechanics simulator is used to analyze walking dynamics, study sports performance, simulate surgicalprocedures, analyze joint loads, design medical devices, and animate human and animal movement.A neuromechanical simulator that combines biomechanical and biologically realistic neural network simulation. Itallows the user to test hypotheses on the neural basis of behavior in a physically accurate 3-D virtual environment.

City and urban simulationA city simulator can be a city-building game but can also be a tool used by urban planners to understand how citiesare likely to evolve in response to various policy decisions. AnyLogic is an example of modern, large-scale urbansimulators designed for use by urban planners. City simulators are generally agent-based simulations with explicitrepresentations for land use and transportation. UrbanSim and LEAM are examples of large-scale urban simulationmodels that are used by metropolitan planning agencies and military bases for land use and transportation planning.

Classroom of the futureThe "classroom of the future" will probably contain several kinds of simulators, in addition to textual and visuallearning tools. This will allow students to enter the clinical years better prepared, and with a higher skill level. Theadvanced student or postgraduate will have a more concise and comprehensive method of retraining — or ofincorporating new clinical procedures into their skill set — and regulatory bodies and medical institutions will find iteasier to assess the proficiency and competency of individuals.The classroom of the future will also form the basis of a clinical skills unit for continuing education of medicalpersonnel; and in the same way that the use of periodic flight training assists airline pilots, this technology will assistpractitioners throughout their career.The simulator will be more than a "living" textbook, it will become an integral a part of the practice of medicine. Thesimulator environment will also provide a standard platform for curriculum development in institutions of medicaleducation.

Communication Satellite SimulationModern satellite communications systems (SatCom) are often large and complex with many interacting parts andelements. In addition, the need for broadband connectivity on a moving vehicle has increased dramatically in the pastfew years for both commercial and military applications. To accurately predict and deliver high quality of service,satcom system designers have to factor in terrain as well as atmospheric and meteorological conditions in theirplanning. To deal with such complexity, system designers and operators increasingly turn towards computer modelsof their systems to simulate real world operational conditions and gain insights in to usability and requirements priorto final product sign-off. Modeling improves the understanding of the system by enabling the SatCom systemdesigner or planner to simulate real world performance by injecting the models with multiple hypotheticalatmospheric and environmental conditions.

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Digital Lifecycle Simulation

Simulation of airflow over an engine

Simulation solutions are being increasingly integrated with CAx(CAD, CAM, CAE....) solutions and processes. The use of simulationthroughout the product lifecycle, especially at the earlier concept anddesign stages, has the potential of providing substantial benefits. Thesebenefits range from direct cost issues such as reduced prototyping andshorter time-to-market, to better performing products and highermargins. However, for some companies, simulation has not providedthe expected benefits.The research firm Aberdeen Group has found that nearly allbest-in-class manufacturers use simulation early in the design processas compared to 3 or 4 laggards who do not.The successful use of Simulation, early in the lifecycle, has been largely driven by increased integration ofsimulation tools with the entire CAD, CAM and PLM solution-set. Simulation solutions can now function across theextended enterprise in a multi-CAD environment, and include solutions for managing simulation data and processesand ensuring that simulation results are made part of the product lifecycle history. The ability to use simulationacross the entire lifecycle has been enhanced through improved user interfaces such as tailorable user interfaces and"wizards" which allow all appropriate PLM participants to take part in the simulation process.

Disaster Preparedness and Simulation TrainingSimulation training has become a method for preparing people for disasters. Simulations can replicate emergencysituations and track how learners respond thanks to a lifelike experience. Disaster preparedness simulations caninvolve training on how to handle terrorism attacks, natural disasters, pandemic outbreaks, or other life-threateningemergencies.One organization that has used simulation training for disaster preparedness is CADE (Center for Advancement ofDistance Education). CADE[44] has used a video game to prepare emergency workers for multiple types of attacks.As reported by News-Medical.Net, ”The video game is the first in a series of simulations to address bioterrorism,pandemic flu, smallpox and other disasters that emergency personnel must prepare for.[45] ” Developed by a teamfrom the University of Illinois at Chicago (UIC), the game allows learners to practice their emergency skills in asafe, controlled environment.The Emergency Simulation Program (ESP) at the British Columbia Institute of Technology (BCIT), Vancouver,British Columbia, Canada is another example of an organization that uses simulation to train for emergencysituations. ESP uses simulation to train on the following situations: forest fire fighting, oil or chemical spill response,earthquake response, law enforcement, municipal fire fighting, hazardous material handling, military training, andresponse to terrorist attack [46] One feature of the simulation system is the implementation of “Dynamic Run-TimeClock,” which allows simulations to run a 'simulated' time frame, 'speeding up' or 'slowing down' time as desired”[46]

Additionally, the system allows session recordings, picture-icon based navigation, file storage of individualsimulations, multimedia components, and launch external applications.At the University of Québec in Chicoutimi, a research team at the outdoor research and expertise laboratory(Laboratoire d'Expertise et de Recherche en Plein Air - LERPA) specializes in using wilderness backcountryaccident simulations to verify emergency response coordination.Instructionally, the benefits of emergency training through simulations are that learner performance can be tracked through the system. This allows the developer to make adjustments as necessary or alert the educator on topics that may require additional attention. Other advantages are that the learner can be guided or trained on how to respond appropriately before continuing to the next emergency segment—this is an aspect that may not be available in the

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live-environment. Some emergency training simulators also allows for immediate feedback, while other simulationsmay provide a summary and instruct the learner to engage in the learning topic again.In a live-emergency situation, emergency responders do not have time to waste. Simulation-training in thisenvironment provides an opportunity for learners to gather as much information as they can and practice theirknowledge in a safe environment. They can make mistakes without risk of endangering lives and be given theopportunity to correct their errors to prepare for the real-life emergency.

Engineering, technology or process simulationSimulation is an important feature in engineering systems or any system that involves many processes. For examplein electrical engineering, delay lines may be used to simulate propagation delay and phase shift caused by an actualtransmission line. Similarly, dummy loads may be used to simulate impedance without simulating propagation, andis used in situations where propagation is unwanted. A simulator may imitate only a few of the operations andfunctions of the unit it simulates. Contrast with: emulate.[47]

Most engineering simulations entail mathematical modeling and computer assisted investigation. There are manycases, however, where mathematical modeling is not reliable. Simulation of fluid dynamics problems often requireboth mathematical and physical simulations. In these cases the physical models require dynamic similitude. Physicaland chemical simulations have also direct realistic uses,[48] rather than research uses; in chemical engineering, forexample, process simulations are used to give the process parameters immediately used for operating chemicalplants, such as oil refineries.

Economics simulationIn economics and especially macroeconomics, the effects of proposed policy actions, such as fiscal policy changes ormonetary policy changes, are simulated in order to judge their desirability. A mathematical model of the economy,having been fitted to historical economic data, is used as a proxy for the actual economy; proposed values ofgovernment spending, taxation, open market operations, etc. are used as inputs to the simulation of the model, andvarious variables of interest such as the inflation rate, the unemployment rate, the balance of trade deficit, thegovernment budget deficit, etc. are the outputs of the simulation. The simulated values of these variables of interestare compared for different proposed policy inputs to determine which set of outcomes is most desirable.

Finance simulationIn finance, computer simulations are often used for scenario planning. Risk-adjusted net present value, for example,is computed from well-defined but not always known (or fixed) inputs. By imitating the performance of the projectunder evaluation, simulation can provide a distribution of NPV over a range of discount rates and other variables.Simulations are frequently used in financial training to engage participates in experiencing various historical as wellas fictional situations. There are stock market simulations, portfolio simulations, risk management simulations ormodels and forex simulations. Using these simulations in a training program allows for the application of theory intoa something akin to real life. As with other industries, the use of simulations can be technology or case-study driven.

Flight simulationFlight Simulation Training Devices (FSTD) are used to train pilots on the ground. In comparison to training in anactual aircraft, simulation based training allows for the training of maneuvers or situations that may be impractical(or even dangerous) to perform in the aircraft, while keeping the pilot and instructor in a relatively low-riskenvironment on the ground. For example, electrical system failures, instrument failures, hydraulic system failures,and even flight control failures can be simulated without risk to the pilots or an aircraft.

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Instructors can also provide students with a higher concentration of training tasks in a given period of time than isusually possible in the aircraft. For example, conducting multiple instrument approaches in the actual aircraft mayrequire significant time spent repositioning the aircraft, while in a simulation, as soon as one approach has beencompleted, the instructor can immediately preposition the simulated aircraft to an ideal (or less than ideal) locationfrom which to begin the next approach.Flight simulation also provides an economic advantage over training in an actual aircraft. Once fuel, maintenance,and insurance costs are taken into account, the operating costs of an FSTD are usually substantially lower than theoperating costs of the simulated aircraft. For some large transport category airplanes, the operating costs may beseveral times lower for the FSTD than the actual aircraft.Some people who use simulator software, especially flight simulator software, build their own simulator at home.Some people — in order to further the realism of their homemade simulator — buy used cards and racks that run thesame software used by the original machine. While this involves solving the problem of matching hardware andsoftware — and the problem that hundreds of cards plug into many different racks — many still find that solvingthese problems is well worthwhile. Some are so serious about realistic simulation that they will buy real aircraftparts, like complete nose sections of written-off aircraft, at aircraft boneyards. This permits people to simulate ahobby that they are unable to pursue in real life.

Marine simulatorsBearing resemblance to flight simulators, marine simulators train ships' personnel. The most common marinesimulators include:• Ship's bridge simulators• Engine room simulators• Cargo handling simulators• Communication / GMDSS simulators• ROV simulatorsSimulators like these are mostly used within maritime colleges, training institutions and navies. They often consist ofa replication of a ships' bridge, with operating desk(s), and a number of screens on which the virtual surroundings areprojected.

Military simulationsMilitary simulations, also known informally as war games, are models in which theories of warfare can be tested andrefined without the need for actual hostilities. They exist in many different forms, with varying degrees of realism. Inrecent times, their scope has widened to include not only military but also political and social factors (for example,the NationLab series of strategic exercises in Latin America.[49] Whilst many governments make use of simulation,both individually and collaboratively, little is known about the model's specifics outside professional circles.

Robotics simulatorsA robotics simulator is used to create embedded applications for a specific (or not) robot without being dependent onthe 'real' robot. In some cases, these applications can be transferred to the real robot (or rebuilt) withoutmodifications. Robotics simulators allow reproducing situations that cannot be 'created' in the real world because ofcost, time, or the 'uniqueness' of a resource. A simulator also allows fast robot prototyping. Many robot simulatorsfeature physics engines to simulate a robot's dynamics.

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Production simulationSimulations of production systems is mainly a used to examine improvements or investments in a production system.Most often is this done using a static spreadsheet with cycleimes and transportation times. For more sophisticatedsimulations Discrete Event Simulation (DES) is used with the advantages to simulate dynamics in the productionsystem, orgin from variation in production cycle times, machine set-ups, breaks, breakdowns and smallstoppages.[50] There are lots of programs commersely used for discrete event simulation. There is an academicproject investigating the possibilities to use production simulation software for ecology labeling, named EcoProIT.

Sales process simulatorsSimulations are useful in modeling the flow of transactions through business processes, such as in the field of salesprocess engineering, to study and improve the flow of customer orders through various stages of completion (say,from an initial proposal for providing goods/services through order acceptance and installation). Such simulationscan help predict the impact of how improvements in methods might impact variability, cost, labor time, and thequantity of transactions at various stages in the process. A full-featured computerized process simulator can be usedto depict such models, as can simpler educational demonstrations using spreadsheet software, pennies beingtransferred between cups based on the roll of a die, or dipping into a tub of colored beads with a scoop.[51]

Payment and Securities Settlement System SimulationsSimulation techniques have also been applied to payment and securities settlement systems. Among the main usersare central banks who are generally responsible for the oversight of market infrastructure and entitled to contribute tothe smooth functioning of the payment systems.Central Banks have been using payment system simulations to evaluate things such as the adequacy or sufficiency ofliquidity available ( in the form of account balances and intraday credit limits) to participants (mainly banks) toallow efficient settlement of payments.[52] [53] The need for liquidity is also dependent on the availability and thetype of netting procedures in the systems, thus some of the studies have a focus on system comparisons.[54]

Another application is to evaluate risks related to events such as communication network breakdowns or the inabilityof participants to send payments (e.g. in case of possible bank failure).[55] This kind of analysis fall under theconcepts of Stress testing or scenario analysis.A common way to conduct these simulations is to replicate the settlement logics of the real payment or securitiessettlement systems under analysis and then use real observed payment data. In case of system comparison or systemdevelopment, naturally also the other settlement logics need to be implemented.To perform stress testing and scenario analysis, the observed data needs to be altered, e.g. some payments delayed orremoved. To analyze the levels of liquidity, initial liquidity levels are varried. System comparisons(benchmarking)or evaluations of new netting algorithms or rules are performed by running simulations with a fixedset of data and wariating only the system setups.Inference is usually done by comparing the benchmark simulation results to the results of altered simulation setupsby comparing indicators such as unsettled transactions or settlement delays.

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Space Shuttle Countdown Simulation

Firing Room 1 configured for space shuttle launches

Simulation is used at Kennedy Space Center(KSC) to train and certify Space Shuttleengineers during simulated launchcountdown operations. The Space Shuttleengineering community participates in alaunch countdown integrated simulationbefore each shuttle flight. This simulation isa virtual simulation where real peopleinteract with simulated Space Shuttlevehicle and Ground Support Equipment(GSE) hardware. The Shuttle FinalCountdown Phase Simulation, also knownas S0044, involves countdown processesthat integrate many of the Space Shuttlevehicle and GSE systems. Some of the Shuttle systems integrated in the simulation are the Main Propulsion System,Main Engines, Solid Rocket Boosters, ground Liquid Hydrogen and Liquid Oxygen, External Tank, Flight Controls,Navigation, and Avionics.[56] The high-level objectives of the Shuttle Final Countdown Phase Simulation are:

• To demonstrate Firing Room final countdown phase operations.• To provide training for system engineers in recognizing, reporting and evaluating system problems in a time

critical environment.• To exercise the launch teams ability to evaluate, prioritize and respond to problems in an integrated manner

within a time critical environment.• To provide procedures to be used in performing failure/recovery testing of the operations performed in the final

countdown phase.[57]

The Shuttle Final Countdown Phase Simulation takes place at the Kennedy Space Center Launch Control CenterFiring Rooms. The firing room used during the simulation is the same control room where real launch countdownoperations are executed. As a result, equipment used for real launch countdown operations is engaged. Commandand control computers, application software, engineering plotting and trending tools, launch countdown proceduredocuments, launch commit criteria documents, hardware requirement documents, and any other items used by theengineering launch countdown teams during real launch countdown operations are used during the simulation. TheSpace Shuttle vehicle hardware and related GSE hardware is simulated by mathematical models (written in ShuttleGround Operations Simulator (SGOS) modeling language [58] ) that behave and react like real hardware. During theShuttle Final Countdown Phase Simulation, engineers command and control hardware via real application softwareexecuting in the control consoles – just as if they were commanding real vehicle hardware. However, these realsoftware applications do not interface with real Shuttle hardware during simulations. Instead, the applicationsinterface with mathematical model representations of the vehicle and GSE hardware. Consequently, the simulationsbypass sensitive and even dangerous mechanisms while providing engineering measurements detailing how thehardware would have reacted. Since these math models interact with the command and control application software,models and simulations are also used to debug and verify the functionality of application software.[59]

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Satellite Navigation SimulatorsThe only true way to test GNSS receivers (commonly known as Sat-Nav's in the commercial world)is by using anRF Constellation Simulator. A receiver that may for example be used on an aircraft, can be tested under dynamicconditions without the need to take it on a real flight. The test conditions can be repeated exactly, and there is fullcontrol over all the test parameters. this is not possible in the 'real-world' using the actual signals. For testingreceivers that will use the new Galileo (satellite navigation) there is no alternative, as the real signals do not yet exist.

Simulation and gamesStrategy games — both traditional and modern — may be viewed as simulations of abstracted decision-making forthe purpose of training military and political leaders (see History of Go for an example of such a tradition, orKriegsspiel for a more recent example).Many other video games are simulators of some kind. Such games can simulate various aspects of reality, frombusiness, to government, to construction, to piloting vehicles (see above).

Historical usageHistorically, the word had negative connotations:

…for Distinction Sake, a Deceiving by Words, is commonly called a Lye, and a Deceiving by Actions,Gestures, or Behavior, is called Simulation…—Robert South, South, 1697, p.525

However, the connection between simulation and dissembling later faded out and is now only of linguisticinterest.[60]

References[1] In the words of the Simulation article (http:/ / www. modelbenders. com/ encyclopedia/ encyclopedia. html) in Encyclopedia of Computer

Science, "designing a model of a real or imagined system and conducting experiments with that model".[2] Sokolowski, J.A., Banks, C.M. (2009). Principles of Modeling and Simulation. Hoboken, NJ: Wiley. p. 6. ISBN 978-0-470-28943-3.[3] For example in computer graphics SIGGRAPH 2007 | For Attendees | Papers (http:/ / www. siggraph. org/ s2007/ attendees/ papers/ 12. html)

Doc:Tutorials/Physics/BSoD - BlenderWiki (http:/ / wiki. blender. org/ index. php/ BSoD/ Physical_Simulation).[4] Thales defines synthetic environment as "the counterpart to simulated models of sensors, platforms and other active objects" for "the

simulation of the external factors that affect them" (http:/ / www. thalesresearch. com/ Default. aspx?tabid=181) while other vendors use theterm for more visual, virtual reality-style simulators (http:/ / www. cae. com/ www2004/ Products_and_Services/Civil_Simulation_and_Training/ Simulation_Equipment/ Visual_Solutions/ Synthetic_Environments/ index. shtml).

[5] For a popular research project in the field of biochemistry where "computer simulation is particularly well suited to address these questions"Folding@home - Main (http:/ / folding. stanford. edu/ Pande/ Main), see Folding@Home.

[6] For an academic take on a training simulator, see e.g. Towards Building an Interactive, Scenario-based Training Simulator (http:/ / gel. msu.edu/ magerko/ papers/ 11TH-CGF-058. pdf), for medical application Medical Simulation Training Benefits (http:/ / www. immersion. com/medical/ benefits1. php) as presented by a simulator vendor and for military practice A civilian's guide to US defense and security assistanceto Latin America and the Caribbean (http:/ / ciponline. org/ facts/ exe. htm) published by Center for International Policy.

[7] http:/ / www. simschool. org[8] Classification used by the Defense Modeling and Simulation Office.[9] "High Versus Low Fidelity Simulations: Does the Type of Format Affect Candidates' Performance or Perceptions?" (http:/ / www. ipmaac.

org/ conf/ 03/ havighurst. pdf)[10] For example All India management association (http:/ / www. aima-ind. org/ ) maintains that playing to win, participants "imbibe new forms

of competitive behavior that are ideal for today's highly chaotic business conditions" (http:/ / www. aima-ind. org/ management_games. asp)and IBM claims that "the skills honed playing massive multiplayer dragon-slaying games like World of Warcraft can be useful whenmanaging modern multinationals".

[11] "Reacting to the Past Home Page" (http:/ / www. barnard. columbia. edu/ reacting/ )[12] "Carana," at 'PaxSims' blog, 27 January 2009 (http:/ / paxsims. wordpress. com/ 2009/ 01/ 27/ carana/ )[13] Sherman, W.R., Craig, A.B. (2003). Understanding Virtual Reality. San Francisco, CA: Morgan Kaufmann. ISBN 978-1-55860-353-0.

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[14] Leeb, R., Lee, F., Keinrath, C., Schere, R., Bischof, H., Pfurtscheller, G. (2007). "Brain-Computer Communication: Motivation, Aim, andImpact of Exploring a Virtual Apartment". IEEE Transactions on Neural Systems and Rehabilitation Engineering 15 (4): 473–481.doi:10.1109/TNSRE.2007.906956.

[15] Zahraee, A.H., Szewczyk, J., Paik, J.K., Guillaume, M. (2010). Robotic hand-held surgical device: evaluation of end-effector’s kinematicsand development of proof-of-concept prototypes. Proceedings of the 13th International Conference on Medical Image Computing andComputer Assisted Intervention, Beijing, China.

[16] Ahmed K, Keeling AN, Fakhry M, Ashrafian H, Aggarwal R, Naughton PA, Darzi A, Cheshire N, et al. (January 2010). "Role of VirtualReality Simulation in Teaching and Assessing Technical Skills in Endovascular Intervention". J Vasc Interv Radiol 21.

[17] Narayan, Roger; Kumta, Prashant; Sfeir, Charles; Lee, Dong-Hyun; Choi, Daiwon; Olton, Dana (October 2004). "Nanostructured ceramicsin medical devices: Applications and prospects" (http:/ / www. ingentaconnect. com/ content/ tms/ jom/ 2004/ 00000056/ 00000010/art00011). JOM 56 (10): 38–43. Bibcode 2004JOM....56j..38N. doi:10.1007/s11837-004-0289-x. PMID 11196953. .

[18] Couvreur P, Vauthier C (July 2006). "Nanotechnology: intelligent design to treat complex disease". Pharm. Res. 23 (7): 1417–50.doi:10.1007/s11095-006-0284-8. PMID 16779701.

[19] Hede S, Huilgol N (2006). ""Nano": the new nemesis of cancer" (http:/ / www. cancerjournal. net/ article.asp?issn=0973-1482;year=2006;volume=2;issue=4;spage=186;epage=195;aulast=Hede). J Cancer Res Ther 2 (4): 186–95.doi:10.4103/0973-1482.29829. PMID 17998702. .

[20] Leary SP, Liu CY, Apuzzo ML (June 2006). "Toward the emergence of nanoneurosurgery: part III—nanomedicine: targeted nanotherapy,nanosurgery, and progress toward the realization of nanoneurosurgery" (http:/ / meta. wkhealth. com/ pt/ pt-core/ template-journal/lwwgateway/ media/ landingpage. htm?issn=0148-396X& volume=58& issue=6& spage=1009). Neurosurgery 58 (6): 1009–26; discussion1009–26. doi:10.1227/01.NEU.0000217016.79256.16. PMID 16723880. .

[21] http:/ / vam. anest. ufl. edu/ wip. html[22] Nishisaki A, Keren R, Nadkarni V (June 2007). "Does simulation improve patient safety? Self-efficacy, competence, operational

performance, and patient safety" (http:/ / linkinghub. elsevier. com/ retrieve/ pii/ S1932-2275(07)00025-0). Anesthesiol Clin 25 (2): 225–36.doi:10.1016/j.anclin.2007.03.009. PMID 17574187. .

[23] Meller, G. (1997). "A Typology of Simulators for Medical Education" (http:/ / www. medsim. com/ profile/ article1. html). Journal ofDigital Imaging. .

[24] Cooper Jeffery B, Taqueti VR (2008-12). "A brief history of the development of mannequin simulators for clinical education and training"(http:/ / pmj. bmj. com/ content/ 84/ 997/ 563. long). Postgrad Med J. 84 (997): 563–570. doi:10.1136/qshc.2004.009886. PMID 19103813. .Retrieved 2011-05-24.

[25] Murphy D, Challacombe B, Nedas T, Elhage O, Althoefer K, Seneviratne L, Dasgupta P. (May 2007). "[Equipment and technology inrobotics]" (in Spanish; Castilian). Arch. Esp. Urol. 60 (4): 349–55. PMID 17626526.

[26] "In Cybertherapy, Avatars Assist With Healing" (http:/ / www. nytimes. com/ 2010/ 11/ 23/ science/ 23avatar. html?_r=1& ref=science).New York Times. 2010-11-22. . Retrieved 2010-11-23.

[27] Dagger, Jacob (May–June 2008). Update: "The New Game Theory" (http:/ / www. dukemagazine. duke. edu/ dukemag/ issues/ 050608/depupd. html). 94. Duke Magazine. . Retrieved 2011-02-08.

[28] Steinberg, Scott (2011-01-31). "How video games can make you smarter" (http:/ / articles. cnn. com/ 2011-01-31/ tech/ video. games.smarter. steinberg_1_video-games-interactive-simulations-digital-world?_s=PM:TECH). Cable News Network (CNN Tech). . Retrieved2011-02-08.

[29] Vlaovic PD, Sargent ER, Boker JR, et al. (2008). "Immediate impact of an intensive one-week laparoscopy training program onlaparoscopic skills among postgraduate urologists" (http:/ / openurl. ingenta. com/ content/ nlm?genre=article& issn=1086-8089&volume=12& issue=1& spage=1& aulast=Vlaovic). JSLS 12 (1): 1–8. PMC 3016039. PMID 18402731. .

[30] Leung J, Foster E (April 2008). "How do we ensure that trainees learn to perform biliary sphincterotomy safely, appropriately, andeffectively?" (http:/ / www. current-reports. com/ article_frame. cfm?PubID=GR10-2-2-03& Type=Abstract). Curr Gastroenterol Rep 10 (2):163–8. doi:10.1007/s11894-008-0038-3. PMID 18462603. .

[31] http:/ / www. pong-story. com/ intro. htm Archived (http:/ / www. webcitation. org/ 5seElSgBC) at WebCite[32] http:/ / homepages. vvm. com/ ~jhunt/ compupedia/ History%20of%20Computers/ history_of_computers_1980. htm[33] "Video Game Console Timeline - Video Game History - Xbox 360 - TIME Magazine" (http:/ / www. time. com/ time/ covers/ 1101050523/

console_timeline/ ). Time. 2005-05-23. . Retrieved 2010-05-23.[34] http:/ / design. osu. edu/ carlson/ history/ tron. html[35] http:/ / www. beanblossom. in. us/ larryy/ cgi. html[36] http:/ / www. starksravings. com/ linktrainer/ linktrainer. htm[37] http:/ / www. trudang. com/ simulatr/ simulatr. html[38] http:/ / open-site. org/ Games/ Video_Games/ Simulation[39] http:/ / www. ibisworld. com/ industry/ retail. aspx?indid=2003& chid=1[40] http:/ / www. sciencedaily. com/ articles/ c/ computer-generated_imagery. htm[41] http:/ / www. awn. com/ mag/ issue4. 02/ 4. 02pages/ kenyonspiderman. php3[42] Benedettini, O., Tjahjono, B. (2008). "Towards an improved tool to facilitate simulation modeling of complex manufacturing systems".

International Journal of Advanced Manufacturing Technology 43 (1/2): 191–9. doi:10.1007/s00170-008-1686-z.

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[43] Banks, J., Carson J., Nelson B.L., Nicol, D. (2005). Discrete-event system simulation (4th ed.). Upper Saddle River, NJ: Pearson PrenticeHall. ISBN 978-0-13-088702-3.

[44] CADE- http:/ / www. uic. edu/ sph/ cade/[45] News-Medical.Net article- http:/ / www. news-medical. net/ news/ 2005/ 10/ 27/ 14106. aspx[46] http:/ / www. straylightmm. com/[47] Federal Standard 1037C[48] D. Passeri et al. (May 2009). "Analysis of 3D stacked fully functional CMOS Active Pixel Sensor detectors" (http:/ / meroli. web. cern. ch/

meroli/ Analysisof3Dstacked. html). Journal of Instrumentation 4 (4): 4009. doi:10.1088/1748-0221/4/04/P04009. .[49] See, for example, United States Joint Forces Command "Multinational Experiment 4" (http:/ / www. jfcom. mil/ about/ experiments/ mne4.

htm)[50] Ulf, Eriksson (2005). Diffusion of Discrete Event Simulation in Swedish Industry. Gothenburg: Doktorsavhandlingar vid Chalmers tekniska

högskola. ISBN 91-7291-577-3.[51] Paul H. Selden (1997). Sales Process Engineering: A Personal Workshop. Milwaukee, WI: ASQ Quality Press. ISBN 978-0-87389-418-0.[52] Leinonen (ed.): Simulation studies of liquidity needs, risks and efficiency in payment networks (Bank of Finland Studies E:39/2007)

Simulation publications (http:/ / pss. bof. fi/ Pages/ Publications. aspx)[53] Neville Arjani: Examining the Trade-Off between Settlement Delay and Intraday Liquidity in Canada's LVTS: A Simulation Approach

(Working Paper 2006-20, Bank of Canada) Simulation publications (http:/ / pss. bof. fi/ Pages/ Publications. aspx)[54] Johnson, K. - McAndrews, J. - Soramäki, K. 'Economizing on Liquidity with Deferred Settlement Mechanisms' (Reserve Bank of New York

Economic Policy Review, December 2004)[55] H. Leinonen (ed.): Simulation analyses and stress testing of payment networks (Bank of Finland Studies E:42/2009) Simulation publications

(http:/ / pss. bof. fi/ Pages/ Publications. aspx)[56] Sikora, E.A. (2010, July 27). Space Shuttle Main Propulsion System expert, John F. Kennedy Space Center. Interview.[57] Shuttle Final Countdown Phase Simulation. National Aeronautics and Space Administration KSC Document # RTOMI S0044, Revision

AF05, 2009.[58] Shuttle Ground Operations Simulator (SGOS) Summary Description Manual. National Aeronautics and Space Administration KSC

Document # KSC-LPS-SGOS-1000, Revision 3 CHG-A, 1995.[59] Math Model Main Propulsion System (MPS) Requirements Document, National Aeronautics and Space Administration KSC Document #

KSCL-1100-0522, Revision 9, June 2009.[60] South, in the passage quoted, was speaking of the differences between a falsehood and an honestly mistaken statement; the difference being

that in order for the statement to be a lie the truth must be known, and the opposite of the truth must have been knowingly uttered. And, fromthis, to the extent to which a lie involves deceptive words, a simulation involves deceptive actions, deceptive gestures, or deceptive behavior.Thus, it would seem, if a simulation is false, then the truth must be known (in order for something other than the truth to be presented in itsstead); and, for the simulation to simulate. Because, otherwise, one would not know what to offer up in simulation. Bacon’s essay OfSimulation and Dissimulation (http:/ / www. authorama. com/ essays-of-francis-bacon-7. html) expresses somewhat similar views; it is alsosignificant that Samuel Johnson thought so highly of South's definition, that he used it in the entry for simulation in his Dictionary of theEnglish Language.

Further reading• C. Aldrich (2003). Learning by Doing : A Comprehensive Guide to Simulations, Computer Games, and Pedagogy

in e-Learning and Other Educational Experiences. San Francisco: Pfeifer — John Wiley & Sons.ISBN 978-0-7879-7735-1.

• C. Aldrich (2004). Simulations and the future of learning: an innovative (and perhaps revolutionary) approach toe-learning. San Francisco: Pfeifer — John Wiley & Sons. ISBN 978-0-7879-6962-2.

• Steve Cohen (2006). Virtual Decisions. Mahwah, NJ: Lawrence Erlbaum Associates. ISBN 978-0-8058-4994-3.• R. Frigg, S. Hartmann (2007). "Models in Science" (http:/ / plato. stanford. edu/ entries/ models-science/ ).

Stanford Encyclopedia of Philosophy.• S. Hartmann (1996). "The World as a Process: Simulations in the Natural and Social Sciences" (http:/ /

philsci-archive. pitt. edu/ archive/ 00002412/ ). In R. Hegselmann, et al.. Modelling and Simulation in the SocialSciences from the Philosophy of Science Point of View. Theory and Decision Library. Dordrecht: Kluwer.pp. 77–100.

• J.P. Hertel (2002). Using Simulations to Promote Learning in Higher Education. Sterling, Virginia: Stylus.ISBN 978-1-57922-052-5.

• P. Humphreys (2004). Extending Ourselves: Computational Science, Empiricism, and Scientific Method. Oxford:Oxford University Press. ISBN 978-0-19-515870-0.

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• F. Percival, S. Lodge, D. Saunders (1993). The Simulation and Gaming Yearbook: Developing Transferable Skillsin Education and Training. London: Kogan Page.

• D. Saunders, ed (2000). The International Simulation and Gaming Research Yearbook. London: Kogan Page.• Roger D. Smith: Simulation Article (http:/ / www. modelbenders. com/ encyclopedia/ encyclopedia. html),

Encyclopedia of Computer Science, Nature Publishing Group, ISBN 978-0-333-77879-1.• Roger D. Smith: "Simulation: The Engine Behind the Virtual World" (http:/ / www. modelbenders. com/

Bookshop/ techpapers. html), eMatter, December, 1999.• R. South (1688). "A Sermon Delivered at Christ-Church, Oxon., Before the University, Octob. 14. 1688: Prov.

XII.22 Lying Lips are abomination to the Lord", pp. 519–657 in South, R., Twelve Sermons Preached UponSeveral Occasions (Second Edition), Volume I, Printed by S.D. for Thomas Bennet, (London), 1697.

• Eric Winsberg (1999) Sanctioning Models: The epistemology of simulation (http:/ / www. cas. usf. edu/ ~ewinsb/SiC_Eric_Winsberg. pdf), in Sismondo, Sergio and Snait Gissis (eds.) (1999), Modeling and Simulation. SpecialIssue of Science in Context 12.

• Eric Winsberg (2001). "Simulations, Models and Theories: Complex Physical Systems and theirRepresentations". Philosophy of Science 68: 442–454.

• Eric Winsberg (2003). "Simulated Experiments: Methodology for a Virtual World" (http:/ / www. cas. usf. edu/~ewinsb/ methodology. pdf) (PDF). Philosophy of Science 70: 105–125. doi:10.1086/367872.

• Joseph Wolfe, David Crookall (1998). "Developing a scientific knowledge of simulation/gaming" (http:/ / sag.sagepub. com/ cgi/ reprint/ 29/ 1/ 7). Simulation & Gaming: an International Journal of Theory, Design andResearch 29 (1): 7–19.

• Ellen K. Levy (2004). "Synthetic Lighting: Complex Simulations of Nature". Photography Quarterly (88): 5–9.• James J. Nutaro, Building Software for Simulation: Theory and Algorithms, with Applications in C++. Wiley,

2010.

External links• Bibliographies containing more references (http:/ / www. unice. fr/ sg/ resources/ bibliographies. htm) to be

found on the website of the journal Simulation & Gaming (http:/ / www. unice. fr/ sg/ ).• What are simulations good for? (http:/ / www. learningsutras. com/ why-are-simulations-good) A Focalworks

Q&A.

Artificial intelligence 67

Artificial intelligence

TOPIO, a humanoid robot, played table tennis at Tokyo International Robot Exhibition(IREX) 2009.[1]

Artificial intelligence (AI) is theintelligence of machines and thebranch of computer science that aimsto create it. AI textbooks define thefield as "the study and design ofintelligent agents"[2] where anintelligent agent is a system thatperceives its environment and takesactions that maximize its chances ofsuccess.[3] John McCarthy, who coinedthe term in 1956,[4] defines it as "thescience and engineering of makingintelligent machines."[5]

The field was founded on the claimthat a central property of humans,intelligence—the sapience of Homosapiens—can be so precisely described that it can be simulated by a machine.[6] This raises philosophical issuesabout the nature of the mind and the ethics of creating artificial beings, issues which have been addressed by myth,fiction and philosophy since antiquity.[7] Artificial intelligence has been the subject of optimism,[8] but has alsosuffered setbacks[9] and, today, has become an essential part of the technology industry, providing the heavy liftingfor many of the most difficult problems in computer science.[10]

AI research is highly technical and specialized, and deeply divided into subfields that often fail to communicate witheach other.[11] Subfields have grown up around particular institutions, the work of individual researchers, thesolution of specific problems, longstanding differences of opinion about how AI should be done and the applicationof widely differing tools. The central problems of AI include such traits as reasoning, knowledge, planning, learning,communication, perception and the ability to move and manipulate objects.[12] General intelligence (or "strong AI")is still among the field's long term goals.[13]

HistoryThinking machines and artificial beings appear in Greek myths, such as Talos of Crete, the bronze robot ofHephaestus, and Pygmalion's Galatea.[14] Human likenesses believed to have intelligence were built in every majorcivilization: animated cult images were worshipped in Egypt and Greece[15] and humanoid automatons were built byYan Shi, Hero of Alexandria and Al-Jazari.[16] It was also widely believed that artificial beings had been created byJābir ibn Hayyān, Judah Loew and Paracelsus.[17] By the 19th and 20th centuries, artificial beings had become acommon feature in fiction, as in Mary Shelley's Frankenstein or Karel Čapek's R.U.R. (Rossum's UniversalRobots).[18] Pamela McCorduck argues that all of these are examples of an ancient urge, as she describes it, "to forgethe gods".[7] Stories of these creatures and their fates discuss many of the same hopes, fears and ethical concerns thatare presented by artificial intelligence.Mechanical or "formal" reasoning has been developed by philosophers and mathematicians since antiquity. The study of logic led directly to the invention of the programmable digital electronic computer, based on the work of mathematician Alan Turing and others. Turing's theory of computation suggested that a machine, by shuffling symbols as simple as "0" and "1", could simulate any conceivable act of mathematical deduction.[19] [20] This, along with concurrent discoveries in neurology, information theory and cybernetics, inspired a small group of researchers

Artificial intelligence 68

to begin to seriously consider the possibility of building an electronic brain.[21]

The field of AI research was founded at a conference on the campus of Dartmouth College in the summer of1956.[22] The attendees, including John McCarthy, Marvin Minsky, Allen Newell and Herbert Simon, became theleaders of AI research for many decades.[23] They and their students wrote programs that were, to most people,simply astonishing:[24] computers were solving word problems in algebra, proving logical theorems and speakingEnglish.[25] By the middle of the 1960s, research in the U.S. was heavily funded by the Department of Defense[26]

and laboratories had been established around the world.[27] AI's founders were profoundly optimistic about the futureof the new field: Herbert Simon predicted that "machines will be capable, within twenty years, of doing any work aman can do" and Marvin Minsky agreed, writing that "within a generation ... the problem of creating 'artificialintelligence' will substantially be solved".[28]

They had failed to recognize the difficulty of some of the problems they faced.[29] In 1974, in response to thecriticism of Sir James Lighthill and ongoing pressure from the US Congress to fund more productive projects, boththe U.S. and British governments cut off all undirected exploratory research in AI. The next few years, when fundingfor projects was hard to find, would later be called the "AI winter".[30]

In the early 1980s, AI research was revived by the commercial success of expert systems,[31] a form of AI programthat simulated the knowledge and analytical skills of one or more human experts. By 1985 the market for AI hadreached over a billion dollars. At the same time, Japan's fifth generation computer project inspired the U.S andBritish governments to restore funding for academic research in the field.[32] However, beginning with the collapseof the Lisp Machine market in 1987, AI once again fell into disrepute, and a second, longer lasting AI winterbegan.[33]

In the 1990s and early 21st century, AI achieved its greatest successes, albeit somewhat behind the scenes. Artificialintelligence is used for logistics, data mining, medical diagnosis and many other areas throughout the technologyindustry.[10] The success was due to several factors: the increasing computational power of computers (see Moore'slaw), a greater emphasis on solving specific subproblems, the creation of new ties between AI and other fieldsworking on similar problems, and a new commitment by researchers to solid mathematical methods and rigorousscientific standards.[34]

On 11 May 1997, Deep Blue became the first computer chess-playing system to beat a reigning world chesschampion, Garry Kasparov.[35] In 2005, a Stanford robot won the DARPA Grand Challenge by drivingautonomously for 131 miles along an unrehearsed desert trail.[36] In February 2011, in a Jeopardy! quiz showexhibition match, IBM's question answering system, Watson, defeated the two greatest Jeopardy! champions, BradRutter and Ken Jennings, by a significant margin.[37]

The leading-edge definition of artificial intelligence research is changing over time. One pragmatic definition is: "AIresearch is that which computing scientists do not know how to do cost-effectively today." For example, in 1956optical character recognition (OCR) was considered AI, but today, sophisticated OCR software with acontext-sensitive spell checker and grammar checker software comes for free with most image scanners. No onewould any longer consider already-solved computing science problems like OCR "artificial intelligence" today.Low-cost entertaining chess-playing software is commonly available for tablet computers. DARPA no longerprovides significant funding for chess-playing computing system development. The Kinect which provides a3D-body-motion interface for the Xbox 360 uses algorithms that emerged from lengthy AI research,[38] but fewconsumers realize the technology source.AI applications are no longer the exclusive domain of Department of defense R&D, but are now common placeconsumer items and inexpensive intelligent toys.In common usage, the term "AI" no longer seems to apply to off-the-shelf solved computing-science problems,which may have originally emerged out of years of AI research.

Artificial intelligence 69

Problems" Can a machine act intelligently?" is still an open problem. Taking "A machine can act intelligently." as a workinghypothesis, many researchers have attempted to build such a machine.The general problem of simulating (or creating) intelligence has been broken down into a number of specificsub-problems. These consist of particular traits or capabilities that researchers would like an intelligent system todisplay. The traits described below have received the most attention.[12]

For some of these traits, the sub-problem of how to build a machine that exhibits that trait has been completelysolved. The rest remain open problems.

Deduction, reasoning, problem solvingEarly AI researchers developed algorithms that imitated the step-by-step reasoning that humans were often assumedto use when they solve puzzles, play board games or make logical deductions.[39] By the late 1980s and '90s, AIresearch had also developed highly successful methods for dealing with uncertain or incomplete information,employing concepts from probability and economics.[40]

For difficult problems, most of these algorithms can require enormous computational resources — most experience a"combinatorial explosion": the amount of memory or computer time required becomes astronomical when theproblem goes beyond a certain size. The search for more efficient problem solving algorithms is a high priority forAI research.[41]

Human beings solve most of their problems using fast, intuitive judgments rather than the conscious, step-by-stepdeduction that early AI research was able to model.[42] AI has made some progress at imitating this kind of"sub-symbolic" problem solving: embodied agent approaches emphasize the importance of sensorimotor skills tohigher reasoning; neural net research attempts to simulate the structures inside human and animal brains that giverise to this skill.

Knowledge representation

An ontology represents knowledge as a set ofconcepts within a domain and the relationships

between those concepts.

Knowledge representation[43] and knowledge engineering[44] arecentral to AI research. Many of the problems machines are expected tosolve will require extensive knowledge about the world. Among thethings that AI needs to represent are: objects, properties, categories andrelations between objects;[45] situations, events, states and time;[46]

causes and effects;[47] knowledge about knowledge (what we knowabout what other people know);[48] and many other, less wellresearched domains. A representation of "what exists" is an ontology(borrowing a word from traditional philosophy), of which the mostgeneral are called upper ontologies.[49]

Among the most difficult problems in knowledge representation are:Default reasoning and the qualification problem

Many of the things people know take the form of "workingassumptions." For example, if a bird comes up in conversation,people typically picture an animal that is fist sized, sings, and flies. None of these things are true about allbirds. John McCarthy identified this problem in 1969[50] as the qualification problem: for any commonsenserule that AI researchers care to represent, there tend to be a huge number of exceptions. Almost nothing issimply true or false in the way that abstract logic requires. AI research has explored a number of solutions tothis problem.[51]

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The breadth of commonsense knowledgeThe number of atomic facts that the average person knows is astronomical. Research projects that attempt tobuild a complete knowledge base of commonsense knowledge (e.g., Cyc) require enormous amounts oflaborious ontological engineering — they must be built, by hand, one complicated concept at a time.[52] Amajor goal is to have the computer understand enough concepts to be able to learn by reading from sourceslike the internet, and thus be able to add to its own ontology.

The subsymbolic form of some commonsense knowledgeMuch of what people know is not represented as "facts" or "statements" that they could express verbally. Forexample, a chess master will avoid a particular chess position because it "feels too exposed"[53] or an art criticcan take one look at a statue and instantly realize that it is a fake.[54] These are intuitions or tendencies that arerepresented in the brain non-consciously and sub-symbolically.[55] Knowledge like this informs, supports andprovides a context for symbolic, conscious knowledge. As with the related problem of sub-symbolicreasoning, it is hoped that situated AI or computational intelligence will provide ways to represent this kind ofknowledge.[55]

Planning

A hierarchical control system is a form of controlsystem in which a set of devices and governing

software is arranged in a hierarchy.

Intelligent agents must be able to set goals and achieve them.[56] Theyneed a way to visualize the future (they must have a representation ofthe state of the world and be able to make predictions about how theiractions will change it) and be able to make choices that maximize theutility (or "value") of the available choices.[57]

In classical planning problems, the agent can assume that it is the onlything acting on the world and it can be certain what the consequencesof its actions may be.[58] However, if this is not true, it mustperiodically check if the world matches its predictions and it mustchange its plan as this becomes necessary, requiring the agent to reasonunder uncertainty.[59]

Multi-agent planning uses the cooperation and competition of manyagents to achieve a given goal. Emergent behavior such as this is usedby evolutionary algorithms and swarm intelligence.[60]

Learning

Machine learning[61] has been central to AI research from the beginning.[62] In 1956, at the original Dartmouth AIsummer conference, Ray Solomonoff wrote a report on unsupervised probabilistic machine learning: "An InductiveInference Machine".[63] Unsupervised learning is the ability to find patterns in a stream of input. Supervised learningincludes both classification and numerical regression. Classification is used to determine what category somethingbelongs in, after seeing a number of examples of things from several categories. Regression takes a set of numericalinput/output examples and attempts to discover a continuous function that would generate the outputs from theinputs. In reinforcement learning[64] the agent is rewarded for good responses and punished for bad ones. These canbe analyzed in terms of decision theory, using concepts like utility. The mathematical analysis of machine learningalgorithms and their performance is a branch of theoretical computer science known as computational learningtheory.[65]

Artificial intelligence 71

Natural language processing

A parse tree represents the syntactic structure of asentence according to some formal grammar.

Natural language processing[66] gives machines the ability to read andunderstand the languages that humans speak. Many researchers hopethat a sufficiently powerful natural language processing system wouldbe able to acquire knowledge on its own, by reading the existing textavailable over the internet. Some straightforward applications ofnatural language processing include information retrieval (or textmining) and machine translation.[67]

Motion and manipulation

The field of robotics[68] is closely related to AI. Intelligence is requiredfor robots to be able to handle such tasks as object manipulation[69] andnavigation, with sub-problems of localization (knowing where youare), mapping (learning what is around you) and motion planning(figuring out how to get there).[70]

Perception

Machine perception[71] is the ability to use input from sensors (such as cameras, microphones, sonar and others moreexotic) to deduce aspects of the world. Computer vision[72] is the ability to analyze visual input. A few selectedsubproblems are speech recognition,[73] facial recognition and object recognition.[74]

Social intelligence

Kismet, a robot with rudimentary social skills

Emotion and social skills[75] play two roles for an intelligent agent.First, it must be able to predict the actions of others, by understandingtheir motives and emotional states. (This involves elements of gametheory, decision theory, as well as the ability to model human emotionsand the perceptual skills to detect emotions.) Also, for goodhuman-computer interaction, an intelligent machine also needs todisplay emotions. At the very least it must appear polite and sensitiveto the humans it interacts with. At best, it should have normal emotionsitself.

Creativity

A sub-field of AI addresses creativity both theoretically (from a philosophical and psychological perspective) andpractically (via specific implementations of systems that generate outputs that can be considered creative, or systemsthat identify and assess creativity). A related area of computational research is Artificial Intuition and ArtificialImagination.

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General intelligenceMost researchers hope that their work will eventually be incorporated into a machine with general intelligence(known as strong AI), combining all the skills above and exceeding human abilities at most or all of them.[13] A fewbelieve that anthropomorphic features like artificial consciousness or an artificial brain may be required for such aproject.[76] [77]

Many of the problems above are considered AI-complete: to solve one problem, you must solve them all. Forexample, even a straightforward, specific task like machine translation requires that the machine follow the author'sargument (reason), know what is being talked about (knowledge), and faithfully reproduce the author's intention(social intelligence). Machine translation, therefore, is believed to be AI-complete: it may require strong AI to bedone as well as humans can do it.[78]

ApproachesThere is no established unifying theory or paradigm that guides AI research. Researchers disagree about manyissues.[79] A few of the most long standing questions that have remained unanswered are these: should artificialintelligence simulate natural intelligence by studying psychology or neurology? Or is human biology as irrelevant toAI research as bird biology is to aeronautical engineering?[80] Can intelligent behavior be described using simple,elegant principles (such as logic or optimization)? Or does it necessarily require solving a large number ofcompletely unrelated problems?[81] Can intelligence be reproduced using high-level symbols, similar to words andideas? Or does it require "sub-symbolic" processing?[82] John Haugeland, who coined the term GOFAI (GoodOld-Fashioned Artificial Intelligence), also proposed that AI should more properly be referred to as syntheticintelligence,[83] a term which has since been adopted by some non-GOFAI researchers.[84][85][86]

Cybernetics and brain simulation

There is currently no consensus on how closelythe brain should be simulated.

In the 1940s and 1950s, a number of researchers explored theconnection between neurology, information theory, and cybernetics.Some of them built machines that used electronic networks to exhibitrudimentary intelligence, such as W. Grey Walter's turtles and theJohns Hopkins Beast. Many of these researchers gathered for meetingsof the Teleological Society at Princeton University and the Ratio Clubin England.[21] By 1960, this approach was largely abandoned,although elements of it would be revived in the 1980s.

Symbolic

When access to digital computers became possible in the middle1950s, AI research began to explore the possibility that humanintelligence could be reduced to symbol manipulation. The researchwas centered in three institutions: CMU, Stanford and MIT, and each one developed its own style of research. JohnHaugeland named these approaches to AI "good old fashioned AI" or "GOFAI".[87]

Cognitive simulationEconomist Herbert Simon and Allen Newell studied human problem solving skills and attempted to formalizethem, and their work laid the foundations of the field of artificial intelligence, as well as cognitive science,operations research and management science. Their research team used the results of psychologicalexperiments to develop programs that simulated the techniques that people used to solve problems. Thistradition, centered at Carnegie Mellon University would eventually culminate in the development of the Soararchitecture in the middle 80s.[88] [89]

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Logic-basedUnlike Newell and Simon, John McCarthy felt that machines did not need to simulate human thought, butshould instead try to find the essence of abstract reasoning and problem solving, regardless of whether peopleused the same algorithms.[80] His laboratory at Stanford (SAIL) focused on using formal logic to solve a widevariety of problems, including knowledge representation, planning and learning.[90] Logic was also focus ofthe work at the University of Edinburgh and elsewhere in Europe which led to the development of theprogramming language Prolog and the science of logic programming.[91]

"Anti-logic" or "scruffy"Researchers at MIT (such as Marvin Minsky and Seymour Papert)[92] found that solving difficult problems invision and natural language processing required ad-hoc solutions – they argued that there was no simple andgeneral principle (like logic) that would capture all the aspects of intelligent behavior. Roger Schank describedtheir "anti-logic" approaches as "scruffy" (as opposed to the "neat" paradigms at CMU and Stanford).[81]

Commonsense knowledge bases (such as Doug Lenat's Cyc) are an example of "scruffy" AI, since they mustbe built by hand, one complicated concept at a time.[93]

Knowledge-basedWhen computers with large memories became available around 1970, researchers from all three traditionsbegan to build knowledge into AI applications.[94] This "knowledge revolution" led to the development anddeployment of expert systems (introduced by Edward Feigenbaum), the first truly successful form of AIsoftware.[31] The knowledge revolution was also driven by the realization that enormous amounts ofknowledge would be required by many simple AI applications.

Sub-symbolicDuring the 1960s, symbolic approaches had achieved great success at simulating high-level thinking in smalldemonstration programs. Approaches based on cybernetics or neural networks were abandoned or pushed into thebackground.[95] By the 1980s, however, progress in symbolic AI seemed to stall and many believed that symbolicsystems would never be able to imitate all the processes of human cognition, especially perception, robotics, learningand pattern recognition. A number of researchers began to look into "sub-symbolic" approaches to specific AIproblems.[82]

Bottom-up, embodied, situated, behavior-based or nouvelle AIResearchers from the related field of robotics, such as Rodney Brooks, rejected symbolic AI and focused onthe basic engineering problems that would allow robots to move and survive.[96] Their work revived thenon-symbolic viewpoint of the early cybernetics researchers of the 50s and reintroduced the use of controltheory in AI. This coincided with the development of the embodied mind thesis in the related field of cognitivescience: the idea that aspects of the body (such as movement, perception and visualization) are required forhigher intelligence.

Computational IntelligenceInterest in neural networks and "connectionism" was revived by David Rumelhart and others in the middle1980s.[97] These and other sub-symbolic approaches, such as fuzzy systems and evolutionary computation, arenow studied collectively by the emerging discipline of computational intelligence.[98]

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StatisticalIn the 1990s, AI researchers developed sophisticated mathematical tools to solve specific subproblems. These toolsare truly scientific, in the sense that their results are both measurable and verifiable, and they have been responsiblefor many of AI's recent successes. The shared mathematical language has also permitted a high level of collaborationwith more established fields (like mathematics, economics or operations research). Stuart Russell and Peter Norvigdescribe this movement as nothing less than a "revolution" and "the victory of the neats."[34]

Integrating the approachesIntelligent agent paradigm

An intelligent agent is a system that perceives its environment and takes actions which maximizes its chancesof success. The simplest intelligent agents are programs that solve specific problems. More complicated agentsinclude human beings and organizations of human beings (such as firms). The paradigm gives researcherslicense to study isolated problems and find solutions that are both verifiable and useful, without agreeing onone single approach. An agent that solves a specific problem can use any approach that works — some agentsare symbolic and logical, some are sub-symbolic neural networks and others may use new approaches. Theparadigm also gives researchers a common language to communicate with other fields—such as decisiontheory and economics—that also use concepts of abstract agents. The intelligent agent paradigm becamewidely accepted during the 1990s.[3]

Agent architectures and cognitive architecturesResearchers have designed systems to build intelligent systems out of interacting intelligent agents in amulti-agent system.[99] A system with both symbolic and sub-symbolic components is a hybrid intelligentsystem, and the study of such systems is artificial intelligence systems integration. A hierarchical controlsystem provides a bridge between sub-symbolic AI at its lowest, reactive levels and traditional symbolic AI atits highest levels, where relaxed time constraints permit planning and world modelling.[100] Rodney Brooks'subsumption architecture was an early proposal for such a hierarchical system.[101]

ToolsIn the course of 50 years of research, AI has developed a large number of tools to solve the most difficult problemsin computer science. A few of the most general of these methods are discussed below.

Search and optimizationMany problems in AI can be solved in theory by intelligently searching through many possible solutions:[102]

Reasoning can be reduced to performing a search. For example, logical proof can be viewed as searching for a paththat leads from premises to conclusions, where each step is the application of an inference rule.[103] Planningalgorithms search through trees of goals and subgoals, attempting to find a path to a target goal, a process calledmeans-ends analysis.[104] Robotics algorithms for moving limbs and grasping objects use local searches inconfiguration space.[69] Many learning algorithms use search algorithms based on optimization.Simple exhaustive searches[105] are rarely sufficient for most real world problems: the search space (the number ofplaces to search) quickly grows to astronomical numbers. The result is a search that is too slow or never completes.The solution, for many problems, is to use "heuristics" or "rules of thumb" that eliminate choices that are unlikely tolead to the goal (called "pruning the search tree"). Heuristics supply the program with a "best guess" for what paththe solution lies on.[106]

A very different kind of search came to prominence in the 1990s, based on the mathematical theory of optimization. For many problems, it is possible to begin the search with some form of a guess and then refine the guess incrementally until no more refinements can be made. These algorithms can be visualized as blind hill climbing: we

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begin the search at a random point on the landscape, and then, by jumps or steps, we keep moving our guess uphill,until we reach the top. Other optimization algorithms are simulated annealing, beam search and randomoptimization.[107]

Evolutionary computation uses a form of optimization search. For example, they may begin with a population oforganisms (the guesses) and then allow them to mutate and recombine, selecting only the fittest to survive eachgeneration (refining the guesses). Forms of evolutionary computation include swarm intelligence algorithms (such asant colony or particle swarm optimization)[108] and evolutionary algorithms (such as genetic algorithms and geneticprogramming).[109]

LogicLogic[110] is used for knowledge representation and problem solving, but it can be applied to other problems as well.For example, the satplan algorithm uses logic for planning[111] and inductive logic programming is a method forlearning.[112]

Several different forms of logic are used in AI research. Propositional or sentential logic[113] is the logic ofstatements which can be true or false. First-order logic[114] also allows the use of quantifiers and predicates, and canexpress facts about objects, their properties, and their relations with each other. Fuzzy logic,[115] is a version offirst-order logic which allows the truth of a statement to be represented as a value between 0 and 1, rather thansimply True (1) or False (0). Fuzzy systems can be used for uncertain reasoning and have been widely used inmodern industrial and consumer product control systems. Subjective logic[116] models uncertainty in a different andmore explicit manner than fuzzy-logic: a given binomial opinion satisfies belief + disbelief + uncertainty = 1 withina Beta distribution. By this method, ignorance can be distinguished from probabilistic statements that an agent makeswith high confidence.Default logics, non-monotonic logics and circumscription[51] are forms of logic designed to help with defaultreasoning and the qualification problem. Several extensions of logic have been designed to handle specific domainsof knowledge, such as: description logics;[45] situation calculus, event calculus and fluent calculus (for representingevents and time);[46] causal calculus;[47] belief calculus; and modal logics.[48]

Probabilistic methods for uncertain reasoningMany problems in AI (in reasoning, planning, learning, perception and robotics) require the agent to operate withincomplete or uncertain information. AI researchers have devised a number of powerful tools to solve theseproblems using methods from probability theory and economics.[117]

Bayesian networks[118] are a very general tool that can be used for a large number of problems: reasoning (using theBayesian inference algorithm),[119] learning (using the expectation-maximization algorithm),[120] planning (usingdecision networks)[121] and perception (using dynamic Bayesian networks).[122] Probabilistic algorithms can also beused for filtering, prediction, smoothing and finding explanations for streams of data, helping perception systems toanalyze processes that occur over time (e.g., hidden Markov models or Kalman filters).[122]

A key concept from the science of economics is "utility": a measure of how valuable something is to an intelligentagent. Precise mathematical tools have been developed that analyze how an agent can make choices and plan, usingdecision theory, decision analysis,[123] information value theory.[57] These tools include models such as Markovdecision processes,[124] dynamic decision networks,[122] game theory and mechanism design.[125]

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Classifiers and statistical learning methodsThe simplest AI applications can be divided into two types: classifiers ("if shiny then diamond") and controllers ("ifshiny then pick up"). Controllers do however also classify conditions before inferring actions, and thereforeclassification forms a central part of many AI systems. Classifiers are functions that use pattern matching todetermine a closest match. They can be tuned according to examples, making them very attractive for use in AI.These examples are known as observations or patterns. In supervised learning, each pattern belongs to a certainpredefined class. A class can be seen as a decision that has to be made. All the observations combined with theirclass labels are known as a data set. When a new observation is received, that observation is classified based onprevious experience.[126]

A classifier can be trained in various ways; there are many statistical and machine learning approaches. The mostwidely used classifiers are the neural network,[127] kernel methods such as the support vector machine,[128] k-nearestneighbor algorithm,[129] Gaussian mixture model,[130] naive Bayes classifier,[131] and decision tree.[132] Theperformance of these classifiers have been compared over a wide range of tasks. Classifier performance dependsgreatly on the characteristics of the data to be classified. There is no single classifier that works best on all givenproblems; this is also referred to as the "no free lunch" theorem. Determining a suitable classifier for a given problemis still more an art than science.[133]

Neural networks

A neural network is an interconnected group ofnodes, akin to the vast network of neurons in the

human brain.

The study of artificial neural networks[127] began in the decade beforethe field AI research was founded, in the work of Walter Pitts andWarren McCullough. Other important early researchers were FrankRosenblatt, who invented the perceptron and Paul Werbos whodeveloped the backpropagation algorithm.[134]

The main categories of networks are acyclic or feedforward neuralnetworks (where the signal passes in only one direction) and recurrentneural networks (which allow feedback). Among the most popularfeedforward networks are perceptrons, multi-layer perceptrons andradial basis networks.[135] Among recurrent networks, the most famousis the Hopfield net, a form of attractor network, which was firstdescribed by John Hopfield in 1982.[136] Neural networks can beapplied to the problem of intelligent control (for robotics) or learning,using such techniques as Hebbian learning and competitivelearning.[137]

Hierarchical temporal memory is an approach that models some of the structural and algorithmic properties of theneocortex.[138]

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Control theoryControl theory, the grandchild of cybernetics, has many important applications, especially in robotics.[139]

LanguagesAI researchers have developed several specialized languages for AI research, including Lisp[140] and Prolog.[141]

Evaluating progressIn 1950, Alan Turing proposed a general procedure to test the intelligence of an agent now known as the Turing test.This procedure allows almost all the major problems of artificial intelligence to be tested. However, it is a verydifficult challenge and at present all agents fail.[142]

Artificial intelligence can also be evaluated on specific problems such as small problems in chemistry, hand-writingrecognition and game-playing. Such tests have been termed subject matter expert Turing tests. Smaller problemsprovide more achievable goals and there are an ever-increasing number of positive results.[143]

The broad classes of outcome for an AI test are: (1) Optimal: it is not possible to perform better. (2) Strongsuper-human: performs better than all humans. (3) Super-human: performs better than most humans. (4) Sub-human:performs worse than most humans. For example, performance at draughts is optimal,[144] performance at chess issuper-human and nearing strong super-human (see Computer chess#Computers versus humans) and performance atmany everyday tasks (such as recognizing a face or crossing a room without bumping into something) is sub-human.A quite different approach measures machine intelligence through tests which are developed from mathematicaldefinitions of intelligence. Examples of these kinds of tests start in the late nineties devising intelligence tests usingnotions from Kolmogorov complexity and data compression.[145] Two major advantages of mathematical definitionsare their applicability to nonhuman intelligences and their absence of a requirement for human testers.

Applications

An automated online assistant providingcustomer service on a web page - one of many

applications of artificial intelligence.

Artificial intelligence techniques are pervasive and are too numerous tolist. Frequently, when a technique reaches mainstream use, it is nolonger considered artificial intelligence; this phenomenon is describedas the AI effect.[146]

Competitions and prizes

There are a number of competitions and prizes to promote research inartificial intelligence. The main areas promoted are: general machineintelligence, conversational behavior, data-mining, driverless cars,robot soccer and games.

Platforms

A platform (or "computing platform") is defined as "some sort ofhardware architecture or software framework (including applicationframeworks), that allows software to run." As Rodney Brooks[147]

pointed out many years ago, it is not just the artificial intelligencesoftware that defines the AI features of the platform, but rather theactual platform itself that affects the AI that results, i.e., we need to be working out AI problems on real-worldplatforms rather than in isolation.

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A wide variety of platforms has allowed different aspects of AI to develop, ranging from expert systems, albeitPC-based but still an entire real-world system, to various robot platforms such as the widely available Roomba withopen interface.[148]

PhilosophyArtificial intelligence, by claiming to be able to recreate the capabilities of the human mind, is both a challenge andan inspiration for philosophy. Are there limits to how intelligent machines can be? Is there an essential differencebetween human intelligence and artificial intelligence? Can a machine have a mind and consciousness? A few of themost influential answers to these questions are given below.[149]

Turing's "polite convention"If a machine acts as intelligently as a human being, then it is as intelligent as a human being. Alan Turingtheorized that, ultimately, we can only judge the intelligence of a machine based on its behavior. This theoryforms the basis of the Turing test.[142]

The Dartmouth proposal"Every aspect of learning or any other feature of intelligence can be so precisely described that a machine canbe made to simulate it." This conjecture was printed in the proposal for the Dartmouth Conference of 1956,and represents the position of most working AI researchers.[150]

Newell and Simon's physical symbol system hypothesis"A physical symbol system has the necessary and sufficient means of general intelligent action." Newell andSimon argue that intelligences consists of formal operations on symbols.[151] Hubert Dreyfus argued that, onthe contrary, human expertise depends on unconscious instinct rather than conscious symbol manipulation andon having a "feel" for the situation rather than explicit symbolic knowledge. (See Dreyfus' critique of AI.)[152]

[153]

Gödel's incompleteness theoremA formal system (such as a computer program) can not prove all true statements.[154] Roger Penrose is amongthose who claim that Gödel's theorem limits what machines can do. (See The Emperor's New Mind.)[155]

Searle's strong AI hypothesis"The appropriately programmed computer with the right inputs and outputs would thereby have a mind inexactly the same sense human beings have minds."[156] John Searle counters this assertion with his Chineseroom argument, which asks us to look inside the computer and try to find where the "mind" might be.[157]

The artificial brain argumentThe brain can be simulated. Hans Moravec, Ray Kurzweil and others have argued that it is technologicallyfeasible to copy the brain directly into hardware and software, and that such a simulation will be essentiallyidentical to the original.[77]

PredictionsArtificial Intelligence is a common topic in both science fiction and projections about the future of technology andsociety. The existence of an artificial intelligence that rivals human intelligence raises difficult ethical issues, and thepotential power of the technology inspires both hopes and fears.In fiction, Artificial Intelligence has appeared fulfilling many roles, including a servant (R2D2 in Star Wars), a lawenforcer (K.I.T.T. "Knight Rider"), a comrade (Lt. Commander Data in Star Trek: The Next Generation), aconqueror/overlord (The Matrix), a dictator (With Folded Hands), a benevolent provider/de facto ruler (The Culture),an assassin (Terminator), a sentient race (Battlestar Galactica/Transformers), an extension to human abilities (Ghostin the Shell) and the savior of the human race (R. Daneel Olivaw in the Asimov's Robot Series).

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Mary Shelley's Frankenstein considers a key issue in the ethics of artificial intelligence: if a machine can be createdthat has intelligence, could it also feel? If it can feel, does it have the same rights as a human? The idea also appearsin modern science fiction, including the films I Robot, Blade Runner and A.I.: Artificial Intelligence, in whichhumanoid machines have the ability to feel human emotions. This issue, now known as "robot rights", is currentlybeing considered by, for example, California's Institute for the Future, although many critics believe that thediscussion is premature.[158] The subject is profoundly discussed in the 2010 documentary film Plug & Pray.[159]

Martin Ford, author of The Lights in the Tunnel: Automation, Accelerating Technology and the Economy of theFuture,[160] and others argue that specialized artificial intelligence applications, robotics and other forms ofautomation will ultimately result in significant unemployment as machines begin to match and exceed the capabilityof workers to perform most routine and repetitive jobs. Ford predicts that many knowledge-based occupations—andin particular entry level jobs—will be increasingly susceptible to automation via expert systems, machinelearning[161] and other AI-enhanced applications. AI-based applications may also be used to amplify the capabilitiesof low-wage offshore workers, making it more feasible to outsource knowledge work.[162]

Joseph Weizenbaum wrote that AI applications can not, by definition, successfully simulate genuine human empathyand that the use of AI technology in fields such as customer service or psychotherapy[163] was deeply misguided.Weizenbaum was also bothered that AI researchers (and some philosophers) were willing to view the human mind asnothing more than a computer program (a position now known as computationalism). To Weizenbaum these pointssuggest that AI research devalues human life.[164]

Many futurists believe that artificial intelligence will ultimately transcend the limits of progress. Ray Kurzweil hasused Moore's law (which describes the relentless exponential improvement in digital technology) to calculate thatdesktop computers will have the same processing power as human brains by the year 2029. He also predicts that by2045 artificial intelligence will reach a point where it is able to improve itself at a rate that far exceeds anythingconceivable in the past, a scenario that science fiction writer Vernor Vinge named the "singularity".[165]

Robot designer Hans Moravec, cyberneticist Kevin Warwick and inventor Ray Kurzweil have predicted that humansand machines will merge in the future into cyborgs that are more capable and powerful than either.[166] This idea,called transhumanism, which has roots in Aldous Huxley and Robert Ettinger, has been illustrated in fiction as well,for example in the manga Ghost in the Shell and the science-fiction series Dune.Edward Fredkin argues that "artificial intelligence is the next stage in evolution," an idea first proposed by SamuelButler's "Darwin among the Machines" (1863), and expanded upon by George Dyson in his book of the same namein 1998.[167]

Pamela McCorduck writes that all these scenarios are expressions of the ancient human desire to, as she calls it,"forge the gods".[7]

References

Notes[1] TOPIO:

• "A Ping-Pong-Playing Terminator" (http:/ / www. popsci. com/ technology/ article/ 2010-02/ ping-pong-playing-terminator). PopularScience. .

• "Best robot 2009" (http:/ / www. gadgetrivia. com/ 8164-best_robot_international_robot_exhibition). www.gadgetrivia.com. .[2] Definition of AI as the study of intelligent agents:

• Poole, Mackworth & Goebel 1998, p. 1 (http:/ / people. cs. ubc. ca/ ~poole/ ci/ ch1. pdf), which provides the version that is used in thisarticle. Note that they use the term "computational intelligence" as a synonym for artificial intelligence.

• Russell & Norvig (2003) (who prefer the term "rational agent") and write "The whole-agent view is now widely accepted in the field"(Russell & Norvig 2003, p. 55).

• Nilsson 1998[3] The intelligent agent paradigm:

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• Russell & Norvig 2003, pp. 27, 32–58, 968–972• Poole, Mackworth & Goebel 1998, pp. 7–21• Luger & Stubblefield 2004, pp. 235–240The definition used in this article, in terms of goals, actions, perception and environment, is due to Russell & Norvig (2003). Other definitionsalso include knowledge and learning as additional criteria.

[4] Although there is some controversy on this point (see Crevier (1993, p. 50)), McCarthy states unequivocally "I came up with the term" in ac|net interview. (Skillings 2006)

[5] McCarthy's definition of AI:

• McCarthy 2007[6] See the Dartmouth proposal, under Philosophy, below.[7] This is a central idea of Pamela McCorduck's Machines That Think. She writes: "I like to think of artificial intelligence as the scientific

apotheosis of a venerable cultural tradition." (McCorduck 2004, p. 34) "Artificial intelligence in one form or another is an idea that haspervaded Western intellectual history, a dream in urgent need of being realized." (McCorduck 2004, p. xviii) "Our history is full ofattempts—nutty, eerie, comical, earnest, legendary and real—to make artificial intelligences, to reproduce what is the essential us—bypassingthe ordinary means. Back and forth between myth and reality, our imaginations supplying what our workshops couldn't, we have engaged for along time in this odd form of self-reproduction." (McCorduck 2004, p. 3) She traces the desire back to its Hellenistic roots and calls it the urgeto "forge the Gods." (McCorduck 2004, pp. 340–400)

[8] The optimism referred to includes the predictions of early AI researchers (see optimism in the history of AI) as well as the ideas of moderntranshumanists such as Ray Kurzweil.

[9] The "setbacks" referred to include the ALPAC report of 1966, the abandonment of perceptrons in 1970, the Lighthill Report of 1973 and thecollapse of the lisp machine market in 1987.

[10] AI applications widely used behind the scenes:

• Russell & Norvig 2003, p. 28• Kurzweil 2005, p. 265• NRC 1999, pp. 216–222

[11] Pamela McCorduck (2004, pp. 424) writes of "the rough shattering of AI in subfields—vision, natural language, decision theory, geneticalgorithms, robotics ... and these with own sub-subfield—that would hardly have anything to say to each other."

[12] This list of intelligent traits is based on the topics covered by the major AI textbooks, including:

• Russell & Norvig 2003• Luger & Stubblefield 2004• Poole, Mackworth & Goebel 1998• Nilsson 1998

[13] General intelligence (strong AI) is discussed in popular introductions to AI:

• Kurzweil 1999 and Kurzweil 2005[14] AI in myth:

• McCorduck 2004, pp. 4–5• Russell & Norvig 2003, p. 939

[15] Cult images as artificial intelligence:

• Crevier (1993, p. 1) (statue of Amun)• McCorduck (2004, pp. 6–9)These were the first machines to be believed to have true intelligence and consciousness. Hermes Trismegistus expressed the common beliefthat with these statues, craftsman had reproduced "the true nature of the gods", their sensus and spiritus. McCorduck makes the connectionbetween sacred automatons and Mosaic law (developed around the same time), which expressly forbids the worship of robots (McCorduck2004, pp. 6–9)

[16] Humanoid automata:Yan Shi:

• Needham 1986, p. 53Hero of Alexandria:

• McCorduck 2004, p. 6Al-Jazari:

• "A Thirteenth Century Programmable Robot" (http:/ / www. shef. ac. uk/ marcoms/ eview/ articles58/ robot. html). Shef.ac.uk. . Retrieved2009-04-25.

Wolfgang von Kempelen:

• McCorduck 2004, p. 17[17] Artificial beings:

Jābir ibn Hayyān's Takwin:

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• O'Connor, Kathleen Malone (1994). The alchemical creation of life (takwin) and other concepts of Genesis in medieval Islam (http:/ /repository. upenn. edu/ dissertations/ AAI9503804). University of Pennsylvania. . Retrieved 2007-01-10.

Judah Loew's Golem:

• McCorduck 2004, pp. 15–16• Buchanan 2005, p. 50Paracelsus' Homunculus:

• McCorduck 2004, pp. 13–14[18] AI in early science fiction.

• McCorduck 2004, pp. 17–25[19] This insight, that digital computers can simulate any process of formal reasoning, is known as the Church-Turing thesis.[20] Formal reasoning:

• Berlinski, David (2000). The Advent of the Algorithm. Harcourt Books. ISBN 0-15-601391-6. OCLC 46890682.[21] AI's immediate precursors:

• McCorduck 2004, pp. 51–107• Crevier 1993, pp. 27–32• Russell & Norvig 2003, pp. 15, 940• Moravec 1988, p. 3See also Cybernetics and early neural networks (in History of artificial intelligence). Among the researchers who laid the foundations of AIwere Alan Turing, John Von Neumann, Norbert Wiener, Claude Shannon, Warren McCullough, Walter Pitts and Donald Hebb.

[22] Dartmouth conference:

• McCorduck 2004, pp. 111–136• Crevier 1993, pp. 47–49, who writes "the conference is generally recognized as the official birthdate of the new science."• Russell & Norvig 2003, p. 17, who call the conference "the birth of artificial intelligence."• NRC 1999, pp. 200–201

[23] Hegemony of the Dartmouth conference attendees:

• Russell & Norvig 2003, p. 17, who write "for the next 20 years the field would be dominated by these people and their students."• McCorduck 2004, pp. 129–130

[24] Russell and Norvig write "it was astonishing whenever a computer did anything kind of smartish." Russell & Norvig 2003, p. 18[25] "Golden years" of AI (successful symbolic reasoning programs 1956-1973):

• McCorduck 2004, pp. 243–252• Crevier 1993, pp. 52–107• Moravec 1988, p. 9• Russell & Norvig 2003, pp. 18–21The programs described are Daniel Bobrow's STUDENT, Newell and Simon's Logic Theorist and Terry Winograd's SHRDLU.

[26] DARPA pours money into undirected pure research into AI during the 1960s:

• McCorduck 2004, pp. 131• Crevier 1993, pp. 51, 64–65• NRC 1999, pp. 204–205

[27] AI in England:

• Howe 1994[28] Optimism of early AI:

• Herbert Simon quote: Simon 1965, p. 96 quoted in Crevier 1993, p. 109.• Marvin Minsky quote: Minsky 1967, p. 2 quoted in Crevier 1993, p. 109.

[29] See The problems (in History of artificial intelligence)[30] First AI Winter, Mansfield Amendment, Lighthill report

• Crevier 1993, pp. 115–117• Russell & Norvig 2003, p. 22• NRC 1999, pp. 212–213• Howe 1994

[31] Expert systems:

• ACM 1998, I.2.1,• Russell & Norvig 2003, pp. 22−24• Luger & Stubblefield 2004, pp. 227–331,• Nilsson 1998, chpt. 17.4• McCorduck 2004, pp. 327–335, 434–435• Crevier 1993, pp. 145–62, 197−203

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[32] Boom of the 1980s: rise of expert systems, Fifth Generation Project, Alvey, MCC, SCI:

• McCorduck 2004, pp. 426–441• Crevier 1993, pp. 161–162,197–203, 211, 240• Russell & Norvig 2003, p. 24• NRC 1999, pp. 210–211

[33] Second AI winter:

• McCorduck 2004, pp. 430–435• Crevier 1993, pp. 209–210• NRC 1999, pp. 214–216

[34] Formal methods are now preferred ("Victory of the neats"):

• Russell & Norvig 2003, pp. 25–26• McCorduck 2004, pp. 486–487

[35] McCorduck 2004, pp. 480–483[36] DARPA Grand Challenge -- home page (http:/ / www. darpa. mil/ grandchallenge/ )[37] Markoff, John (16 February 2011). "On 'Jeopardy!' Watson Win Is All but Trivial" (http:/ / www. nytimes. com/ 2011/ 02/ 17/ science/

17jeopardy-watson. html). The New York Times. .[38] Kinect's AI breakthrough explained (http:/ / www. i-programmer. info/ news/ 105-artificial-intelligence/

2176-kinects-ai-breakthrough-explained. html)[39] Problem solving, puzzle solving, game playing and deduction:

• Russell & Norvig 2003, chpt. 3-9,• Poole, Mackworth & Goebel 1998, chpt. 2,3,7,9,• Luger & Stubblefield 2004, chpt. 3,4,6,8,• Nilsson 1998, chpt. 7-12

[40] Uncertain reasoning:

• Russell & Norvig 2003, pp. 452–644,• Poole, Mackworth & Goebel 1998, pp. 345–395,• Luger & Stubblefield 2004, pp. 333–381,• Nilsson 1998, chpt. 19

[41] Intractability and efficiency and the combinatorial explosion:

• Russell & Norvig 2003, pp. 9, 21–22[42] Psychological evidence of sub-symbolic reasoning:

• Wason & Shapiro (1966) showed that people do poorly on completely abstract problems, but if the problem is restated to allow the use ofintuitive social intelligence, performance dramatically improves. (See Wason selection task)

• Kahneman, Slovic & Tversky (1982) have shown that people are terrible at elementary problems that involve uncertain reasoning. (See listof cognitive biases for several examples).

• Lakoff & Núñez (2000) have controversially argued that even our skills at mathematics depend on knowledge and skills that come from"the body", i.e. sensorimotor and perceptual skills. (See Where Mathematics Comes From)

[43] Knowledge representation:

• ACM 1998, I.2.4,• Russell & Norvig 2003, pp. 320–363,• Poole, Mackworth & Goebel 1998, pp. 23–46, 69–81, 169–196, 235–277, 281–298, 319–345,• Luger & Stubblefield 2004, pp. 227–243,• Nilsson 1998, chpt. 18

[44] Knowledge engineering:

• Russell & Norvig 2003, pp. 260–266,• Poole, Mackworth & Goebel 1998, pp. 199–233,• Nilsson 1998, chpt. ~17.1-17.4

[45] Representing categories and relations: Semantic networks, description logics, inheritance (including frames and scripts):

• Russell & Norvig 2003, pp. 349–354,• Poole, Mackworth & Goebel 1998, pp. 174–177,• Luger & Stubblefield 2004, pp. 248–258,• Nilsson 1998, chpt. 18.3

[46] Representing events and time:Situation calculus, event calculus, fluent calculus (including solving the frame problem):

• Russell & Norvig 2003, pp. 328–341,• Poole, Mackworth & Goebel 1998, pp. 281–298,• Nilsson 1998, chpt. 18.2

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[47] Causal calculus:

• Poole, Mackworth & Goebel 1998, pp. 335–337[48] Representing knowledge about knowledge: Belief calculus, modal logics:

• Russell & Norvig 2003, pp. 341–344,• Poole, Mackworth & Goebel 1998, pp. 275–277

[49] Ontology:

• Russell & Norvig 2003, pp. 320–328[50] Qualification problem:

• McCarthy & Hayes 1969• Russell & Norvig 2003While McCarthy was primarily concerned with issues in the logical representation of actions, Russell & Norvig 2003 apply the term to themore general issue of default reasoning in the vast network of assumptions underlying all our commonsense knowledge.

[51] Default reasoning and default logic, non-monotonic logics, circumscription, closed world assumption, abduction (Poole et al. placesabduction under "default reasoning". Luger et al. places this under "uncertain reasoning"):

• Russell & Norvig 2003, pp. 354–360,• Poole, Mackworth & Goebel 1998, pp. 248–256, 323–335,• Luger & Stubblefield 2004, pp. 335–363,• Nilsson 1998, ~18.3.3

[52] Breadth of commonsense knowledge:

• Russell & Norvig 2003, p. 21,• Crevier 1993, pp. 113–114,• Moravec 1988, p. 13,• Lenat & Guha 1989 (Introduction)

[53] Dreyfus & Dreyfus 1986[54] Gladwell 2005[55] Expert knowledge as embodied intuition:

• Dreyfus & Dreyfus 1986 (Hubert Dreyfus is a philosopher and critic of AI who was among the first to argue that most useful humanknowledge was encoded sub-symbolically. See Dreyfus' critique of AI)

• Gladwell 2005 (Gladwell's Blink is a popular introduction to sub-symbolic reasoning and knowledge.)• Hawkins & Blakeslee 2005 (Hawkins argues that sub-symbolic knowledge should be the primary focus of AI research.)

[56] Planning:

• ACM 1998, ~I.2.8,• Russell & Norvig 2003, pp. 375–459,• Poole, Mackworth & Goebel 1998, pp. 281–316,• Luger & Stubblefield 2004, pp. 314–329,• Nilsson 1998, chpt. 10.1-2, 22

[57] Information value theory:

• Russell & Norvig 2003, pp. 600–604[58] Classical planning:

• Russell & Norvig 2003, pp. 375–430,• Poole, Mackworth & Goebel 1998, pp. 281–315,• Luger & Stubblefield 2004, pp. 314–329,• Nilsson 1998, chpt. 10.1-2, 22

[59] Planning and acting in non-deterministic domains: conditional planning, execution monitoring, replanning and continuous planning:

• Russell & Norvig 2003, pp. 430–449[60] Multi-agent planning and emergent behavior:

• Russell & Norvig 2003, pp. 449–455[61] Learning:

• ACM 1998, I.2.6,• Russell & Norvig 2003, pp. 649–788,• Poole, Mackworth & Goebel 1998, pp. 397–438,• Luger & Stubblefield 2004, pp. 385–542,• Nilsson 1998, chpt. 3.3 , 10.3, 17.5, 20

[62] Alan Turing discussed the centrality of learning as early as 1950, in his classic paper Computing Machinery and Intelligence. (Turing 1950)[63] (pdf scanned copy of the original) (http:/ / world. std. com/ ~rjs/ indinf56. pdf) (version published in 1957, An Inductive Inference

Machine," IRE Convention Record, Section on Information Theory, Part 2, pp. 56-62)

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[64] Reinforcement learning:

• Russell & Norvig 2003, pp. 763–788• Luger & Stubblefield 2004, pp. 442–449

[65] Computational learning theory:

• CITATION IN PROGRESS.[66] Natural language processing:

• ACM 1998, I.2.7• Russell & Norvig 2003, pp. 790–831• Poole, Mackworth & Goebel 1998, pp. 91–104• Luger & Stubblefield 2004, pp. 591–632

[67] Applications of natural language processing, including information retrieval (i.e. text mining) and machine translation:

• Russell & Norvig 2003, pp. 840–857,• Luger & Stubblefield 2004, pp. 623–630

[68] Robotics:

• ACM 1998, I.2.9,• Russell & Norvig 2003, pp. 901–942,• Poole, Mackworth & Goebel 1998, pp. 443–460

[69] Moving and configuration space:

• Russell & Norvig 2003, pp. 916–932[70] Robotic mapping (localization, etc):

• Russell & Norvig 2003, pp. 908–915[71] Machine perception:

• Russell & Norvig 2003, pp. 537–581, 863–898• Nilsson 1998, ~chpt. 6

[72] Computer vision:

• ACM 1998, I.2.10• Russell & Norvig 2003, pp. 863–898• Nilsson 1998, chpt. 6

[73] Speech recognition:

• ACM 1998, ~I.2.7• Russell & Norvig 2003, pp. 568–578

[74] Object recognition:

• Russell & Norvig 2003, pp. 885–892[75] Emotion and affective computing:

• Minsky 2006• Picard 1997

[76] Gerald Edelman, Igor Aleksander and others have both argued that artificial consciousness is required for strong AI. (Aleksander 1995;Edelman 2007)

[77] Artificial brain arguments: AI requires a simulation of the operation of the human brain

• Russell & Norvig 2003, p. 957• Crevier 1993, pp. 271 and 279A few of the people who make some form of the argument:

• Moravec 1988• Kurzweil 2005, p. 262• Hawkins & Blakeslee 2004The most extreme form of this argument (the brain replacement scenario) was put forward by Clark Glymour in the mid-70s and was touchedon by Zenon Pylyshyn and John Searle in 1980.

[78] AI complete: Shapiro 1992, p. 9[79] Nils Nilsson writes: "Simply put, there is wide disagreement in the field about what AI is all about" (Nilsson 1983, p. 10).[80] Biological intelligence vs. intelligence in general:

• Russell & Norvig 2003, pp. 2–3, who make the analogy with aeronautical engineering.• McCorduck 2004, pp. 100–101, who writes that there are "two major branches of artificial intelligence: one aimed at producing intelligent

behavior regardless of how it was accomplioshed, and the other aimed at modeling intelligent processes found in nature, particularlyhuman ones."

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• Kolata 1982, a paper in Science, which describes McCathy's indifference to biological models. Kolata quotes McCarthy as writing: "Thisis AI, so we don't care if it's psychologically real" (http:/ / books. google. com/ books?id=PEkqAAAAMAAJ& q="we+ don't+ care+ if+it's+ psychologically+ real"& dq="we+ don't+ care+ if+ it's+ psychologically+ real"& output=html& pgis=1). McCarthy recentlyreiterated his position at the AI@50 conference where he said "Artificial intelligence is not, by definition, simulation of humanintelligence" (Maker 2006).

[81] Neats vs. scruffies:

• McCorduck 2004, pp. 421–424, 486–489• Crevier 1993, pp. 168• Nilsson 1983, pp. 10–11

[82] Symbolic vs. sub-symbolic AI:

• Nilsson (1998, p. 7), who uses the term "sub-symbolic".[83] Haugeland 1985, p. 255.[84] http:/ / citeseerx. ist. psu. edu/ viewdoc/ download?doi=10. 1. 1. 38. 8384& rep=rep1& type=pdf[85] http:/ / books. google. com/ books?hl=en& lr=& id=a_ZR81Z25z0C& oi=fnd& pg=PA63& dq=related:iCpuTpmGk6gJ:scholar. google.

com/ & ots=n1-TsvqXGJ& sig=1xthOJ2WQmKtdSt7ISFH2yu2l3Q#v=onepage& q& f=false[86] http:/ / www. imagination-engines. com/[87] Haugeland 1985, pp. 112–117[88] Cognitive simulation, Newell and Simon, AI at CMU (then called Carnegie Tech):

• McCorduck 2004, pp. 139–179, 245–250, 322–323 (EPAM)• Crevier 1993, pp. 145–149

[89] Soar (history):

• McCorduck 2004, pp. 450–451• Crevier 1993, pp. 258–263

[90] McCarthy and AI research at SAIL and SRI:

• McCorduck 2004, pp. 251–259• Crevier 1993

[91] AI research at Edinburgh and in France, birth of Prolog:

• Crevier 1993, pp. 193–196• Howe 1994

[92] AI at MIT under Marvin Minsky in the 1960s :

• McCorduck 2004, pp. 259–305• Crevier 1993, pp. 83–102, 163–176• Russell & Norvig 2003, p. 19

[93] Cyc:

• McCorduck 2004, p. 489, who calls it "a determinedly scruffy enterprise"• Crevier 1993, pp. 239−243• Russell & Norvig 2003, p. 363−365• Lenat & Guha 1989

[94] Knowledge revolution:

• McCorduck 2004, pp. 266–276, 298–300, 314, 421• Russell & Norvig 2003, pp. 22–23

[95] The most dramatic case of sub-symbolic AI being pushed into the background was the devastating critique of perceptrons by Marvin Minskyand Seymour Papert in 1969. See History of AI, AI winter, or Frank Rosenblatt.

[96] Embodied approaches to AI:

• McCorduck 2004, pp. 454–462• Brooks 1990• Moravec 1988

[97] Revival of connectionism:

• Crevier 1993, pp. 214–215• Russell & Norvig 2003, p. 25

[98] Computational intelligence

• IEEE Computational Intelligence Society (http:/ / www. ieee-cis. org/ )[99] Agent architectures, hybrid intelligent systems:

• Russell & Norvig (2003, pp. 27, 932, 970–972)• Nilsson (1998, chpt. 25)

[100] Hierarchical control system:

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• Albus, J. S. 4-D/RCS reference model architecture for unmanned ground vehicles. (http:/ / www. isd. mel. nist. gov/ documents/ albus/4DRCS. pdf) In G Gerhart, R Gunderson, and C Shoemaker, editors, Proceedings of the SPIE AeroSense Session on Unmanned GroundVehicle Technology, volume 3693, pages 11—20

[101] Subsumption architecture:

• CITATION IN PROGRESS.[102] Search algorithms:

• Russell & Norvig 2003, pp. 59–189• Poole, Mackworth & Goebel 1998, pp. 113–163• Luger & Stubblefield 2004, pp. 79–164, 193–219• Nilsson 1998, chpt. 7-12

[103] Forward chaining, backward chaining, Horn clauses, and logical deduction as search:

• Russell & Norvig 2003, pp. 217–225, 280–294• Poole, Mackworth & Goebel 1998, pp. ~46–52• Luger & Stubblefield 2004, pp. 62–73• Nilsson 1998, chpt. 4.2, 7.2

[104] State space search and planning:

• Russell & Norvig 2003, pp. 382–387• Poole, Mackworth & Goebel 1998, pp. 298–305• Nilsson 1998, chpt. 10.1-2

[105] Uninformed searches (breadth first search, depth first search and general state space search):

• Russell & Norvig 2003, pp. 59–93• Poole, Mackworth & Goebel 1998, pp. 113–132• Luger & Stubblefield 2004, pp. 79–121• Nilsson 1998, chpt. 8

[106] Heuristic or informed searches (e.g., greedy best first and A*):

• Russell & Norvig 2003, pp. 94–109,• Poole, Mackworth & Goebel 1998, pp. pp. 132–147,• Luger & Stubblefield 2004, pp. 133–150,• Nilsson 1998, chpt. 9

[107] Optimization searches:

• Russell & Norvig 2003, pp. 110–116,120–129• Poole, Mackworth & Goebel 1998, pp. 56–163• Luger & Stubblefield 2004, pp. 127–133

[108] Artificial life and society based learning:

• Luger & Stubblefield 2004, pp. 530–541[109] Genetic programming and genetic algorithms:

• Luger & Stubblefield 2004, pp. 509–530,• Nilsson 1998, chpt. 4.2.• Holland, John H. (1975). Adaptation in Natural and Artificial Systems. University of Michigan Press. ISBN 0262581116.• Koza, John R. (1992). Genetic Programming. MIT Press. ISBN 0262111705.• Poli, R., Langdon, W. B., McPhee, N. F. (2008). A Field Guide to Genetic Programming. Lulu.com, freely available from http:/ / www.

gp-field-guide. org. uk/ . & #32;ISBN& nbsp;978-1-4092-0073-4.[110] Logic:

• ACM 1998, ~I.2.3,• Russell & Norvig 2003, pp. 194–310,• Luger & Stubblefield 2004, pp. 35–77,• Nilsson 1998, chpt. 13-16

[111] Satplan:

• Russell & Norvig 2003, pp. 402–407,• Poole, Mackworth & Goebel 1998, pp. 300–301,• Nilsson 1998, chpt. 21

[112] Explanation based learning, relevance based learning, inductive logic programming, case based reasoning:

• Russell & Norvig 2003, pp. 678–710,• Poole, Mackworth & Goebel 1998, pp. 414–416,• Luger & Stubblefield 2004, pp. ~422–442,• Nilsson 1998, chpt. 10.3, 17.5

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[113] Propositional logic:

• Russell & Norvig 2003, pp. 204–233,• Luger & Stubblefield 2004, pp. 45–50• Nilsson 1998, chpt. 13

[114] First-order logic and features such as equality:

• ACM 1998, ~I.2.4,• Russell & Norvig 2003, pp. 240–310,• Poole, Mackworth & Goebel 1998, pp. 268–275,• Luger & Stubblefield 2004, pp. 50–62,• Nilsson 1998, chpt. 15

[115] Fuzzy logic:

• Russell & Norvig 2003, pp. 526–527[116] Subjective logic:

• CITATION IN PROGRESS.[117] Stochastic methods for uncertain reasoning:

• ACM 1998, ~I.2.3,• Russell & Norvig 2003, pp. 462–644,• Poole, Mackworth & Goebel 1998, pp. 345–395,• Luger & Stubblefield 2004, pp. 165–191, 333–381,• Nilsson 1998, chpt. 19

[118] Bayesian networks:

• Russell & Norvig 2003, pp. 492–523,• Poole, Mackworth & Goebel 1998, pp. 361–381,• Luger & Stubblefield 2004, pp. ~182–190, ~363–379,• Nilsson 1998, chpt. 19.3-4

[119] Bayesian inference algorithm:

• Russell & Norvig 2003, pp. 504–519,• Poole, Mackworth & Goebel 1998, pp. 361–381,• Luger & Stubblefield 2004, pp. ~363–379,• Nilsson 1998, chpt. 19.4 & 7

[120] Bayesian learning and the expectation-maximization algorithm:

• Russell & Norvig 2003, pp. 712–724,• Poole, Mackworth & Goebel 1998, pp. 424–433,• Nilsson 1998, chpt. 20

[121] Bayesian decision theory snd Bayesian decision networks:

• Russell & Norvig 2003, pp. 597–600[122] Stochastic temporal models:

• Russell & Norvig 2003, pp. 537–581Dynamic Bayesian networks:

• Russell & Norvig 2003, pp. 551–557Hidden Markov model:

• (Russell & Norvig 2003, pp. 549–551)Kalman filters:

• Russell & Norvig 2003, pp. 551–557[123] decision theory and decision analysis:

• Russell & Norvig 2003, pp. 584–597,• Poole, Mackworth & Goebel 1998, pp. 381–394

[124] Markov decision processes and dynamic decision networks:

• Russell & Norvig 2003, pp. 613–631[125] Game theory and mechanism design:

• Russell & Norvig 2003, pp. 631–643[126] Statistical learning methods and classifiers:

• Russell & Norvig 2003, pp. 712–754,• Luger & Stubblefield 2004, pp. 453–541

[127] Neural networks and connectionism:

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• Russell & Norvig 2003, pp. 736–748,• Poole, Mackworth & Goebel 1998, pp. 408–414,• Luger & Stubblefield 2004, pp. 453–505,• Nilsson 1998, chpt. 3

[128] Kernel methods:

• Russell & Norvig 2003, pp. 749–752[129] K-nearest neighbor algorithm:

• Russell & Norvig 2003, pp. 733–736[130] Gaussian mixture model:

• Russell & Norvig 2003, pp. 725–727[131] Naive Bayes classifier:

• Russell & Norvig 2003, pp. 718[132] Decision tree:

• Russell & Norvig 2003, pp. 653–664,• Poole, Mackworth & Goebel 1998, pp. 403–408,• Luger & Stubblefield 2004, pp. 408–417

[133] Classifier performance:

• van der Walt & Bernard 2006[134] Backpropagation:

• Russell & Norvig 2003, pp. 744–748,• Luger & Stubblefield 2004, pp. 467–474,• Nilsson 1998, chpt. 3.3

[135] Feedforward neural networks, perceptrons and radial basis networks:

• Russell & Norvig 2003, pp. 739–748, 758• Luger & Stubblefield 2004, pp. 458–467

[136] Recurrent neural networks, Hopfield nets:

• Russell & Norvig 2003, p. 758• Luger & Stubblefield 2004, pp. 474–505

[137] Competitive learning, Hebbian coincidence learning, Hopfield networks and attractor networks:

• Luger & Stubblefield 2004, pp. 474–505[138] Hierarchical temporal memory:

• Hawkins & Blakeslee 2004[139] Control theory:

• ACM 1998, ~I.2.8,• Russell & Norvig 2003, pp. 926–932

[140] Lisp:

• Luger & Stubblefield 2004, pp. 723–821• Crevier 1993, pp. 59–62,• Russell & Norvig 2003, p. 18

[141] Prolog:

• Poole, Mackworth & Goebel 1998, pp. 477–491,• Luger & Stubblefield 2004, pp. 641–676, 575–581

[142] The Turing test:Turing's original publication:

• Turing 1950Historical influence and philosophical implications:

• Haugeland 1985, pp. 6–9• Crevier 1993, p. 24• McCorduck 2004, pp. 70–71• Russell & Norvig 2003, pp. 2–3 and 948

[143] Subject matter expert Turing test:

• CITATION IN PROGRESS.[144] Game AI:

• CITATION IN PROGRESS.[145] Mathematical definitions of intelligence:

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• Jose Hernandez-Orallo (2000). "Beyond the Turing Test" (http:/ / citeseerx. ist. psu. edu/ viewdoc/ summary?doi=10. 1. 1. 44. 8943).Journal of Logic, Language and Information 9 (4): 447–466. doi:10.1023/A:1008367325700. . Retrieved 2009-07-21.

• D L Dowe and A R Hajek (1997). "A computational extension to the Turing Test" (http:/ / www. csse. monash. edu. au/ publications/1997/ tr-cs97-322-abs. html). Proceedings of the 4th Conference of the Australasian Cognitive Science jSociety. . Retrieved 2009-07-21.

• J Hernandez-Orallo and D L Dowe (2010). "Measuring Universal Intelligence: Towards an Anytime Intelligence Test". ArtificialIntelligence Journal 174 (18): 1508–1539. doi:10.1016/j.artint.2010.09.006.

[146] "AI set to exceed human brain power" (http:/ / www. cnn. com/ 2006/ TECH/ science/ 07/ 24/ ai. bostrom/ ) (web article). CNN.com.2006-07-26. . Retrieved 2008-02-26.

[147] Brooks, R.A., "How to build complete creatures rather than isolated cognitive simulators," in K. VanLehn (ed.), Architectures forIntelligence, pp. 225–239, Lawrence Erlbaum Associates, Hillsdale, NJ, 1991.

[148] Hacking Roomba » Search Results » atmel (http:/ / hackingroomba. com/ ?s=atmel)[149] Philosophy of AI. All of these positions in this section are mentioned in standard discussions of the subject, such as:

• Russell & Norvig 2003, pp. 947–960• Fearn 2007, pp. 38–55

[150] Dartmouth proposal:

• McCarthy et al. 1955 (the original proposal)• Crevier 1993, p. 49 (historical significance)

[151] The physical symbol systems hypothesis:

• Newell & Simon 1976, p. 116• McCorduck 2004, p. 153• Russell & Norvig 2003, p. 18

[152] Dreyfus criticized the necessary condition of the physical symbol system hypothesis, which he called the "psychological assumption": "Themind can be viewed as a device operating on bits of information according to formal rules". (Dreyfus 1992, p. 156)

[153] Dreyfus' critique of artificial intelligence:

• Dreyfus 1972, Dreyfus & Dreyfus 1986• Crevier 1993, pp. 120–132• McCorduck 2004, pp. 211–239• Russell & Norvig 2003, pp. 950–952,

[154] This is a paraphrase of the relevant implication of Gödel's theorems.[155] The Mathematical Objection:

• Russell & Norvig 2003, p. 949• McCorduck 2004, pp. 448–449Making the Mathematical Objection:

• Lucas 1961• Penrose 1989Refuting Mathematical Objection:

• Turing 1950 under "(2) The Mathematical Objection"• Hofstadter 1979Background:

• Gödel 1931, Church 1936, Kleene 1935, Turing 1937[156] This version is from Searle (1999), and is also quoted in Dennett 1991, p. 435. Searle's original formulation was "The appropriately

programmed computer really is a mind, in the sense that computers given the right programs can be literally said to understand and have othercognitive states." (Searle 1980, p. 1). Strong AI is defined similarly by Russell & Norvig (2003, p. 947): "The assertion that machines couldpossibly act intelligently (or, perhaps better, act as if they were intelligent) is called the 'weak AI' hypothesis by philosophers, and the assertionthat machines that do so are actually thinking (as opposed to simulating thinking) is called the 'strong AI' hypothesis."

[157] Searle's Chinese Room argument:

• Searle 1980. Searle's original presentation of the thought experiment.• Searle 1999.Discussion:

• Russell & Norvig 2003, pp. 958–960• McCorduck 2004, pp. 443–445• Crevier 1993, pp. 269–271

[158] Robot rights:

• Russell & Norvig 2003, p. 964• "Robots could demand legal rights" (http:/ / news. bbc. co. uk/ 2/ hi/ technology/ 6200005. stm). BBC News. 21 December 2006. .

Retrieved 3 February 2011.

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Prematurity of:

• Henderson, Mark (24 April 2007). "Human rights for robots? We're getting carried away" (http:/ / www. timesonline. co. uk/ tol/ news/ uk/science/ article1695546. ece). The Times Online (London). .

In fiction:

• McCorduck (2004, p. 190-25) discusses Frankenstein and identifies the key ethical issues as scientific hubris and the suffering of themonster, i.e. robot rights.

[159] http:/ / www. plugandpray-film. de/ en/ content. html Independent documentary Plug & Pray, featuring Joseph Weizenbaum and RaymondKurzweil

[160] Ford, Martin R. (2009), The Lights in the Tunnel: Automation, Accelerating Technology and the Economy of the Future (http:/ / www.thelightsinthetunnel. com), Acculant Publishing, ISBN 978-1448659814, . ( e-book available free online (http:/ / www. thelightsinthetunnel.com/ ).)

[161] "Machine Learning: A Job Killer?" (http:/ / econfuture. wordpress. com/ 2011/ 04/ 14/ machine-learning-a-job-killer/ )[162] AI could decrease the demand for human labor:

• Russell & Norvig 2003, pp. 960–961• Ford, Martin (2009). The Lights in the Tunnel: Automation, Accelerating Technology and the Economy of the Future (http:/ / www.

thelightsinthetunnel. com). Acculant Publishing. ISBN 978-1-4486-5981-4. .[163] In the early 70s, Kenneth Colby presented a version of Weizenbaum's ELIZA known as DOCTOR which he promoted as a serious

therapeutic tool. (Crevier & 1993 132−144)[164] Joseph Weizenbaum's critique of AI:

• Weizenbaum 1976• Crevier 1993, pp. 132−144• McCorduck 2004, pp. 356–373• Russell & Norvig 2003, p. 961Weizenbaum (the AI researcher who developed the first chatterbot program, ELIZA) argued in 1976 that the misuse of artificial intelligencehas the potential to devalue human life.

[165] Technological singularity:

• Vinge 1983• Kurzweil 2005• Russell & Norvig 2003, p. 963

[166] Transhumanism:

• Moravec 1988• Kurzweil 2005• Russell & Norvig 2003, p. 963

[167] AI as evolution:

• Edward Fredkin is quoted in McCorduck (2004, p. 401).• Butler, Samuel (13 June 1863). the Press (Christchurch, New Zealand). http:/ / www. nzetc. org/ tm/ scholarly/

tei-ButFir-t1-g1-t1-g1-t4-body. html, Letter to the Editor.• Dyson, George (1998). Darwin among the Machiens. Allan Lane Science. ISBN 0738200301.

AI textbooks• Luger, George; Stubblefield, William (2004). Artificial Intelligence: Structures and Strategies for Complex

Problem Solving (http:/ / www. cs. unm. edu/ ~luger/ ai-final/ tocfull. html) (5th ed.). The Benjamin/CummingsPublishing Company, Inc.. ISBN 0-8053-4780-1.

• Nilsson, Nils (1998). Artificial Intelligence: A New Synthesis. Morgan Kaufmann Publishers.ISBN 978-1-55860-467-4.

• Russell, Stuart J.; Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (http:/ / aima. cs. berkeley.edu/ ) (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0-13-790395-2

• Poole, David; Mackworth, Alan; Goebel, Randy (1998). Computational Intelligence: A Logical Approach (http:/ /www. cs. ubc. ca/ spider/ poole/ ci. html). New York: Oxford University Press. ISBN 0195102703.

• Winston, Patrick Henry (1984). Artificial Intelligence. Reading, Massachusetts: Addison-Wesley.ISBN 0201082594.

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History of AI• Crevier, Daniel (1993), AI: The Tumultuous Search for Artificial Intelligence, New York, NY: BasicBooks, ISBN

0-465-02997-3• McCorduck, Pamela (2004), Machines Who Think (http:/ / www. pamelamc. com/ html/ machines_who_think.

html) (2nd ed.), Natick, MA: A. K. Peters, Ltd., ISBN 1-56881-205-1

Other sources• "ACM Computing Classification System: Artificial intelligence" (http:/ / www. acm. org/ class/ 1998/ I. 2. html).

ACM. 1998. Retrieved 2007-08-30.• Aleksander, Igor (1995). Artificial Neuroconsciousness: An Update (http:/ / web. archive. org/ web/

19970302014628/ http:/ / www. ee. ic. ac. uk/ research/ neural/ publications/ iwann. html). IWANN. Archivedfrom the original (http:/ / www. ee. ic. ac. uk/ research/ neural/ publications/ iwann. html) on 1997-03-02. BibTex(http:/ / dblp. uni-trier. de/ rec/ bibtex/ conf/ iwann/ Aleksander95) Internet Archive (http:/ / web. archive. org/web/ 19970302014628/ http:/ / www. ee. ic. ac. uk/ research/ neural/ publications/ iwann. html)

• Brooks, Rodney (1990). "Elephants Don't Play Chess" (http:/ / people. csail. mit. edu/ brooks/ papers/ elephants.pdf) (PDF). Robotics and Autonomous Systems 6: 3–15. doi:10.1016/S0921-8890(05)80025-9. Retrieved2007-08-30..

• Buchanan, Bruce G. (Winter 2005). "A (Very) Brief History of Artificial Intelligence" (http:/ / www. aaai. org/AITopics/ assets/ PDF/ AIMag26-04-016. pdf) (PDF). AI Magazine: 53–60. Retrieved 2007-08-30.

• Dennett, Daniel (1991). Consciousness Explained. The Penguin Press. ISBN 0-7139-9037-6.• Dreyfus, Hubert (1972). What Computers Can't Do. New York: MIT Press. ISBN 0060110821.• Dreyfus, Hubert (1979). What Computers Still Can't Do. New York: MIT Press. ISBN 0262041340.• Dreyfus, Hubert; Dreyfus, Stuart (1986). Mind over Machine: The Power of Human Intuition and Expertise in the

Era of the Computer. Oxford, UK: Blackwell. ISBN 0029080606.• Dreyfus, Hubert (1992). What Computers Still Can't Do. New York: MIT Press. ISBN 0-262-54067-3.• Edelman, Gerald (23 November 2007). "Gerald Edelman - Neural Darwinism and Brain-based Devices" (http:/ /

lis. epfl. ch/ resources/ podcast/ 2007/ 11/ gerald-edelman-neural-darwinism-and. html). Talking Robots.• Fearn, Nicholas (2007). The Latest Answers to the Oldest Questions: A Philosophical Adventure with the World's

Greatest Thinkers. New York: Grove Press. ISBN 0802118399.• Forster, Dion (2006). "Self validating consciousness in strong artificial intelligence: An African theological

contribution" (http:/ / www. spirituality. org. za/ files/ D Forster doctorate. pdf). Pretoria: University of SouthAfrica.

• Gladwell, Malcolm (2005). Blink. New York: Little, Brown and Co.. ISBN 0-316-17232-4.• Haugeland, John (1985). Artificial Intelligence: The Very Idea. Cambridge, Mass.: MIT Press.

ISBN 0-262-08153-9.• Hawkins, Jeff; Blakeslee, Sandra (2004). On Intelligence. New York, NY: Owl Books. ISBN 0-8050-7853-3.• Hofstadter, Douglas (1979). Gödel, Escher, Bach: an Eternal Golden Braid. New York, NY: Vintage Books.

ISBN 0394745027.• Howe, J. (November 1994). "Artificial Intelligence at Edinburgh University: a Perspective" (http:/ / www. inf. ed.

ac. uk/ about/ AIhistory. html). Retrieved 30 August 2007..• Kahneman, Daniel; Slovic, D.; Tversky, Amos (1982). Judgment under uncertainty: Heuristics and biases. New

York: Cambridge University Press. ISBN 0521284147.• Kolata, G. (1982). "How can computers get common sense?". Science 217 (4566): 1237–1238.

doi:10.1126/science.217.4566.1237. PMID 17837639.• Kurzweil, Ray (1999). The Age of Spiritual Machines. Penguin Books. ISBN 0-670-88217-8.• Kurzweil, Ray (2005). The Singularity is Near. Penguin Books. ISBN 0-670-03384-7.

Artificial intelligence 92

• Lakoff, George (1987). Women, Fire, and Dangerous Things: What Categories Reveal About the Mind.University of Chicago Press. ISBN 0-226-46804-6.

• Lakoff, George; Núñez, Rafael E. (2000). Where Mathematics Comes From: How the Embodied Mind BringsMathematics into Being. Basic Books. ISBN 0-465-03771-2..

• Lenat, Douglas; Guha, R. V. (1989). Building Large Knowledge-Based Systems. Addison-Wesley.ISBN 0201517523.

• Lighthill, Professor Sir James (1973). "Artificial Intelligence: A General Survey". Artificial Intelligence: a papersymposium. Science Research Council.

• Lucas, John (1961). "Minds, Machines and Gödel" (http:/ / users. ox. ac. uk/ ~jrlucas/ Godel/ mmg. html). InAnderson, A.R.. Minds and Machines. Retrieved 30 August 2007.

• Maker, Meg Houston (2006). "AI@50: AI Past, Present, Future" (http:/ / www. engagingexperience. com/ 2006/07/ ai50_ai_past_pr. html). Dartmouth College. Retrieved 16 October 2008.

• McCarthy, John; Minsky, Marvin; Rochester, Nathan; Shannon, Claude (1955). "A Proposal for the DartmouthSummer Research Project on Artificial Intelligence" (http:/ / www-formal. stanford. edu/ jmc/ history/ dartmouth/dartmouth. html). Retrieved 30 August 2007..

• McCarthy, John; Hayes, P. J. (1969). "Some philosophical problems from the standpoint of artificial intelligence"(http:/ / www-formal. stanford. edu/ jmc/ mcchay69. html). Machine Intelligence 4: 463–502. Retrieved 30August 2007.

• McCarthy, John (November 12, 2007). "What Is Artificial Intelligence?" (http:/ / www-formal. stanford. edu/ jmc/whatisai/ whatisai. html).

• Minsky, Marvin (1967). Computation: Finite and Infinite Machines. Englewood Cliffs, N.J.: Prentice-Hall.ISBN 0131654497.

• Minsky, Marvin (2006). The Emotion Machine. New York, NY: Simon & Schusterl. ISBN 0-7432-7663-9.• Moravec, Hans (1976). "The Role of Raw Power in Intelligence" (http:/ / www. frc. ri. cmu. edu/ users/ hpm/

project. archive/ general. articles/ 1975/ Raw. Power. html). Retrieved 30 August 2007.• Moravec, Hans (1988). Mind Children. Harvard University Press. ISBN 0674576160.• NRC, (United States National Research Council) (1999). "Developments in Artificial Intelligence". Funding a

Revolution: Government Support for Computing Research. National Academy Press.• Needham, Joseph (1986). Science and Civilization in China: Volume 2. Caves Books Ltd..• Newell, Allen; Simon, H. A. (1963). "GPS: A Program that Simulates Human Thought". In Feigenbaum, E.A.;

Feldman, J.. Computers and Thought. New York: McGraw-Hill.• Newell, Allen; Simon, H. A. (1976). "Computer Science as Empirical Inquiry: Symbols and Search" (http:/ /

www. rci. rutgers. edu/ ~cfs/ 472_html/ AI_SEARCH/ PSS/ PSSH4. html). Communications of the ACM. 19..• Nilsson, Nils (1983), "Artificial Intelligence Prepares for 2001" (http:/ / ai. stanford. edu/ ~nilsson/

OnlinePubs-Nils/ General Essays/ AIMag04-04-002. pdf), AI Magazine 1 (1), Presidential Address to theAssociation for the Advancement of Artificial Intelligence.

• Searle, John (1980). "Minds, Brains and Programs" (http:/ / www. bbsonline. org/ documents/ a/ 00/ 00/ 04/ 84/bbs00000484-00/ bbs. searle2. html). Behavioral and Brain Sciences 3 (3): 417–457.doi:10.1017/S0140525X00005756.

• Searle, John (1999). Mind, language and society. New York, NY: Basic Books. ISBN 0465045219.OCLC 231867665 43689264.

• Serenko, Alexander; Detlor, Brian (2004). "Intelligent agents as innovations" (http:/ / foba. lakeheadu. ca/serenko/ papers/ Serenko_Detlor_AI_and_Society. pdf). AI and Society 18 (4): 364–381.doi:10.1007/s00146-004-0310-5.

• Serenko, Alexander; Ruhi, Umar; Cocosila, Mihail (2007). "Unplanned effects of intelligent agents on Internet use: Social Informatics approach" (http:/ / foba. lakeheadu. ca/ serenko/ papers/

AI_Society_Serenko_Social_Impacts_of_Agents. pdf). AI and Society 21 (1–2): 141–166.

Artificial intelligence 93

doi:10.1007/s00146-006-0051-8.• Shapiro, Stuart C. (1992). "Artificial Intelligence" (http:/ / www. cse. buffalo. edu/ ~shapiro/ Papers/ ai. pdf). In

Shapiro, Stuart C.. Encyclopedia of Artificial Intelligence (2nd ed.). New York: John Wiley. pp. 54–57.ISBN 0471503061.

• Simon, H. A. (1965). The Shape of Automation for Men and Management. New York: Harper & Row.• Skillings, Jonathan (3 July 2006). "Getting Machines to Think Like Us" (http:/ / news. cnet. com/

Getting-machines-to-think-like-us/ 2008-11394_3-6090207. html). cnet. Retrieved February 3, 2011.• Turing, Alan (October 1950), "Computing Machinery and Intelligence" (http:/ / loebner. net/ Prizef/

TuringArticle. html), Mind LIX (236): 433–460, doi:10.1093/mind/LIX.236.433, ISSN 0026-4423, retrieved2008-08-18.

• van der Walt, Christiaan; Bernard, Etienne (2006<!––year is presumed based on acknowledgements at the end ofthe article––>). "Data characteristics that determine classifier performance" (http:/ / www. patternrecognition. co.za/ publications/ cvdwalt_data_characteristics_classifiers. pdf) (PDF). Retrieved 5 August 2009.

• Wason, P. C.; Shapiro, D. (1966). "Reasoning". In Foss, B. M.. New horizons in psychology. Harmondsworth:Penguin.

• Weizenbaum, Joseph (1976). Computer Power and Human Reason. San Francisco: W.H. Freeman & Company.ISBN 0716704641.

Further reading• TechCast Article Series, John Sagi, Framing Consciousness (http:/ / www. techcast. org/ Upload/ PDFs/

634146249446122137_Consciousness-Sagifinalversion. pdf)• Boden, Margaret, Mind As Machine, Oxford University Press, 2006• Johnston, John (2008) "The Allure of Machinic Life: Cybernetics, Artificial Life, and the New AI", MIT Press• Myers, Courtney Boyd ed. (2009). The AI Report (http:/ / www. forbes. com/ 2009/ 06/ 22/

singularity-robots-computers-opinions-contributors-artificial-intelligence-09_land. html). Forbes June 2009• Serenko, Alexander (2010). "The development of an AI journal ranking based on the revealed preference

approach" (http:/ / foba. lakeheadu. ca/ serenko/ papers/ JOI_Serenko_AI_Journal_Ranking_Published. pdf)(PDF). Journal of Informetrics 4 (4): 447–459. doi:10.1016/j.joi.2010.04.001.

• Sun, R. & Bookman, L. (eds.), Computational Architectures: Integrating Neural and Symbolic Processes. KluwerAcademic Publishers, Needham, MA. 1994.

External links• What Is AI? (http:/ / www-formal. stanford. edu/ jmc/ whatisai/ whatisai. html) — An introduction to artificial

intelligence by AI founder John McCarthy.• Logic and Artificial Intelligence (http:/ / plato. stanford. edu/ entries/ logic-ai) entry by Richmond Thomason in

the Stanford Encyclopedia of Philosophy• AI (http:/ / www. dmoz. org/ Computers/ Artificial_Intelligence/ / ) at the Open Directory Project• AITopics (http:/ / aaai. org/ AITopics/ ) — A large directory of links and other resources maintained by the

Association for the Advancement of Artificial Intelligence, the leading organization of academic AI researchers.• Artificial Intelligence Discussion group (https:/ / www. researchgate. net/ group/ Artificial_Intelligence)

Probability theory 94

Probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The centralobjects of probability theory are random variables, stochastic processes, and events: mathematical abstractions ofnon-deterministic events or measured quantities that may either be single occurrences or evolve over time in anapparently random fashion. If an individual coin toss or the roll of die is considered to be a random event, then ifrepeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted.Two representative mathematical results describing such patterns are the law of large numbers and the central limittheorem.As a mathematical foundation for statistics, probability theory is essential to many human activities that involvequantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complexsystems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentiethcentury physics was the probabilistic nature of physical phenomena at atomic scales, described in quantummechanics.

HistoryThe mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano inthe sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the"problem of points"). Christiaan Huygens published a book on the subject in 1657.[2]

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial.Eventually, analytical considerations compelled the incorporation of continuous variables into the theory.This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov.Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory andpresented his axiom system for probability theory in 1933. Fairly quickly this became the mostly undisputedaxiomatic basis for modern probability theory but alternatives exist, in particular the adoption of finite rather thancountable additivity by Bruno de Finetti.[3]

TreatmentMost introductions to probability theory treat discrete probability distributions and continuous probabilitydistributions separately. The more mathematically advanced measure theory based treatment of probability coversboth the discrete, the continuous, any mix of these two and more.

MotivationConsider an experiment that can produce a number of outcomes. The collection of all results is called the samplespace of the experiment. The power set of the sample space is formed by considering all different collections ofpossible results. For example, rolling a die produces one of six possible results. One collection of possible resultscorresponds to getting an odd number. Thus, the subset {1,3,5} is an element of the power set of the sample space ofdie rolls. These collections are called events. In this case, {1,3,5} is the event that the die falls on some odd number.If the results that actually occur fall in a given event, that event is said to have occurred.Probability is a way of assigning every "event" a value between zero and one, with the requirement that the eventmade up of all possible results (in our example, the event {1,2,3,4,5,6}) be assigned a value of one. To qualify as aprobability distribution, the assignment of values must satisfy the requirement that if you look at a collection ofmutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are allmutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilitiesof all the individual events.[4]

Probability theory 95

The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that theprobability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five beingrolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1 -absolute certainty. For convenience's sake, we ignore the possibility that the die, once rolled, will be obliteratedbefore it can hit the table.

Discrete probability distributionsDiscrete probability theory deals with events that occur in countable sample spaces.Examples: Throwing dice, experiments with decks of cards, and random walk.Classical definition: Initially the probability of an event to occur was defined as number of cases favorable for theevent, over the number of total outcomes possible in an equiprobable sample space: see Classical definition ofprobability.

For example, if the event is "occurrence of an even number when a die is rolled", the probability is given by ,since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.Modern definition: The modern definition starts with a finite or countable set called the sample space, whichrelates to the set of all possible outcomes in classical sense, denoted by . It is then assumed that for each element

, an intrinsic "probability" value is attached, which satisfies the following properties:

1.

2.

That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and thesum of f(x) over all values x in the sample space Ω is equal to 1. An event is defined as any subset of the samplespace . The probability of the event is defined as

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

The function mapping a point in the sample space to the "probability" value is called a probability mass

function abbreviated as pmf. The modern definition does not try to answer how probability mass functions areobtained; instead it builds a theory that assumes their existence.

Continuous probability distributionsContinuous probability theory deals with events that occur in a continuous sample space.Classical definition: The classical definition breaks down when confronted with the continuous case. See Bertrand'sparadox.Modern definition: If the outcome space of a random variable X is the set of real numbers ( ) or a subset thereof,then a function called the cumulative distribution function (or cdf) exists, defined by .That is, F(x) returns the probability that X will be less than or equal to x.The cdf necessarily satisfies the following properties.1. is a monotonically non-decreasing, right-continuous function;

2.3.If is absolutely continuous, i.e., its derivative exists and integrating the derivative gives us the cdf back again, then the random variable X is said to have a probability density function or pdf or simply density

Probability theory 96

For a set , the probability of the random variable X being in is

In case the probability density function exists, this can be written as

Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables (includingdiscrete random variables) that take values in These concepts can be generalized for multidimensional cases on and other continuous sample spaces.

Measure-theoretic probability theoryThe raison d'être of the measure-theoretic treatment of probability is that it unifies the discrete and the continuouscases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that areneither discrete nor continuous nor mixtures of the two.An example of such distributions could be a mix of discrete and continuous distributions—for example, a randomvariable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. Itcan still be studied to some extent by considering it to have a pdf of , where is the Diracdelta function.Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for anysingle point, neither does it have a density. The modern approach to probability theory solves these problems usingmeasure theory to define the probability space:Given any set , (also called sample space) and a σ-algebra on it, a measure defined on is called aprobability measure if If is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on for any cdf,and vice versa. The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with thepmf for discrete variables, and pdf for continuous variables, making the measure-theoretic approach free of fallacies.The probability of a set in the σ-algebra is defined as

where the integration is with respect to the measure induced by Along with providing better understanding and unification of discrete and continuous probabilities,measure-theoretic treatment also allows us to work on probabilities outside , as in the theory of stochasticprocesses. For example to study Brownian motion, probability is defined on a space of functions.

Probability distributionsCertain random variables occur very often in probability theory because they well describe many natural or physicalprocesses. Their distributions therefore have gained special importance in probability theory. Some fundamentaldiscrete distributions are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometricdistributions. Important continuous distributions include the continuous uniform, normal, exponential, gamma andbeta distributions.

Probability theory 97

Convergence of random variablesIn probability theory, there are several notions of convergence for random variables. They are listed below in theorder of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of thepreceding notions.

Weak convergence: A sequence of random variables converges weakly to the randomvariable if their respective cumulative distribution functions converge to the cumulativedistribution function of , wherever is continuous. Weak convergence is also called convergence indistribution.

Most common short hand notation: Convergence in probability: The sequence of random variables is said to converge towards therandom variable in probability if for every ε > 0.

Most common short hand notation: Strong convergence: The sequence of random variables is said to converge towards the randomvariable strongly if . Strong convergence is also known as almost sure

convergence.Most common short hand notation:

As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence impliesconvergence in probability, and convergence in probability implies weak convergence. The reverse statements arenot always true.

Law of large numbersCommon intuition suggests that if a fair coin is tossed many times, then roughly half of the time it will turn upheads, and the other half it will turn up tails. Furthermore, the more often the coin is tossed, the more likely it shouldbe that the ratio of the number of heads to the number of tails will approach unity. Modern probability provides aformal version of this intuitive idea, known as the law of large numbers. This law is remarkable because it isnowhere assumed in the foundations of probability theory, but instead emerges out of these foundations as atheorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world,the law of large numbers is considered as a pillar in the history of statistical theory.[5]

The law of large numbers (LLN) states that the sample average of (independent

and identically distributed random variables with finite expectation ) converges towards the theoreticalexpectation It is in the different forms of convergence of random variables that separates the weak and the strong law of largenumbers

It follows from LLN that if an event of probability p is observed repeatedly during independent experiments, theratio of the observed frequency of that event to the total number of repetitions converges towards p.

Putting this in terms of random variables and LLN we have are independent Bernoulli random variablestaking values 1 with probability p and 0 with probability 1-p. for all i and it follows from LLN that

converges to p almost surely.

Probability theory 98

Central limit theorem"The central limit theorem (CLT) is one of the great results of mathematics." (Chapter 18 in.[6] ) It explains theubiquitous occurrence of the normal distribution in nature.The theorem states that the average of many independent and identically distributed random variables with finitevariance tends towards a normal distribution irrespective of the distribution followed by the original randomvariables. Formally, let be independent random variables with mean and variance Thenthe sequence of random variables

converges in distribution to a standard normal random variable.

Notes[1] Probability theory, Encyclopaedia Britannica (http:/ / www. britannica. com/ ebc/ article-9375936)[2] Grinstead, Charles Miller; James Laurie Snell. "Introduction". Introduction to Probability. pp. vii.[3] "The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk (http:/ / www. probabilityandfinance. com/

articles/ 04. pdf)[4] Ross, Sheldon. A First course in Probability, 8th Edition. Page 26-27.[5] http:/ / www. leithner. com. au/ circulars/ circular17. htm[6] David Williams, "Probability with martingales", Cambridge 1991/2008

References• Pierre Simon de Laplace (1812). Analytical Theory of Probability.

The first major treatise blending calculus with probability theory, originally in French: ThéorieAnalytique des Probabilités.

• Andrei Nikolajevich Kolmogorov (1950). Foundations of the Theory of Probability.The modern measure-theoretic foundation of probability theory; the original German version(Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933.

• Patrick Billingsley (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons.• Olav Kallenberg; Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN

0-387-95313-2• Henk Tijms (2004). Understanding Probability. Cambridge Univ. Press.

A lively introduction to probability theory for the beginner.• Olav Kallenberg; Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510

pp. ISBN 0-387-25115-4• Gut, Allan (2005). Probability: A Graduate Course. Springer-Verlag. ISBN 0387228330.

Linear programming 99

Linear programmingLinear programming (LP, or linear optimization) is a mathematical method for determining a way to achieve thebest outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirementsrepresented as linear relationships. Linear programming is a specific case of mathematical programming(mathematical optimization).More formally, linear programming is a technique for the optimization of a linear objective function, subject to linearequality and linear inequality constraints. Its feasible region is a convex polyhedron, which is a set defined as theintersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is areal-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in thepolyhedron where this function has the smallest (or largest) value if such point exists.Linear programs are problems that can be expressed in canonical form:

where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients and A is a(known) matrix of coefficients. The expression to be maximized or minimized is called the objective function (cTx inthis case). The equations Ax ≤ b are the constraints which specify a convex polytope over which the objectivefunction is to be optimized. (In this context, two vectors are comparable when every entry in one is less-than orequal-to the corresponding entry in the other. Otherwise, they are incomparable.)Linear programming can be applied to various fields of study. It is used most extensively in business and economics,but can also be utilized for some engineering problems. Industries that use linear programming models includetransportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types ofproblems in planning, routing, scheduling, assignment, and design.

HistoryThe problem of solving a system of linear inequalities dates back at least as far as Fourier, after whom the method ofFourier-Motzkin elimination is named. Linear programming itself was first developed by Leonid Kantorovich, aRussian mathematician, in 1939[1] . It was used during World War II to plan expenditures and returns in order toreduce costs to the army and increase losses to the enemy. The method was kept secret until 1947 when George B.Dantzig published the simplex method and John von Neumann developed the theory of duality. Postwar, manyindustries found its use in their daily planning.The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979,but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced anew interior-point method for solving linear-programming problems.Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linearprogramming. The computing power required to test all the permutations to select the best assignment is vast; thenumber of possible configurations exceeds the number of particles in the universe. However, it takes only a momentto find the optimum solution by posing the problem as a linear program and applying the Simplex algorithm. Thetheory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.

Linear programming 100

UsesLinear programming is a considerable field of optimization for several reasons. Many practical problems inoperations research can be expressed as linear programming problems. Certain special cases of linear programming,such as network flow problems and multicommodity flow problems are considered important enough to havegenerated much research on specialized algorithms for their solution. A number of algorithms for other types ofoptimization problems work by solving LP problems as sub-problems. Historically, ideas from linear programminghave inspired many of the central concepts of optimization theory, such as duality, decomposition, and theimportance of convexity and its generalizations. Likewise, linear programming is heavily used in microeconomicsand company management, such as planning, production, transportation, technology and other issues. Although themodern management issues are ever-changing, most companies would like to maximize profits or minimize costswith limited resources. Therefore, many issues can be characterized as linear programming problems.

Standard formStandard form is the usual and most intuitive form of describing a linear programming problem. It consists of thefollowing four parts:• A linear function to be maximized

e.g. • Problem constraints of the following form

e.g.

• Non-negative variables

e.g.

• Non-negative right hand side constants

The problem is usually expressed in matrix form, and then becomes:

Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problemsinvolving negative variables can always be rewritten into an equivalent problem in standard form.

ExampleSuppose that a farmer has a piece of farm land, say L km2, to be planted with either wheat or barley or somecombination of the two. The farmer has a limited amount of fertilizer, F kilograms, and insecticide, P kilograms.Every square kilometer of wheat requires F1 kilograms of fertilizer, and P1 kilograms of insecticide, while everysquare kilometer of barley requires F2 kilograms of fertilizer, and P2 kilograms of insecticide. Let S1 be the sellingprice of wheat per square kilometer, and S2 be the price of barley. If we denote the area of land planted with wheatand barley by x1 and x2 respectively, then profit can be maximized by choosing optimal values for x1 and x2. Thisproblem can be expressed with the following linear programming problem in the standard form:

Linear programming 101

Maximize: S1x1 + S2x2(maximize the revenue—revenue is the "objective function")

Subject to: 0 ≤ x1 + x2 ≤ L (limit on total area)

0 ≤ F1x1 + F2x2 ≤ F (limit on fertilizer)

0 ≤ P1x1 + P2x2 ≤ P (limit on insecticide)

x1 ≥ 0, x2 ≥ 0 (cannot plant a negative area).

Which in matrix form becomes:

maximize

subject to

Augmented form (slack form)Linear programming problems must be converted into augmented form before being solved by the simplexalgorithm. This form introduces non-negative slack variables to replace inequalities with equalities in theconstraints. The problem can then be written in the following block matrix form:

Maximize Z:

x, xs ≥ 0where xs are the newly introduced slack variables, and Z is the variable to be maximized.

ExampleThe example above is converted into the following augmented form:

Maximize: S1x1 + S2x2(objective function)

Subject to: x1 + x2 + x3 = L (augmented constraint)

F1x1 + F2x2 + x4 = F (augmented constraint)

P1x1 + P2x2 + x5 = P (augmented constraint)

x1, x2, x3, x4, x5 ≥ 0.

where x3, x4, x5 are (non-negative) slack variables, representing in this example the unused area, the amount ofunused fertilizer, and the amount of unused insecticide.In matrix form this becomes:

Maximize Z:

Linear programming 102

DualityEvery linear programming problem, referred to as a primal problem, can be converted into a dual problem, whichprovides an upper bound to the optimal value of the primal problem. In matrix form, we can express the primalproblem as:

Maximize cTx subject to Ax ≤ b, x ≥ 0;with the corresponding symmetric dual problem,

Minimize bTy subject to ATy ≥ c, y ≥ 0.An alternative primal formulation is:

Maximize cTx subject to Ax ≤ b;with the corresponding asymmetric dual problem,

Minimize bTy subject to ATy = c, y ≥ 0.There are two ideas fundamental to duality theory. One is the fact that (for the symmetric dual) the dual of a duallinear program is the original primal linear program. Additionally, every feasible solution for a linear program givesa bound on the optimal value of the objective function of its dual. The weak duality theorem states that the objectivefunction value of the dual at any feasible solution is always greater than or equal to the objective function value ofthe primal at any feasible solution. The strong duality theorem states that if the primal has an optimal solution, x*,then the dual also has an optimal solution, y*, such that cTx*=bTy*.A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then thedual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must beinfeasible. However, it is possible for both the dual and the primal to be infeasible (See also Farkas' lemma).

ExampleRevisit the above example of the farmer who may grow wheat and barley with the set provision of some L land, Ffertilizer and P insecticide. Assume now that unit prices for each of these means of production (inputs) are set by aplanning board. The planning board's job is to minimize the total cost of procuring the set amounts of inputs whileproviding the farmer with a floor on the unit price of each of his crops (outputs), S1 for wheat and S2 for barley. Thiscorresponds to the following linear programming problem:

Minimize: LyL + FyF + PyP(minimize the total cost of the means of production as the "objective function")

Subject to: yL + F1yF + P1yP ≥ S1(the farmer must receive no less than S1 for his wheat)

yL + F2 yF + P2yP ≥ S2(the farmer must receive no less than S2 for his barley)

yL ≥ 0, yF ≥ 0, yP ≥ 0 (prices cannot be negative).

Which in matrix form becomes:

Minimize:

Subject to:

The primal problem deals with physical quantities. With all inputs available in limited quantities, and assuming theunit prices of all outputs is known, what quantities of outputs to produce so as to maximize total revenue? The dualproblem deals with economic values. With floor guarantees on all output unit prices, and assuming the availablequantity of all inputs is known, what input unit pricing scheme to set so as to minimize total expenditure?

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To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by outputtype. To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by inputtype.The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space,input quantities in this example. The coefficients used to compute the objective in the primal space bound theinequalities in the dual space, output unit prices in this example.Both the primal and the dual problems make use of the same matrix. In the primal space, this matrix expresses theconsumption of physical quantities of inputs necessary to produce set quantities of outputs. In the dual space, itexpresses the creation of the economic values associated with the outputs from set input unit prices.Since each inequality can be replaced by an equality and a slack variable, this means each primal variablecorresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable. This relationallows us to complementary slackness.

Another exampleSometimes, one may find it more intuitive to obtain the dual program without looking at program matrix. Considerthe following linear program:

minimize

subject to ,

,

,

We have m + n conditions and all variables are non-negative. We shall define m + n dual variables: yj and si. We get:

minimize

subject to ,

,

,

,

Since this is a minimization problem, we would like to obtain a dual program that is a lower bound of the primal. Inother words, we would like the sum of all right hand side of the constraints to be the maximal under the conditionthat for each primal variable the sum of its coefficients do not exceed its coefficient in the linear function. Forexample, x1 appears in n + 1 constraints. If we sum its constraints' coefficients we geta1,1y1 + a1,2y2 + ... + a1,nyn + f1s1. This sum must be at most c1. As a result we get:

Linear programming 104

maximize

subject to ,

,

,

Note that we assume in our calculations steps that the program is in standard form. However, any linear programmay be transformed to standard form and it is therefore not a limiting factor.

Covering-packing dualitiesA covering LP is a linear program of the form:

Minimize: bTy,Subject to: ATy ≥ c, y ≥ 0,

such that the matrix A and the vectors b and c are non-negative.The dual of a covering LP is a packing LP, a linear program of the form:

Maximize: cTx,Subject to: Ax ≤ b, x ≥ 0,

such that the matrix A and the vectors b and c are non-negative.

ExamplesCovering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and areimportant in the study of approximation algorithms.[2] For example, the LP relaxations of the set packing problem,the independent set problem, and the matching problem are packing LPs. The LP relaxations of the set coverproblem, the vertex cover problem, and the dominating set problem are also covering LPs.Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one constraint foreach vertex of the graph and one variable for each independent set of the graph.

Complementary slacknessIt is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using thecomplementary slackness theorem. The theorem states:Suppose that x = (x1, x2, ... , xn) is primal feasible and that y = (y1, y2, ... , ym) is dual feasible. Let (w1, w2, ..., wm)denote the corresponding primal slack variables, and let (z1, z2, ... , zn) denote the corresponding dual slackvariables. Then x and y are optimal for their respective problems if and only if• xjzj = 0, for j = 1, 2, ... , n, and• wiyi = 0, for i = 1, 2, ... , m.So if the i-th slack variable of the primal is not zero, then the i-th variable of the dual is equal zero. Likewise, if thej-th slack variable of the dual is not zero, then the j-th variable of the primal is equal to zero.This necessary condition for optimality conveys a fairly simple economic principle. In standard form (whenmaximizing), if there is slack in a constrained primal resource (i.e., there are "leftovers"), then additional quantitiesof that resource must have no value. Likewise, if there is slack in the dual (shadow) price non-negativity constraintrequirement , i.e., the price is not zero, then there must be scarce supplies (no "leftovers").

Linear programming 105

Theory

Existence of optimal solutionsGeometrically, the linear constraints define the feasible region, which is a convex polyhedron. A linear function is aconvex function, which implies that every local minimum is a global minimum; similarly, a linear function is aconcave function, which implies that every local maximum is a global maximum.Optimal solution need not exist, for two reasons. First, if two constraints are inconsistent, then no feasible solutionexists: For instance, the constraints x ≥ 2 and x ≤ 1 cannot be satisfied jointly; in this case, we say that the LP isinfeasible. Second, when the polytope is unbounded in the direction of the gradient of the objective function (wherethe gradient of the objective function is the vector of the coefficients of the objective function), then no optimalvalue is attained.

Optimal vertices (and rays) of polyhedraOtherwise, if a feasible solution exists and if the (linear) objective function is bounded, then the optimum value isalways attained on the boundary of optimal level-set, by the maximum principle for convex functions (alternatively,by the minimum principle for concave functions): Recall that linear functions are both convex and concave.However, some problems have distinct optimal solutions: For example, the problem of finding a feasible solution toa system of linear inequalities is a linear programming problem in which the objective function is the zero function(that is, the constant function taking the value zero everywhere): For this feasibility problem with the zero-functionfor its objective-function, if there are two distinct solutions, then every convex combination of the solutions is asolution.The vertices of the polytope are also called basic feasible solutions. The reason for this choice of name is as follows.Let d denote the number of variables. Then the fundamental theorem of linear inequalities implies (for feasibleproblems) that for every vertex x* of the LP feasible region, there exists a set of d (or fewer) inequality constraintsfrom the LP such that, when we treat those d constraints as equalities, the unique solution is x*. Thereby we canstudy these vertices by means of looking at certain subsets of the set of all constraints (a discrete set), rather than thecontinuum of LP solutions. This principle underlies the simplex algorithm for solving linear programs.

Algorithms

A series of linear constraints on two variablesproduces a region of possible values for those

variables. Solvable problems will have a feasibleregion in the shape of a simple polygon.

Basis exchange algorithms

Simplex algorithm of Dantzig

The simplex algorithm, developed by George Dantzig in 1947, solvesLP problems by constructing a feasible solution at a vertex of thepolytope and then walking along a path on the edges of the polytope tovertices with non-decreasing values of the objective function until anoptimum is reached. In many practical problems, "stalling" occurs:Many pivots are made with no increase in the objective function.[3] [4]

In rare practical problems, the usual versions of the simplex algorithmmay actually "cycle".[4] To avoid cycles, researchers developed newpivoting rules .[5] [6] [] [] [7] [8]

Linear programming 106

In practice, the simplex algorithm is quite efficient and can be guaranteed to find the global optimum if certainprecautions against cycling are taken. The simplex algorithm has been proved to solve "random" problemsefficiently, i.e. in a cubic number of steps,[9] which is similar to its behavior on practical problems.[3] [10]

However, the simplex algorithm has poor worst-case behavior: Klee and Minty constructed a family of linearprogramming problems for which the simplex method takes a number of steps exponential in the problem size.[3] [][6] In fact, for some time it was not known whether the linear programming problem was solvable in polynomial time(complexity class P).

Criss-cross algorithm

Like the simplex algorithm of Dantzig, the criss-cross algorithm is a basis-exchange algorithm that pivots betweenbases. However, the criss-cross algorithm need not maintain feasibility, but can pivot rather from a feasible basis toan infeasible basis. The criss-cross algorithm does not have polynomial time-complexity for linear programming.Both algorithms visit all 2D corners of a (perturbed) cube in dimension D, the Klee–Minty cube (after Victor Kleeand George J. Minty), in the worst case.[11] [8]

Interior point

Ellipsoid algorithm, following Khachiyan

This is the first worst-case polynomial-time algorithm for linear programming. To solve a problem which has nvariables and can be encoded in L input bits, this algorithm uses O(n4L) pseudo-arithmetic operations on numberswith O(L) digits. Khachiyan's algorithm and his long standing issue was resolved by Leonid Khachiyan in 1979 withthe introduction of the ellipsoid method. The convergence analysis have (real-number) predecessors, notably theiterative methods developed by Naum Z. Shor and the approximation algorithms by Arkadi Nemirovski and D.Yudin.

Projective algorithm of Karmarkar

Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linearprograms. The algorithm was not a computational break-through, as the simplex method is more efficient for all butspecially constructed families of linear programs.However, Khachiyan's algorithm inspired new lines of research in linear programming. In 1984, N. Karmarkarproposed a projective method for linear programming. Karmarkar's algorithm improved on Khachiyan's worst-casepolynomial bound (giving ). Karmarkar claimed that his algorithm was much faster in practical LP thanthe simplex method, a claim that created great interest in interior-point methods. Its projective geometry isinteresting.[12]

Path-following algorithms

In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on apolyhedral set, interior-point methods move through the interior of the feasible region. Since then, manyinterior-point methods have been proposed and analyzed. Early successful implementations were based on affinescaling variants of the method. For both theoretical and practical purposes, barrier function or path-followingmethods have been the most popular since the 1990s.[13]

Linear programming 107

Comparison of interior-point methods versus simplex algorithmsThe current opinion is that the efficiency of good implementations of simplex-based methods and interior pointmethods are similar for routine applications of linear programming.[13] However, for specific types of LP problems,it may be that one type of solver is better than another (sometimes much better).LP solvers are in widespread use for optimization of various problems in industry, such as optimization of flow intransportation networks.[14]

Open problems and recent workThere are several open problems in the theory of linear programming, the solution of which would representfundamental breakthroughs in mathematics and potentially major advances in our ability to solve large-scale linearprograms.• Does LP admit a strongly polynomial-time algorithm?• Does LP admit a strongly polynomial algorithm to find a strictly complementary solution?• Does LP admit a polynomial algorithm in the real number (unit cost) model of computation?This closely related set of problems has been cited by Stephen Smale as among the 18 greatest unsolved problems ofthe 21st century. In Smale's words, the third version of the problem "is the main unsolved problem of linearprogramming theory." While algorithms exist to solve linear programming in weakly polynomial time, such as theellipsoid methods and interior-point techniques, no algorithms have yet been found that allow stronglypolynomial-time performance in the number of constraints and the number of variables. The development of suchalgorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well.Although the Hirsch conjecture was recently disproved for higher dimensions, it still leaves the following questionsopen.• Are there pivot rules which lead to polynomial-time Simplex variants?• Do all polytopal graphs have polynomially-bounded diameter?These questions relate to the performance analysis and development of Simplex-like methods. The immenseefficiency of the Simplex algorithm in practice despite its exponential-time theoretical performance hints that theremay be variations of Simplex that run in polynomial or even strongly polynomial time. It would be of great practicaland theoretical significance to know whether any such variants exist, particularly as an approach to deciding if LPcan be solved in strongly polynomial time.The Simplex algorithm and its variants fall in the family of edge-following algorithms, so named because they solvelinear programming problems by moving from vertex to vertex along edges of a polytope. This means that theirtheoretical performance is limited by the maximum number of edges between any two vertices on the LP polytope.As a result, we are interested in knowing the maximum graph-theoretical diameter of polytopal graphs. It has beenproved that all polytopes have subexponential diameter. The recent disprove of the Hirsch conjecture is the first stepto prove whether any polytope has superpolynomial diameter. If any such polytopes exist, then no edge-followingvariant can run in polynomial time. Questions about polytope diameter are of independent mathematical interest.Simplex pivot methods preserve primal (or dual) feasibility. On the other hand, criss-cross pivot methods do notpreserve (primal or dual) feasibility—they may visit primal feasible, dual feasible or primal-and-dual infeasiblebases in any order. Pivot methods of this type have been studied since the 1970s. Essentially, these methods attemptto find the shortest pivot path on the arrangement polytope under the linear programming problem. In contrast topolytopal graphs, graphs of arrangement polytopes are known to have small diameter, allowing the possibility ofstrongly polynomial-time criss-cross pivot algorithm without resolving questions about the diameter of generalpolytopes.[8]

Linear programming 108

Integer unknownsIf the unknown variables are all required to be integers, then the problem is called an integer programming (IP) orinteger linear programming (ILP) problem. In contrast to linear programming, which can be solved efficiently inthe worst case, integer programming problems are in many practical situations (those with bounded variables)NP-hard. 0-1 integer programming or binary integer programming (BIP) is the special case of integerprogramming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classifiedas NP-hard, and in fact the decision version was one of Karp's 21 NP-complete problems.If only some of the unknown variables are required to be integers, then the problem is called a mixed integerprogramming (MIP) problem. These are generally also NP-hard.There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notablyproblems where the constraint matrix is totally unimodular and the right-hand sides of the constraints are integers.Advanced algorithms for solving integer linear programs include:• cutting-plane method• branch and bound• branch and cut• branch and price• if the problem has some extra structure, it may be possible to apply delayed column generation.Such integer-programming algorithms are discussed by Padberg and in Beasley.

Integral Linear ProgramsA linear program in real variables is said to be integral if it has at least one optimal solution which is integral.Likewise, a polyhedron is said to be integral if for all bounded feasible objective functionsc, the linear program has an optimum with integer coordinates. As observed by Edmondsand Giles in 1977, one can equivalently say that a polyhedron is integral if for every bounded feasible integralobjective function c, the optimal value of the linear progam is an integer.Integral linear programs are of central importance in the polyhedral aspect of combinatorial optimization since theyprovide an alternate characterization of a problem. Specifically, for any problem, the convex hull of the solutions isan integral polyhedron; if this polyhedron has a nice/compact description, then we can efficiently find the optimalfeasible solution under any linear objective. Conversely, if we can prove that a linear programming relaxation isintegral, then it is the desired description of the convex hull of feasible (integral) solutions.Note that terminology is not consistent throughout the literature, so one should be careful to distinguish thefollowing two concepts,• in an integer linear program, described in the previous section, variables are forcibly constrained to be integers,

and this problem is NP-hard in general,• in an integral linear program, described in this section, variables are not constrained to be integers but rather one

has proven somehow that the continuous problem always has an integral optimal value (assuming c is integral),and this optimal value may be found efficiently since all polynomial-size linear programs can be solved inpolynomial time.

One common way of proving that a polyhedron is integral is to show that it is totally unimodular. There are othergeneral methods including the integer decomposition property and total dual integrality. Other specific well-knownintegral LPs include the matching polytope, lattice polyhedra, submodular flow polyhedra, and the intersection of 2generalized polymatroids/g-polymatroids --- e.g. see Schrijver 2003.A bounded integral polyhedron is sometimes called a convex lattice polytope, particularly in two dimensions.

Linear programming 109

Solvers and scripting (programming) languagesFree open-source permissive licenses:

Name License Brief info

OpenOpt BSD Universal cross-platform numerical optimization framework,see its LP [15] page and other problems [16] involved

pulp-or [17] BSD Python module for modeling and solving linear programming problems

Pyomo [18] BSD Python module for formulating linear programming problems with abstract models

Free open-source copyleft (reciprocal) licenses:

Name License Brief info

LP_Solve [19] LGPL User-friendly linear and integer programming solver. Also provides DLL for program integration.

Cassowaryconstraint solver

LGPL an incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities.

CVXOPT [20] GPL general purpose convex optimization solver written in Python, with a C API, and calls external routines (e.g. BLAS,LAPACK, FFTW) for numerical computations. Has its own solvers, but can also call glpk or MOSEK[21] ifinstalled

glpk GPL GNU Linear Programming Kit, a free LP/MILP solver. Uses GNU MathProg modelling language.

Qoca GPL a library for incrementally solving systems of linear equations with various goal functions

CBC [22] CPL a MIP solver from COIN-OR

CLP CPL an LP solver from COIN-OR

R-Project GPL a programming language and software environment for statistical computing and graphics

CVX [23] GPL MATLAB based modeling system for convex optimization, including linear programs; calls either SDPT3 orSeDuMi as a solver

CVXMOD [24] GPL Python based modeling system, similar to CVX. It calls CVXOPT as its solver. It is still in alpha release, as of 2009

SDPT3 [25] GPL MATLAB based convex optimization solver

SeDuMi [26] GPL MATLAB based convex optimization solver

MINTO (Mixed Integer Optimizer, an integer programming solver which uses branch and bound algorithm) haspublicly available source code[27] but not open source.Proprietary:

Linear programming 110

Name Brief info

APMonitor

AIMMS

AMPL A popular modeling language for large-scale linear, mixed integer and nonlinear optimisation with a free student versionavailable.

Analytica [28] Optimization modeling software that incorporates state-of-the-art algorithms for linear and nonlinear optimization. SupportsLP, NLP, QP, continuous and integer optimization.

CPLEX Popular solver with an API for several programming languages, and also has a modelling language and works with AIMMS,AMPL, GAMS, MPL, OpenOpt, OPL Development Studio, and TOMLAB

EXCEL SolverFunction

FortMP

GAMS

GIPALS

Gurobi

IMSL NumericalLibraries

Collections of math and statistical algorithms available in C/C++, Fortran, Java and C#/.NET. Optimization routines in theIMSL Libraries include unconstrained, linearly and nonlinearly constrained minimizations, and linear programmingalgorithms.

Lingo

LPL

LiPS [29]

(freeware)

Linear Program Solver (LiPS) is intended for solving linear programming problems. Main features: easy to use graphicalinterface, sensitivity analysis, goal and mixed integer programming solver. LiPS supports MPS and simple LP format (likelpsolve).

MATLAB A general-purpose and matrix-oriented programming-language for numerical computing. Linear programming in MATLABrequires the Optimization Toolbox in addition to the base MATLAB product; available routines include BINTPROG andLINPROG

Mathematica A general-purpose programming-language for mathematics, including symbolic and numerical capabilities.

MOPS

MOSEK A solver for large scale optimization with API for several languages (C++,java,.net, Matlab and python).

NMath Stats A general-purpose .NET statistical library containing a simplex solver.[30]

OptimJ A Java-based modeling language for optimization with a free version available. [31] [32]

SAS

SCIP A general-purpose constraint integer programming solver with an emphasis on MIP. Free for academic use and available insource code.

Solver Foundation A .NET platform for modeling, scheduling, and optimization.

SoPlex [33] The Sequential object-oriented simPlex: a general-purpose LP solver. Free for academic use and available in source code.

SuanShu A Java-based math library that supports linear programming and other kinds of numerical optimization. [34] .

TOMLAB

VisSim A visual block diagram language for simulation of dynamical systems.

Xpress

Linear programming 111

Notes[1] See his 1940 paper listed below[2] Vazirani (2001, p. 112)[3] Dantzig & Thapa (2003)[4] Padberg (1999)[5] Bland (1977)[6] Murty (1983)[7] Papadimitriou (Steiglitz)[8] Fukuda & Terlaky (1997): Fukuda, Komei; Terlaky, Tamás (1997). "Criss-cross methods: A fresh view on pivot algorithms" (http:/ / www.

cas. mcmaster. ca/ ~terlaky/ files/ crisscross. ps). Mathematical Programming: Series B (Amsterdam: North-Holland Publishing Co.) 79(Papers from the 16th International Symposium on Mathematical Programming held in Lausanne, 1997): pp. 369—395.doi:10.1016/S0025-5610(97)00062-2. MR1464775. .

[9] Borgwardt (1987)[10] Todd (2002)[11] Roos (1990): Roos, C. (1990). "An exponential example for Terlaky's pivoting rule for the criss-cross simplex method" (http:/ / dx. doi. org/

10. 1007/ BF01585729). Mathematical Programming. Series A 46 (1). doi:10.1007/BF01585729. MR1045573. .[12] Strang, Gilbert (1 June 1987). "Karmarkar’s algorithm and its place in applied mathematics" (http:/ / dx. doi. org/ 10. 1007/ BF03025891).

The Mathematical Intelligencer (New York: Springer) 9 (2): 4–10. doi:10.1007/BF03025891. ISSN 0343-6993. MR883185. .[13] Gondzio & Terlaky (1996)[14] For solving network-flow problems in transportation networks, specialized implementations of the simplex algorithm can dramatically

improve its efficiency. Dantzig & Thapa (2003)[15] http:/ / openopt. org/ LP[16] http:/ / openopt. org/ Problems[17] http:/ / pulp-or. googlecode. com[18] https:/ / software. sandia. gov/ pyomo[19] http:/ / lpsolve. sourceforge. net/ 5. 5/[20] http:/ / abel. ee. ucla. edu/ cvxopt/[21] http:/ / www. mosek. com/[22] http:/ / www. coin-or. org/ projects/ Cbc. xml[23] http:/ / cvxr. com/ cvx[24] http:/ / cvxmod. net/[25] http:/ / www. math. nus. edu. sg/ ~mattohkc/ sdpt3. html[26] http:/ / sedumi. ie. lehigh. edu/[27] http:/ / coral. ie. lehigh. edu/ ~minto/ download. html[28] http:/ / www. lumina. com/ products/ analytica-optimizer/[29] http:/ / sourceforge. net/ projects/ lipside/[30] Linear programming page at CenterSpace Software (http:/ / www. centerspace. net/ landing. php?id=lp)[31] http:/ / www. in-ter-trans. eu/ resources/ Zesch_Hellingrath_2010_Integrated+ Production-Distribution+ Planning. pdf OptimJ used in an

optimization model for mixed-model assembly lines, University of Münster[32] http:/ / www. aaai. org/ ocs/ index. php/ AAAI/ AAAI10/ paper/ viewFile/ 1769/ 2076 OptimJ used in an Approximate Subgame-Perfect

Equilibrium Computation Technique for Repeated Games[33] http:/ / soplex. zib. de/[34] http:/ / www. numericalmethod. com

References• L.V. Kantorovich: A new method of solving some classes of extremal problems, Doklady Akad Sci USSR, 28,

1940, 211-214.• G.B Dantzig: Maximization of a linear function of variables subject to linear inequalities, 1947. Published pp.

339-347 in T.C. Koopmans (ed.):Activity Analysis of Production and Allocation, New York-London 1951 (Wiley& Chapman-Hall)

• J. E. Beasley, editor. Advances in Linear and Integer Programming. Oxford Science, 1996. (Collection ofsurveys)

• R. G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2 (1977) 103–107.• Karl-Heinz Borgwardt, The Simplex Algorithm: A Probabilistic Analysis, Algorithms and Combinatorics, Volume

1, Springer-Verlag, 1987. (Average behavior on random problems)

Linear programming 112

• Richard W. Cottle, ed. The Basic George B. Dantzig. Stanford Business Books, Stanford University Press,Stanford, California, 2003. (Selected papers by George B. Dantzig)

• George B. Dantzig and Mukund N. Thapa. 1997. Linear programming 1: Introduction. Springer-Verlag.• George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions.

Springer-Verlag. (Comprehensive, covering e.g. pivoting and interior-point algorithms, large-scale problems,decomposition following Dantzig-Wolfe and Benders, and introducing stochastic programming.)

• Edmonds, J. and Giles, R., "A min-max relation for submodular functions on graphs," Ann. Discrete Math., v1,pp. 185-204, 1977

• Fukuda, Komei; Terlaky, Tamás (1997). "Criss-cross methods: A fresh view on pivot algorithms" (http:/ / www.cas. mcmaster. ca/ ~terlaky/ files/ crisscross. ps). Mathematical Programming: Series B (Amsterdam:North-Holland Publishing Co.) 79 (Papers from the 16th International Symposium on Mathematical Programmingheld in Lausanne, 1997): pp. 369—395. doi:10.1016/S0025-5610(97)00062-2. MR1464775.

• Gondzio, Jacek; Terlaky, Tamás (1996). "3 A computational view of interior point methods" (http:/ / www.maths. ed. ac. uk/ ~gondzio/ CV/ oxford. ps). In J. E. Beasley. Advances in linear and integer programming.Oxford Lecture Series in Mathematics and its Applications. 4. New York: Oxford University Press. pp. 103–144.MR1438311. Postscript file at website of Gondzio (http:/ / www. maths. ed. ac. uk/ ~gondzio/ CV/ oxford. ps)and at McMaster University website of Terlaky (http:/ / www. cas. mcmaster. ca/ ~terlaky/ files/ dut-twi-94-73.ps. gz).

• Murty, Katta G. (1983). Linear programming. New York: John Wiley & Sons, Inc.. pp. xix+482.ISBN 0-471-09725-X. MR720547. (comprehensive reference to classical approaches).

• Evar D. Nering and Albert W. Tucker, 1993, Linear Programs and Related Problems, Academic Press.(elementary)

• M. Padberg, Linear Optimization and Extensions, Second Edition, Springer-Verlag, 1999. (carefully writtenaccount of primal and dual simplex algorithms and projective algorithms, with an introduction to integer linearprogramming --- featuring the traveling salesman problem for Odysseus.)

• Christos H. Papadimitriou and Kenneth Steiglitz, Combinatorial Optimization: Algorithms and Complexity,Corrected republication with a new preface, Dover. (computer science)

• Michael J. Todd (February 2002). "The many facets of linear programming". Mathematical Programming 91 (3).(Invited survey, from the International Symposium on Mathematical Programming.)

• Vazirani, Vijay V. (2001). Approximation Algorithms. Springer-Verlag. ISBN 3-540-65367-8. (Computerscience)

Further readingA reader may consider beginning with Nering and Tucker, with the first volume of Dantzig and Thapa, or withWilliams.• Dmitris Alevras and Manfred W. Padberg, Linear Optimization and Extensions: Problems and Extensions,

Universitext, Springer-Verlag, 2001. (Problems from Padberg with solutions.)• Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry

(2nd revised ed.). Springer-Verlag. ISBN 3-540-65620-0. Chapter 4: Linear Programming: pp. 63–94. Describesa randomized half-plane intersection algorithm for linear programming.

• Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory ofNP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A6: MP1: INTEGER PROGRAMMING, pg.245.(computer science, complexity theory)

• Bernd Gärtner, Jiří Matoušek (2006). Understanding and Using Linear Programming, Berlin: Springer. ISBN3-540-30697-8 (elementary introduction for mathematicians and computer scientists)

• Cornelis Roos, Tamás Terlaky, Jean-Philippe Vial, Interior Point Methods for Linear Optimization, SecondEdition, Springer-Verlag, 2006. (Graduate level)

Linear programming 113

• Alexander Schrijver (2003). Combinatorial optimization: polyhedra and efficiency. Springer.• Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0-471-98232-6

(mathematical)• Robert J. Vanderbei, Linear Programming: Foundations and Extensions (http:/ / www. princeton. edu/ ~rvdb/

LPbook/ ), 3rd ed., International Series in Operations Research & Management Science, Vol. 114, SpringerVerlag, 2008. ISBN 978-0-387-74387-5. (An on-line second edition was formerly available. Vanderbei's site stillcontains extensive materials.)

• H. P. Williams, Model Building in Mathematical Programming, Third revised Edition, 1990. (Modeling)• Stephen J. Wright, 1997, Primal-Dual Interior-Point Methods, SIAM. (Graduate level)• Yinyu Ye, 1997, Interior Point Algorithms: Theory and Analysis, Wiley. (Advanced graduate-level)• Ziegler, Günter M., Chapters 1–3 and 6–7 in Lectures on Polytopes, Springer-Verlag, New York, 1994.

(Geometry)

External links• Guidance on Formulating LP problems (http:/ / people. brunel. ac. uk/ ~mastjjb/ jeb/ or/ lp. html)• Mathematical Programming Glossary (http:/ / glossary. computing. society. informs. org/ )• The linear programming FAQ (http:/ / wiki. mcs. anl. gov/ NEOS/ index. php/ Linear_Programming_FAQ)• 2009 Linear Programming Software Survey (http:/ / www. lionhrtpub. com/ orms/ surveys/ LP/ LP-survey. html)

- OR/MS Today• George Dantzig (http:/ / www. stanford. edu/ group/ SOL/ dantzig. html)• Dual solution of a large marketing optimization problem (http:/ / home. comcast. net/ ~raagnew/ Downloads/

Dual_Solution_Marketing_Optimization. pdf)PDF (105 KB)

Nonlinear programming 114

Nonlinear programmingIn mathematics, nonlinear programming (NLP) is the process of solving a system of equalities and inequalities,collectively termed constraints, over a set of unknown real variables, along with an objective function to bemaximized or minimized, where some of the constraints or the objective function are nonlinear.[1]

ApplicabilityA typical nonconvex problem is that of optimising transportation costs by selection from a set of transportionmethods, one or more of which exhibit economies of scale, with various connectivities and capacity constraints. Anexample would be petroleum product transport given a selection or combination of pipeline, rail tanker, road tanker,river barge, or coastal tankship. Owing to economic batch size the cost functions may have discontinuities inaddition to smooth changes.

Mathematical formulation of the problemThe problem can be stated simply as:

to maximize some variable such as product throughputor

to minimize a cost functionwhere

Methods for solving the problemIf the objective function f is linear and the constrained space is a polytope, the problem is a linear programmingproblem, which may be solved using well known linear programming solutions.If the objective function is concave (maximization problem), or convex (minimization problem) and the constraintset is convex, then the program is called convex and general methods from convex optimisation can be used in mostcases.If the objective function is a ratio of a concave and a convex function (in the maximization case) and the constraintsare convex, then the problem can be transformed to a convex optimization problem using fractional programmingtechniques.Several methods are available for solving nonconvex problems. One approach is to use special formulations of linearprogramming problems. Another method involves the use of branch and bound techniques, where the program isdivided into subclasses to be solved with convex (minimization problem) or linear approximations that form a lowerbound on the overall cost within the subdivision. With subsequent divisions, at some point an actual solution will beobtained whose cost is equal to the best lower bound obtained for any of the approximate solutions. This solution isoptimal, although possibly not unique. The algorithm may also be stopped early, with the assurance that the bestpossible solution is within a tolerance from the best point found; such points are called ε-optimal. Terminating toε-optimal points is typically necessary to ensure finite termination. This is especially useful for large, difficultproblems and problems with uncertain costs or values where the uncertainty can be estimated with an appropriatereliability estimation.Under differentiability and constraint qualifications, the Karush–Kuhn–Tucker (KKT) conditions provide necessaryconditions for a solution to be optimal. Under convexity, these conditions are also sufficient.

Nonlinear programming 115

Examples

2-dimensional example

The intersection of the line with the constrainedspace represents the solution

A simple problem can be defined by the constraintsx1 ≥ 0x2 ≥ 0x1

2 + x22 ≥ 1

x12 + x2

2 ≤ 2with an objective function to be maximized

f(x) = x1 + x2where x = (x1, x2). Solve 2-D Problem [2].

3-dimensional example

The intersection of the top surface with theconstrained space in the center represents the

solution

Another simple problem can be defined by the constraintsx1

2 − x22 + x3

2 ≤ 2x1

2 + x22 + x3

2 ≤ 10with an objective function to be maximized

f(x) = x1x2 + x2x3where x = (x1, x2, x3). Solve 3-D Problem [3].

References[1] Bertsekas, Dimitri P. (1999). Nonlinear Programming (Second ed.). Cambridge,

MA.: Athena Scientific. ISBN 1-886529-00-0.

[2] http:/ / apmonitor. com/ online/ view_pass. php?f=2d. apm[3] http:/ / apmonitor. com/ online/ view_pass. php?f=3d. apm

Further reading• Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN

0-486-43227-0.• Bazaraa, Mokhtar S. and Shetty, C. M. (1979). Nonlinear programming. Theory and algorithms. John Wiley &

Sons. ISBN 0-471-78610-1.• Bertsekas, Dimitri P. (1999). Nonlinear Programming: 2nd Edition. Athena Scientific. ISBN 1-886529-00-0.• Bonnans, J. Frédéric; Gilbert, J. Charles; Lemaréchal, Claude; Sagastizábal, Claudia A. (2006). Numerical

optimization: Theoretical and practical aspects (http:/ / www. springer. com/ mathematics/ applications/ book/978-3-540-35445-1). Universitext (Second revised ed. of translation of 1997 French ed.). Berlin: Springer-Verlag.pp. xiv+490. doi:10.1007/978-3-540-35447-5. ISBN 3-540-35445-X. MR2265882.

• Luenberger, David G.; Ye, Yinyu (2008). Linear and nonlinear programming. International Series in Operations Research & Management Science. 116 (Third ed.). New York: Springer. pp. xiv+546. ISBN 978-0-387-74502-2.

Nonlinear programming 116

MR2423726• Nocedal, Jorge and Wright, Stephen J. (1999). Numerical Optimization. Springer. ISBN 0-387-98793-2.• Jan Brinkhuis and Vladimir Tikhomirov, 'Optimization: Insights and Applications', 2005, Princeton University

Press

External links• Nonlinear programming FAQ (http:/ / www-unix. mcs. anl. gov/ otc/ Guide/ faq/ nonlinear-programming-faq.

html)• Mathematical Programming Glossary (http:/ / glossary. computing. society. informs. org/ )• Nonlinear Programming Survey OR/MS Today (http:/ / www. lionhrtpub. com/ orms/ surveys/ nlp/ nlp. html)• Overview of Optimization in Industry (http:/ / apmonitor. com/ wiki/ index. php/ Main/ Background)

Integer programmingAn integer programming problem is a mathematical optimization or feasibility program in which some or all of thevariables are restricted to be integers. In many settings the term refers to integer linear programming, which is alsoknown as mixed integer programming.Integer programming is NP-hard. A special case, 0-1 integer linear programming, in which unknowns are binary, isone of the Karp's 21 NP-complete problems.

Further reading• George L. Nemhauser; Laurence A. Wolsey (1988). Integer and combinatorial optimization. Wiley.

ISBN 9780471828198.• Alexander Schrijver (1998). Theory of linear and integer programming. John Wiley and Sons.

ISBN 9780471982326.• Laurence A. Wolsey (1998). Integer programming. Wiley. ISBN 9780471283669.• Dimitris Bertsimas; Robert Weismantel (2005). Optimization over integers. Dynamic Ideas.

ISBN 9780975914625.• John K. Karlof (2006). Integer programming: theory and practice. CRC Press. ISBN 9780849319143.• H. Paul Williams (2009). Logic and Integer Programming. Springer. ISBN 9780387922799.• Michael Jünger, Thomas M. Liebling, Denis Naddef, George Nemhauser, William R. Pulleyblank, et al., ed

(2009). 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer.ISBN 9783540682745.

• Der-San Chen; Robert G. Batson; Yu Dang (2010). Applied Integer Programming: Modeling and Solution. JohnWiley and Sons. ISBN 9780470373064.

External links• A Tutorial on Integer Programming [1]

References[1] http:/ / mat. gsia. cmu. edu/ orclass/ integer/ integer. html

Dynamic programming 117

Dynamic programmingIn mathematics and computer science, dynamic programming is a method for solving complex problems bybreaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlappingsubproblems which are only slightly smaller[1] and optimal substructure (described below). When applicable, themethod takes far less time than naive methods.The key idea behind dynamic programming is quite simple. In general, to solve a given problem, we need to solvedifferent parts of the problem (subproblems), then combine the solutions of the subproblems to reach an overallsolution. Often, many of these subproblems are really the same. The dynamic programming approach seeks to solveeach subproblem only once, thus reducing the number of computations. This is especially useful when the number ofrepeating subproblems is exponentially large.Top-down dynamic programming simply means storing the results of certain calculations, which are later used againsince the completed calculation is a sub-problem of a larger calculation. Bottom-up dynamic programming involvesformulating a complex calculation as a recursive series of simpler calculations.

HistoryThe term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process ofsolving problems where one needs to find the best decisions one after another. By 1953, he refined this to themodern meaning, referring specifically to nesting smaller decision problems inside larger decisions,[2] and the fieldwas thereafter recognized by the IEEE as a systems analysis and engineering topic. Bellman's contribution isremembered in the name of the Bellman equation, a central result of dynamic programming which restates anoptimization problem in recursive form.The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and also because itsounded impressive.[3] The word programming referred to the use of the method to find an optimal program, in thesense of a military schedule for training or logistics. This usage is the same as that in the phrases linearprogramming and mathematical programming, a synonym for mathematical optimization.[4]

Overview

Figure 1. Finding the shortest path in a graph usingoptimal substructure; a straight line indicates a single

edge; a wavy line indicates a shortest path between thetwo vertices it connects (other nodes on these paths are

not shown); the bold line is the overall shortest pathfrom start to goal.

Dynamic programming is both a mathematical optimizationmethod and a computer programming method. In both contexts itrefers to simplifying a complicated problem by breaking it downinto simpler subproblems in a recursive manner. While somedecision problems cannot be taken apart this way, decisions thatspan several points in time do often break apart recursively;Bellman called this the "Principle of Optimality". Likewise, incomputer science, a problem that can be broken down recursivelyis said to have optimal substructure.

If subproblems can be nested recursively inside larger problems,so that dynamic programming methods are applicable, then thereis a relation between the value of the larger problem and the valuesof the subproblems.[5] In the optimization literature this

relationship is called the Bellman equation.

Dynamic programming 118

Dynamic programming in mathematical optimizationIn terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking itdown into a sequence of decision steps over time. This is done by defining a sequence of value functions V1 , V2 , ...Vn , with an argument y representing the state of the system at times i from 1 to n. The definition of Vn(y) is thevalue obtained in state y at the last time n. The values Vi at earlier times i=n-1,n-2,...,2,1 can be found by workingbackwards, using a recursive relationship called the Bellman equation. For i=2,...n, Vi -1 at any state y is calculatedfrom Vi by maximizing a simple function (usually the sum) of the gain from decision i-1 and the function Vi at thenew state of the system if this decision is made. Since Vi has already been calculated for the needed states, the aboveoperation yields Vi -1 for those states. Finally, V1 at the initial state of the system is the value of the optimal solution.The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations alreadyperformed.

Dynamic programming in computer programmingThere are two key attributes that a problem must have in order for dynamic programming to be applicable: optimalsubstructure and overlapping subproblems. However, when the overlapping problems are much smaller than theoriginal problem, the strategy is called "divide and conquer" rather than "dynamic programming". This is whymergesort, quicksort, and finding all matches of a regular expression are not classified as dynamic programmingproblems.Optimal substructure means that the solution to a given optimization problem can be obtained by the combination ofoptimal solutions to its subproblems. Consequently, the first step towards devising a dynamic programming solutionis to check whether the problem exhibits such optimal substructure. Such optimal substructures are usually describedby means of recursion. For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibitsoptimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then thepath p1 from u to w and p2 from w to v are indeed the shortest paths between the corresponding vertices (by thesimple cut-and-paste argument described in CLRS). Hence, one can easily formulate the solution for finding shortestpaths in a recursive manner, which is what the Bellman-Ford algorithm does.Overlapping subproblems means that the space of subproblems must be small, that is, any recursive algorithmsolving the problem should solve the same subproblems over and over, rather than generating new subproblems. Forexample, consider the recursive formulation for generating the Fibonacci series: Fi = Fi-1 + Fi-2, with base caseF1=F2=1. Then F43 = F42 + F41, and F42 = F41 + F40. Now F41 is being solved in the recursive subtrees of both F43 aswell as F42. Even though the total number of subproblems is actually small (only 43 of them), we end up solving thesame problems over and over if we adopt a naive recursive solution such as this. Dynamic programming takesaccount of this fact and solves each subproblem only once. Note that the subproblems must be only slightly smaller(typically taken to mean a constant additive factor) than the larger problem; when they are a multiplicative factorsmaller the problem is no longer classified as dynamic programming.

Dynamic programming 119

Figure 2. The subproblemgraph for the Fibonacci

sequence. The fact that it isnot a tree indicates

overlapping subproblems.

This can be achieved in either of two ways:• Top-down approach: This is the direct fall-out of the recursive formulation of any

problem. If the solution to any problem can be formulated recursively using thesolution to its subproblems, and if its subproblems are overlapping, then one caneasily memoize or store the solutions to the subproblems in a table. Whenever weattempt to solve a new subproblem, we first check the table to see if it is alreadysolved. If a solution has been recorded, we can use it directly, otherwise we solvethe subproblem and add its solution to the table.

• Bottom-up approach: This is the more interesting case. Once we formulate thesolution to a problem recursively as in terms of its subproblems, we can tryreformulating the problem in a bottom-up fashion: try solving the subproblems firstand use their solutions to build-on and arrive at solutions to bigger subproblems.This is also usually done in a tabular form by iteratively generating solutions tobigger and bigger subproblems by using the solutions to small subproblems. Forexample, if we already know the values of F41 and F40, we can directly calculate the value of F42.

Some programming languages can automatically memoize the result of a function call with a particular set ofarguments, in order to speed up call-by-name evaluation (this mechanism is referred to as call-by-need). Somelanguages make it possible portably (e.g. Scheme, Common Lisp or Perl), some need special extensions (e.g. C++,see[6] ). Some languages have automatic memoization built in, such as tabled Prolog. In any case, this is onlypossible for a referentially transparent function.

Example: Mathematical optimization

Optimal consumption and savingA mathematical optimization problem that is often used in teaching dynamic programming to economists (because itcan be solved by hand[7] ) concerns a consumer who lives over the periods and must decide howmuch to consume and how much to save in each period.Let be consumption in period , and assume consumption yields utility as long as theconsumer lives. Assume the consumer is impatient, so that he discounts future utility by a factor each period,where . Let be capital in period . Assume initial capital is a given amount , and supposethat this period's capital and consumption determine next period's capital as , where is apositive constant and . Assume capital cannot be negative. Then the consumer's decision problem canbe written as follows:

subject to for all

Written this way, the problem looks complicated, because it involves solving for all the choice variablesand simultaneously. (Note that is not a choice variable—the consumer's

initial capital is taken as given.)The dynamic programming approach to solving this problem involves breaking it apart into a sequence of smallerdecisions. To do so, we define a sequence of value functions , for whichrepresent the value of having any amount of capital at each time . Note that , that is, there is(by assumption) no utility from having capital after death.The value of any quantity of capital at any previous time can be calculated by backward induction using the Bellmanequation. In this problem, for each , the Bellman equation is

Dynamic programming 120

subject to This problem is much simpler than the one we wrote down before, because it involves only two decision variables,

and . Intuitively, instead of choosing his whole lifetime plan at birth, the consumer can take things one stepat a time. At time , his current capital is given, and he only needs to choose current consumption andsaving .To actually solve this problem, we work backwards. For simplicity, the current level of capital is denoted as .

is already known, so using the Bellman equation once we can calculate , and so on until we get to, which is the value of the initial decision problem for the whole lifetime. In other words, once we know

, we can calculate , which is the maximum of ,where is the choice variable and .Working backwards, it can be shown that the value function at time is

where each is a constant, and the optimal amount to consume at time is

which can be simplified to

, and , and , etc.

We see that it is optimal to consume a larger fraction of current wealth as one gets older, finally consuming allremaining wealth in period , the last period of life.

Examples: Computer algorithms

Dijkstra's algorithm for the shortest path problemFrom a dynamic programming point of view, Dijkstra's algorithm for the shortest path problem is a successiveapproximation scheme that solves the dynamic programming functional equation for the shortest path problem by theReaching method.[8] [9] [10]

In fact, Dijkstra's explanation of the logic behind the algorithm,[11] namely

Problem 2. Find the path of minimum total length between two given nodes and .We use the fact that, if is a node on the minimal path from to , knowledge of the latter implies theknowledge of the minimal path from to .

is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.

Fibonacci sequenceHere is a naïve implementation of a function finding the nth member of the Fibonacci sequence, based directly onthe mathematical definition:

function fib(n)

if n = 0 return 0

if n = 1 return 1

return fib(n − 1) + fib(n − 2)

Notice that if we call, say, fib(5), we produce a call tree that calls the function on the same value many differenttimes:

Dynamic programming 121

1. fib(5)

2. fib(4) + fib(3)

3. (fib(3) + fib(2)) + (fib(2) + fib(1))

4. ((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))

5. (((fib(1) + fib(0)) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) +

fib(1))

In particular, fib(2) was calculated three times from scratch. In larger examples, many more values of fib, orsubproblems, are recalculated, leading to an exponential time algorithm.Now, suppose we have a simple map object, m, which maps each value of fib that has already been calculated toits result, and we modify our function to use it and update it. The resulting function requires only O(n) time insteadof exponential time:

var m := map(0 → 0, 1 → 1) function fib(n)

if map m does not contain key n

m[n] := fib(n − 1) + fib(n − 2) return m[n]

This technique of saving values that have already been calculated is called memoization; this is the top-downapproach, since we first break the problem into subproblems and then calculate and store values.In the bottom-up approach we calculate the smaller values of fib first, then build larger values from them. Thismethod also uses O(n) time since it contains a loop that repeats n − 1 times, however it only takes constant (O(1))space, in contrast to the top-down approach which requires O(n) space to store the map.

function fib(n)

var previousFib := 0, currentFib := 1

if n = 0

return 0

else if n = 1

return 1

repeat n − 1 times var newFib := previousFib + currentFib

previousFib := currentFib

currentFib := newFib

return currentFib

In both these examples, we only calculate fib(2) one time, and then use it to calculate both fib(4) andfib(3), instead of computing it every time either of them is evaluated. (Note the calculation of the Fibonaccisequence is used to demonstrate dynamic programming. An O(1) formula exists from which an arbitrary term can becalculated, which is more efficient than any dynamic programming technique.)

Dynamic programming 122

A type of balanced 0-1 matrixConsider the problem of assigning values, either zero or one, to the positions of an n × n matrix, n even, so thateach row and each column contains exactly n / 2 zeros and n / 2 ones. We ask how many different assignmentsthere are for a given . For example, when n = 4, four possible solutions are

There are at least three possible approaches: brute force, backtracking, and dynamic programming.Brute force consists of checking all assignments of zeros and ones and counting those that have balanced rows andcolumns ( zeros and ones). As there are possible assignments, this strategy is not practical except

maybe up to .Backtracking for this problem consists of choosing some order of the matrix elements and recursively placing onesor zeros, while checking that in every row and column the number of elements that have not been assigned plus thenumber of ones or zeros are both at least n / 2. While more sophisticated than brute force, this approach will visitevery solution once, making it impractical for n larger than six, since the number of solutions is already116963796250 for n = 8, as we shall see.Dynamic programming makes it possible to count the number of solutions without visiting them all. Imaginebacktracking values for the first row - what information would we require about the remaining rows, in order to beable to accurately count the solutions obtained for each first row values? We consider k × n boards, where 1 ≤ k≤ n, whose rows contain zeros and ones. The function f to which memoization is applied maps vectorsof n pairs of integers to the number of admissible boards (solutions). There is one pair for each column and its twocomponents indicate respectively the number of ones and zeros that have yet to be placed in that column. We seekthe value of ( arguments or one vector of elements). Theprocess of subproblem creation involves iterating over every one of possible assignments for the top row of

the board, and going through every column, subtracting one from the appropriate element of the pair for that column,depending on whether the assignment for the top row contained a zero or a one at that position. If any one of theresults is negative, then the assignment is invalid and does not contribute to the set of solutions (recursion stops).Otherwise, we have an assignment for the top row of the k × n board and recursively compute the number ofsolutions to the remaining (k - 1) × n board, adding the numbers of solutions for every admissible assignment ofthe top row and returning the sum, which is being memoized. The base case is the trivial subproblem, which occursfor a 1 × n board. The number of solutions for this board is either zero or one, depending on whether the vector is apermutation of n / 2 and n / 2 pairs or not.For example, in the two boards shown above the sequences of vectors would be

((2, 2) (2, 2) (2, 2) (2, 2)) ((2, 2) (2, 2) (2, 2) (2, 2)) k = 4

0 1 0 1 0 0 1 1

((1, 2) (2, 1) (1, 2) (2, 1)) ((1, 2) (1, 2) (2, 1) (2, 1)) k = 3

1 0 1 0 0 0 1 1

((1, 1) (1, 1) (1, 1) (1, 1)) ((0, 2) (0, 2) (2, 0) (2, 0)) k = 2

0 1 0 1 1 1 0 0

((0, 1) (1, 0) (0, 1) (1, 0)) ((0, 1) (0, 1) (1, 0) (1, 0)) k = 1

1 0 1 0 1 1 0 0

Dynamic programming 123

((0, 0) (0, 0) (0, 0) (0, 0)) ((0, 0) (0, 0), (0, 0) (0, 0))

The number of solutions (sequence A058527 [12] in OEIS) is

Links to the Perl source of the backtracking approach, as well as a MAPLE and a C implementation of the dynamicprogramming approach may be found among the external links.

CheckerboardConsider a checkerboard with n × n squares and a cost-function c(i, j) which returns a cost associated with square i, j(i being the row, j being the column). For instance (on a 5 × 5 checkerboard),

5 6 7 4 7 8

4 7 6 1 1 4

3 3 5 7 8 2

2 - 6 7 0 -

1 - - *5* - -

1 2 3 4 5

Thus c(1, 3) = 5Let us say you had a checker that could start at any square on the first rank (i.e., row) and you wanted to know theshortest path (sum of the costs of the visited squares are at a minimum) to get to the last rank, assuming the checkercould move only diagonally left forward, diagonally right forward, or straight forward. That is, a checker on (1,3)can move to (2,2), (2,3) or (2,4).

5

4

3

2 x x x

1 o

1 2 3 4 5

This problem exhibits optimal substructure. That is, the solution to the entire problem relies on solutions tosubproblems. Let us define a function q(i, j) as

q(i, j) = the minimum cost to reach square (i, j)If we can find the values of this function for all the squares at rank n, we pick the minimum and follow that pathbackwards to get the shortest path.Note that q(i, j) is equal to the minimum cost to get to any of the three squares below it (since those are the onlysquares that can reach it) plus c(i, j). For instance:

Dynamic programming 124

5

4 A

3 B C D

2

1

1 2 3 4 5

Now, let us define q(i, j) in somewhat more general terms:

The first line of this equation is there to make the recursive property simpler (when dealing with the edges, so weneed only one recursion). The second line says what happens in the last rank, to provide a base case. The third line,the recursion, is the important part. It is similar to the A,B,C,D example. From this definition we can make astraightforward recursive code for q(i, j). In the following pseudocode, n is the size of the board, c(i, j) is thecost-function, and min() returns the minimum of a number of values:

function minCost(i, j)

if j < 1 or j > n

return infinity

else if i = 1

return c(i, j)

else

return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j)

It should be noted that this function only computes the path-cost, not the actual path. We will get to the path soon.This, like the Fibonacci-numbers example, is horribly slow since it spends mountains of time recomputing the sameshortest paths over and over. However, we can compute it much faster in a bottom-up fashion if we store path-costsin a two-dimensional array q[i, j] rather than using a function. This avoids recomputation; before computingthe cost of a path, we check the array q[i, j] to see if the path cost is already there.We also need to know what the actual shortest path is. To do this, we use another array p[i, j], a predecessorarray. This array implicitly stores the path to any square s by storing the previous node on the shortest path to s, i.e.the predecessor. To reconstruct the path, we lookup the predecessor of s, then the predecessor of that square, then thepredecessor of that square, and so on, until we reach the starting square. Consider the following code:

function computeShortestPathArrays()

for x from 1 to n

q[1, x] := c(1, x)

for y from 1 to n

q[y, 0] := infinity

q[y, n + 1] := infinity

for y from 2 to n

for x from 1 to n

m := min(q[y-1, x-1], q[y-1, x], q[y-1, x+1])

q[y, x] := m + c(y, x)

Dynamic programming 125

if m = q[y-1, x-1]

p[y, x] := -1

else if m = q[y-1, x]

p[y, x] := 0

else

p[y, x] := 1

Now the rest is a simple matter of finding the minimum and printing it.

function computeShortestPath()

computeShortestPathArrays()

minIndex := 1

min := q[n, 1]

for i from 2 to n

if q[n, i] < min

minIndex := i

min := q[n, i]

printPath(n, minIndex)

function printPath(y, x)

print(x)

print("<-")

if y = 2

print(x + p[y, x])

else

printPath(y-1, x + p[y, x])

Sequence alignmentIn genetics, sequence alignment is an important application where dynamic programming is essential.[3] Typically,the problem consists of transforming one sequence into another using edit operations that replace, insert, or removean element. Each operation has an associated cost, and the goal is to find the sequence of edits with the lowest totalcost.The problem can be stated naturally as a recursion, a sequence A is optimally edited into a sequence B by either:1. inserting the first character of B, and performing an optimal alignment of A and the tail of B2. deleting the first character of A, and performing the optimal alignment of the tail of A and B3. replacing the first character of A with the first character of B, and performing optimal alignments of the tails of A

and B.The partial alignments can be tabulated in a matrix, where cell (i,j) contains the cost of the optimal alignment ofA[1..i] to B[1..j]. The cost in cell (i,j) can be calculated by adding the cost of the relevant operations to the cost of itsneighboring cells, and selecting the optimum.Different variants exist, see Smith-Waterman and Needleman-Wunsch.

Dynamic programming 126

Tower of Hanoi Puzzle

A model set of the Towers of Hanoi (with 8 disks)

An animated solution of the Tower of Hanoi puzzle for T(4,3).

The Tower of Hanoi or Towers of Hanoi isa mathematical game or puzzle. It consistsof three rods, and a number of disks ofdifferent sizes which can slide onto any rod.The puzzle starts with the disks in a neatstack in ascending order of size on one rod,the smallest at the top, thus making aconical shape.

The objective of the puzzle is to move theentire stack to another rod, obeying thefollowing rules:• Only one disk may be moved at a time.• Each move consists of taking the upper

disk from one of the rods and sliding itonto another rod, on top of the otherdisks that may already be present on thatrod.

• No disk may be placed on top of asmaller disk.

The dynamic programming solution consistsof solving the functional equation

S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t)where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotesthe third rod (neither h nor t), ";" denotes concatenation, and

S(n, h, t) := solution to a problem consisting of n disks that are to be moved from rod h to rod t.Note that for n=1 the problem is trivial, namely S(1,h,t) = "move a disk from rod h to rod t" (there is only one diskleft).The number of moves required by this solution is 2n -1. If the objective is to maximize the number of moves(without cycling) then the dynamic programming functional equation is slightly more complicated and 3n -1 movesare required.[13]

Egg dropping puzzleThe following is a description of the instance of this famous puzzle involving n=2 eggs and a building with H=36floors:[14]

Suppose that we wish to know which storeys in a 36-storey building are safe to drop eggs from, and which willcause the eggs to break on landing. We make a few assumptions:

• An egg that survives a fall can be used again.• A broken egg must be discarded.• The effect of a fall is the same for all eggs.• If an egg breaks when dropped, then it would break if dropped from a higher window.• If an egg survives a fall then it would survive a shorter fall.• It is not ruled out that the first-floor windows break eggs, nor is it ruled out that the 36th-floor windows do not

cause an egg to break.

Dynamic programming 127

If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carriedout in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second floorwindow. Continue upward until it breaks. In the worst case, this method may require 36 droppings. Suppose 2eggs are available. What is the least number of egg-droppings that is guaranteed to work in all cases?

To derive a dynamic programming functional equation for this puzzle, let the state of the dynamic programmingmodel be a pair s = (n,k), where

n = number of test eggs available, n = 0,1,2,3,...,N-1.k = number of (consecutive) floors yet to be tested, k = 0,1,2,...,H-1.

For instance, s = (2,6) indicates that 2 test eggs are available and 6 (consecutive) floors are yet to be tested. Theinitial state of the process is s = (N,H) where N denotes the number of test eggs available at the commencement ofthe experiment. The process terminates either when there are no more test eggs (n = 0) or when k = 0, whicheveroccurs first. If termination occurs at state s=(0,k) and k>0, then the test failed.Now, let

W(n,k) := minimum number of trials required to identify the value of the critical floor under the Worst CaseScenario given that the process is in state s=(n,k).

Then it can be shown that[15]

W(n,k) = 1 + min{max(W(n-1,x-1) , W(n,k-x)): x in {1,2,...,k}} , n = 2,...,N ; k = 2,3,4,...,Hwith W(n,1) = 1 for all n>0 and W(1,k) = k for all k. It is easy to solve this equation iteratively by systematicallyincreasing the values of n and k.An interactive online facility [16] is available for experimentation with this model as well as with other versions ofthis puzzle (e.g. when the objective is to minimize the expected value of the number of trials.[15]

Algorithms that use dynamic programming• Backward induction as a solution method for finite-horizon discrete-time dynamic optimization problems• Method of undetermined coefficients can be used to solve the Bellman equation in infinite-horizon, discrete-time,

discounted, time-invariant dynamic optimization problems• Many string algorithms including longest common subsequence, longest increasing subsequence, longest

common substring, Levenshtein distance (edit distance).• Many algorithmic problems on graphs can be solved efficiently for graphs of bounded treewidth or bounded

clique-width by using dynamic programming on a tree decomposition of the graph.• The Cocke-Younger-Kasami (CYK) algorithm which determines whether and how a given string can be

generated by a given context-free grammar• Knuth's word wrapping algorithm that minimizes raggedness when word wrapping text• The use of transposition tables and refutation tables in computer chess• The Viterbi algorithm (used for hidden Markov models)• The Earley algorithm (a type of chart parser)• The Needleman-Wunsch and other algorithms used in bioinformatics, including sequence alignment, structural

alignment, RNA structure prediction.• Floyd's All-Pairs shortest path algorithm• Optimizing the order for chain matrix multiplication• Pseudopolynomial time algorithms for the Subset Sum and Knapsack and Partition problem Problems• The dynamic time warping algorithm for computing the global distance between two time series• The Selinger (a.k.a. System R) algorithm for relational database query optimization• De Boor algorithm for evaluating B-spline curves• Duckworth-Lewis method for resolving the problem when games of cricket are interrupted

Dynamic programming 128

• The Value Iteration method for solving Markov decision processes• Some graphic image edge following selection methods such as the "magnet" selection tool in Photoshop• Some methods for solving interval scheduling problems• Some methods for solving word wrap problems• Some methods for solving the travelling salesman problem, either exactly (in exponential time) or approximately

(e.g. via the bitonic tour)• Recursive least squares method• Beat tracking in Music Information Retrieval.• Adaptive Critic training strategy for artificial neural networks• Stereo algorithms for solving the Correspondence problem used in stereo vision.• Seam carving (content aware image resizing)• The Bellman-Ford algorithm for finding the shortest distance in a graph.• Some approximate solution methods for the linear search problem.• Kadane's algorithm for the Maximum subarray problem.

References[1] S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani, 'Algorithms', p173, available at http:/ / www. cs. berkeley. edu/ ~vazirani/ algorithms.

html[2] http:/ / www. wu-wien. ac. at/ usr/ h99c/ h9951826/ bellman_dynprog. pdf[3] Eddy, S. R., What is dynamic programming?, Nature Biotechnology, 22, 909-910 (2004).[4] Nocedal, J.; Wright, S. J.: Numerical Optimization, page 9, Springer, 2006..[5] Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; Stein, C. (2001), Introduction to Algorithms (2nd ed.), MIT Press & McGraw-Hill, ISBN

0-262-03293-7 . pp. 327–8.[6] http:/ / www. apl. jhu. edu/ ~paulmac/ c+ + -memoization. html[7] Stokey et al., 1989, Chap. 1[8] Sniedovich, M. (2006), "Dijkstra’s algorithm revisited: the dynamic programming connexion" (http:/ / matwbn. icm. edu. pl/ ksiazki/ cc/

cc35/ cc3536. pdf) (PDF), Journal of Control and Cybernetics 35 (3): 599–620, . Online version of the paper with interactive computationalmodules. (http:/ / www. ifors. ms. unimelb. edu. au/ tutorial/ dijkstra_new/ index. html)

[9] Denardo, E.V. (2003), Dynamic Programming: Models and Applications, Mineola, NY: Dover Publications, ISBN 978-0486428109[10] Sniedovich, M. (2010), Dynamic Programming: Foundations and Principles, Taylor & Francis, ISBN 9780824740993[11] Dijkstra 1959, p. 270[12] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa058527[13] Moshe Sniedovich (2002), ""OR/MS Games: 2. The Towers of Hanoi Problem," (http:/ / archive. ite. journal. informs. org/ Vol3No1/

Sniedovich/ ), INFORMS Transactions on Education 3(1): 34–51, .[14] Konhauser J.D.E., Velleman, D., and Wagon, S. (1996). Which way did the Bicycle Go? (http:/ / www. cambridge. org/ uk/ catalogue/

catalogue. asp?isbn=9780883853252) Dolciani Mathematical Expositions -- No-18. The Mathematical Association of America.[15] Sniedovich, M. (2003). The joy of egg-dropping in Braunschweig and Hong Kong (http:/ / archive. ite. journal. informs. org/ Vol4No1/

Sniedovich/ index. php). INFORMS Transactions on Education, 4(1) 48-64.[16] http:/ / archive. ite. journal. informs. org/ Vol4No1/ Sniedovich/ index. php

Further reading• Adda, Jerome; Cooper, Russell (2003), Dynamic Economics (http:/ / www. eco. utexas. edu/ ~cooper/ dynprog/

dynprog1. html), MIT Press. An accessible introduction to dynamic programming in economics. The link containssample programs.

• Bellman, Richard (1954), "The theory of dynamic programming", Bulletin of the American Mathematical Society60: 503–516, doi:10.1090/S0002-9904-1954-09848-8, MR0067459. Includes an extensive bibliography of theliterature in the area, up to the year 1954.

• Bellman, Richard (1957), Dynamic Programming, Princeton University Press. Dover paperback edition (2003),ISBN 0486428095.

• Bertsekas, D. P. (2000), Dynamic Programming and Optimal Control (2nd ed.), Athena Scientific,ISBN 1-886529-09-4. In two volumes.

Dynamic programming 129

• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), Introduction to Algorithms(2nd ed.), MIT Press & McGraw-Hill, ISBN 0-262-03293-7. Especially pp. 323–69.

• Dreyfus, Stuart E.; Law, Averill M. (1977), The art and theory of dynamic programming, Academic Press,ISBN 978-0122218606.

• Giegerich, R.; Meyer, C.; Steffen, P. (2004), "A Discipline of Dynamic Programming over Sequence Data" (http:// bibiserv. techfak. uni-bielefeld. de/ adp/ ps/ GIE-MEY-STE-2004. pdf), Science of Computer Programming 51(3): 215–263, doi:10.1016/j.scico.2003.12.005.

• Meyn, Sean (2007), Control Techniques for Complex Networks (https:/ / netfiles. uiuc. edu/ meyn/ www/spm_files/ CTCN/ CTCN. html), Cambridge University Press, ISBN 9780521884419.

• S. S. Sritharan, "Dynamic Programming of the Navier-Stokes Equations," in Systems and Control Letters, Vol.16, No. 4, 1991, pp. 299–307.

• Stokey, Nancy; Lucas, Robert E.; Prescott, Edward (1989), Recursive Methods in Economic Dynamics, HarvardUniv. Press, ISBN 9780674750968.

External links• An Introduction to Dynamic Programming (http:/ / 20bits. com/ articles/ introduction-to-dynamic-programming/ )• Dyna (http:/ / www. dyna. org), a declarative programming language for dynamic programming algorithms• Wagner, David B., 1995, " Dynamic Programming. (http:/ / citeseer. ist. psu. edu/ 268391. html)" An introductory

article on dynamic programming in Mathematica.• Ohio State University: CIS 680: class notes on dynamic programming (http:/ / www. cse. ohio-state. edu/ ~gurari/

course/ cis680/ cis680Ch21. html), by Eitan M. Gurari• A Tutorial on Dynamic programming (http:/ / mat. gsia. cmu. edu/ classes/ dynamic/ dynamic. html)• MIT course on algorithms (http:/ / ocw. mit. edu/ OcwWeb/ Electrical-Engineering-and-Computer-Science/

6-046JFall-2005/ VideoLectures/ detail/ embed15. htm) - Includes a video lecture on DP along with lecture notes• More DP Notes (http:/ / www. csse. monash. edu. au/ ~lloyd/ tildeAlgDS/ Dynamic)• King, Ian, 2002 (1987), " A Simple Introduction to Dynamic Programming in Macroeconomic Models. (http:/ /

researchspace. auckland. ac. nz/ bitstream/ handle/ 2292/ 190/ 230. pdf)" An introduction to dynamicprogramming as an important tool in economic theory.

• Dynamic Programming: from novice to advanced (http:/ / www. topcoder. com/ tc?module=Static&d1=tutorials& d2=dynProg) A TopCoder.com article by Dumitru on Dynamic Programming

• Algebraic Dynamic Programming (http:/ / bibiserv. techfak. uni-bielefeld. de/ adp/ ) - a formalized framework fordynamic programming, including an entry-level course (http:/ / bibiserv. techfak. uni-bielefeld. de/ dpcourse) toDP, University of Bielefeld

• Dreyfus, Stuart, " Richard Bellman on the birth of Dynamic Programming. (http:/ / www. eng. tau. ac. il/ ~ami/cd/ or50/ 1526-5463-2002-50-01-0048. pdf)"

• Dynamic programming tutorial (http:/ / www. avatar. se/ lectures/ molbioinfo2001/ dynprog/ dynamic. html)• A Gentle Introduction to Dynamic Programming and the Viterbi Algorithm (http:/ / www. cambridge. org/

resources/ 0521882672/ 7934_kaeslin_dynpro_new. pdf)• Tabled Prolog BProlog (http:/ / www. probp. com) and XSB (http:/ / xsb. sourceforge. net/ )• Online interactive dynamic programming modules (http:/ / ifors. org/ tutorial/ category/ dynamic-programming/ )

including, shortest path, traveling salesman, knapsack, false coin, egg dropping, bridge and torch, replacement,chained matrix products, and critical path problem.

Goal programming 130

Goal programmingGoal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decisionanalysis (MCDA), also known as multiple-criteria decision making (MCDM). This is an optimization programme. Itcan be thought of as an extension or generalisation of linear programming to handle multiple, normally conflictingobjective measures. Each of these measures is given a goal or target value to be achieved. Unwanted deviations fromthis set of target values are then minimised in an achievement function. This can be a vector or a weighted sumdependent on the goal programming variant used. As satisfaction of the target is deemed to satisfy the decisionmaker(s), an underlying satisficing philosophy is assumed.

HistoryGoal programming was first used by Charnes, Cooper and Ferguson in 1955,[1] although the actual name first appearin a 1961 text by Charnes and Cooper.[2] Seminal works by Lee,[3] Ignizio,[4] Ignizio and Cavalier,[5] and Romero[6]

followed. Schniederjans gives in a bibliography of a large number of pre-1995 articles relating to goalprogramming,[7] and Jones and Tamiz give an annotated bibliography of the period 1990-2000.[8] A recent textbookby Jones and Tamiz .[9] gives a comprehensive overview of the state-of-the-art in goal programming.The first engineering application of goal programming, due to Ignizio in 1962, was the design and placement of theantennas employed on the second stage of the Saturn V. This was used to launch the Apollo space capsule thatlanded the first men on the moon.

VariantsThe initial goal programming formulations ordered the unwanted deviations into a number of priority levels, with theminimisation of a deviation in a higher priority level being infinitely more important than any deviations in lowerpriority levels. This is known as lexicographic or pre-emptive goal programming. Ignizio[4] gives an algorithmshowing how a lexicographic goal programme can be solved as a series of linear programmes. Lexicographic goalprogramming should be used when there exists a clear priority ordering amongst the goals to be achieved.If the decision maker is more interested in direct comparisons of the objectives then Weighted or non pre-emptivegoal programming should be used. In this case all the unwanted deviations are multiplied by weights, reflecting theirrelative importance, and added together as a single sum to form the achievement function. It is important torecognise that deviations measured in different units cannot be summed directly due to the phenomenon ofincommensurability.Hence each unwanted deviation is multiplied by a normalisation constant to allow direct comparison. Popularchoices for normalisation constants are the goal target value of the corresponding objective (hence turning alldeviations into percentages) or the range of the corresponding objective (between the best and the worst possiblevalues, hence mapping all deviations onto a zero-one range).[6] For decision makers more interested in obtaining abalance between the competing objectives, Chebyshev goal programming should be used. Introduced by Flavell in1976,[10] this variant seeks to minimise the maximum unwanted deviation, rather than the sum of deviations. Thisutilises the Chebyshev distance metric, which emphasizes justice and balance rather than ruthless optimisation.

Goal programming 131

Strengths and weaknessesA major strength of goal programming is its simplicity and ease of use. This accounts for the large number of goalprogramming applications in many and diverse fields. Linear Goal programmes can be solved using linearprogramming software as either a single linear programme, or in the case of the lexicographic variant, a series ofconnected linear programmes.Goal programming can hence handle relatively large numbers of variables, constraints and objectives. A debatedweakness is the ability of goal programming to produce solutions that are not Pareto efficient. This violates afundamental concept of decision theory, that is no rational decision maker will knowingly choose a solution that isnot Pareto efficient. However, techniques are available [6] [11] [12] to detect when this occurs and project the solutiononto the Pareto efficient solution in an appropriate manner.The setting of appropriate weights in the goal programming model is another area that has caused debate, with someauthors [13] suggesting the use of the Analytic Hierarchy Process or interactive methods [14] for this purpose.

External links• LiPS [29] — Free easy-to-use GUI program intended for solving linear, integer and goal programming problems.• LINSOLVE [15] - Free Windows command-line window linear programming and linear goal programming]

References[1] A Charnes, WW Cooper, R Ferguson (1955) Optimal estimation of executive compensation by linear programming, Management Science, 1,

138-151.[2] A Charnes, WW Cooper (1961) Management models and industrial applications of linear programming, Wiley, New York[3] SM Lee (1972) Goal programming for decision analysis, Auerback, Philadelphia[4] JP Ignizio (1976) Goal programming and extensions, Lexington Books, Lexington, MA.[5] JP Ignizio, TM Cavalier (1994) Linear programming, Prentice Hall.[6] C Romero (1991) Handbook of critical issues in goal programming, Pergamon Press, Oxford.[7] MJ Scniederjans (1995) Goal programming methodology and applications, Kluwer publishers, Boston.[8] DF Jones, M Tamiz (2002) Goal programming in the period 1990-2000, in Multiple Criteria Optimization: State of the art annotated

bibliographic surveys, M. Ehrgott and X.Gandibleux (Eds.), 129-170. Kluwer[9] Jones DF, Tamiz M (2010) Practical Goal Programming, Springer Books.[10] RB Flavell (1976) A new goal programming formulation, Omega, 4, 731-732.[11] EL Hannan (1980) Non-dominance in goal programming, INFOR, 18, 300-309[12] M Tamiz, SK Mirrazavi, DF Jones (1999) Extensions of Pareto efficiency analysis to integer goal programming, Omega, 27, 179-188.[13] SI Gass (1987) A process for determining priorities and weights for large scale linear goal programmes, Journal of the Operational Research

Society, 37, 779-785.[14] BJ White (1996) Developing Products and Their Rhetoric from a Single Hierarchical Model, 1996 Proceedings of the Annual Conference of

the Society for Technical Communication, 43, 223-224.[15] http:/ / garbo. uwasa. fi/ pub/ pc/ ts/ tslin35c. zip

Information theory 132

Information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification ofinformation. Information theory was developed by Claude E. Shannon to find fundamental limits on signalprocessing operations such as compressing data and on reliably storing and communicating data. Since its inceptionit has broadened to find applications in many other areas, including statistical inference, natural language processing,cryptography generally, networks other than communication networks — as in neurobiology,[1] the evolution[2] andfunction[3] of molecular codes, model selection[4] in ecology, thermal physics,[5] quantum computing, plagiarismdetection[6] and other forms of data analysis.[7]

A key measure of information is known as entropy, which is usually expressed by the average number of bits neededfor storage or communication. Entropy quantifies the uncertainty involved in predicting the value of a randomvariable. For example, specifying the outcome of a fair coin flip (two equally likely outcomes) provides lessinformation (lower entropy) than specifying the outcome from a roll of a die (six equally likely outcomes).Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossydata compression (e.g. MP3s and JPGs), and channel coding (e.g. for DSL lines). The field is at the intersection ofmathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has beencrucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility ofmobile phones, the development of the Internet, the study of linguistics and of human perception, the understandingof black holes, and numerous other fields. Important sub-fields of information theory are source coding, channelcoding, algorithmic complexity theory, algorithmic information theory, information-theoretic security, and measuresof information.

OverviewThe main concepts of information theory can be grasped by considering the most widespread means of humancommunication: language. Two important aspects of a concise language are as follows: First, the most commonwords (e.g., "a", "the", "I") should be shorter than less common words (e.g., "benefit", "generation", "mediocre"), sothat sentences will not be too long. Such a tradeoff in word length is analogous to data compression and is theessential aspect of source coding. Second, if part of a sentence is unheard or misheard due to noise — e.g., a passingcar — the listener should still be able to glean the meaning of the underlying message. Such robustness is asessential for an electronic communication system as it is for a language; properly building such robustness intocommunications is done by channel coding. Source coding and channel coding are the fundamental concerns ofinformation theory.Note that these concerns have nothing to do with the importance of messages. For example, a platitude such as"Thank you; come again" takes about as long to say or write as the urgent plea, "Call an ambulance!" while the lattermay be more important and more meaningful in many contexts. Information theory, however, does not considermessage importance or meaning, as these are matters of the quality of data rather than the quantity and readability ofdata, the latter of which is determined solely by probabilities.Information theory is generally considered to have been founded in 1948 by Claude Shannon in his seminal work, "AMathematical Theory of Communication". The central paradigm of classical information theory is the engineeringproblem of the transmission of information over a noisy channel. The most fundamental results of this theory areShannon's source coding theorem, which establishes that, on average, the number of bits needed to represent theresult of an uncertain event is given by its entropy; and Shannon's noisy-channel coding theorem, which states thatreliable communication is possible over noisy channels provided that the rate of communication is below a certainthreshold, called the channel capacity. The channel capacity can be approached in practice by using appropriateencoding and decoding systems.

Information theory 133

Information theory is closely associated with a collection of pure and applied disciplines that have been investigatedand reduced to engineering practice under a variety of rubrics throughout the world over the past half century ormore: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science,cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory isa broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field ofcoding theory.Coding theory is concerned with finding explicit methods, called codes, of increasing the efficiency and reducing thenet error rate of data communication over a noisy channel to near the limit that Shannon proved is the maximumpossible for that channel. These codes can be roughly subdivided into data compression (source coding) anderror-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon'swork proved were possible. A third class of information theory codes are cryptographic algorithms (both codes andciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptographyand cryptanalysis. See the article ban (information) for a historical application.

Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even inmusical composition.

Historical backgroundThe landmark event that established the discipline of information theory, and brought it to immediate worldwideattention, was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" inthe Bell System Technical Journal in July and October 1948.Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assumingevents of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains atheoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communicationsystem, giving the relation , where W is the speed of transmission of intelligence, m is the numberof different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper,Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability todistinguish one sequence of symbols from any other, thus quantifying information as ,where S was the number of possible symbols, and n the number of symbols in a transmission. The natural unit ofinformation was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale ormeasure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking ofthe German second world war Enigma ciphers.Much of the mathematics behind information theory with events of different probabilities was developed for the fieldof thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between information-theoretic entropyand thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored inEntropy in thermodynamics and information theory.In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at BellLabs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model ofcommunication as a statistical process underlying information theory, opening with the assertion that

"The fundamental problem of communication is that of reproducing at one point, either exactly orapproximately, a message selected at another point."

With it came the ideas of• the information entropy and redundancy of a source, and its relevance through the source coding theorem;• the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free

communication given by the noisy-channel coding theorem;• the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as

Information theory 134

• the bit—a new way of seeing the most fundamental unit of information.

Quantities of informationInformation theory is based on probability theory and statistics. The most important quantities of information areentropy, the information in a random variable, and mutual information, the amount of information in commonbetween two random variables. The former quantity indicates how easily message data can be compressed while thelatter can be used to find the communication rate across a channel.The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. Themost common unit of information is the bit, based on the binary logarithm. Other units include the nat, which isbased on the natural logarithm, and the hartley, which is based on the common logarithm.

In what follows, an expression of the form is considered by convention to be equal to zero wheneverThis is justified because for any logarithmic base.

Entropy

Entropy of a Bernoulli trial as a function of successprobability, often called the binary entropy function,

. The entropy is maximized at 1 bit per trial

when the two possible outcomes are equally probable,as in an unbiased coin toss.

The entropy, , of a discrete random variable is a measureof the amount of uncertainty associated with the value of .

Suppose one transmits 1000 bits (0s and 1s). If these bits are known ahead of transmission (to be a certain value withabsolute probability), logic dictates that no information has been transmitted. If, however, each is equally andindependently likely to be 0 or 1, 1000 bits (in the information theoretic sense) have been transmitted. Between thesetwo extremes, information can be quantified as follows. If is the set of all messages that could be, and is the probability of given some , then the entropy of is defined:[8]

(Here, is the self-information, which is the entropy contribution of an individual message, and is theexpected value.) An important property of entropy is that it is maximized when all the messages in the messagespace are equiprobable ,—i.e., most unpredictable—in which case .The special case of information entropy for a random variable with two outcomes is the binary entropy function,usually taken to the logarithmic base 2:

Information theory 135

Joint entropy

The joint entropy of two discrete random variables and is merely the entropy of their pairing: .This implies that if and are independent, then their joint entropy is the sum of their individual entropies.For example, if represents the position of a chess piece — the row and the column, then the jointentropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.

Despite similar notation, joint entropy should not be confused with cross entropy.

Conditional entropy (equivocation)The conditional entropy or conditional uncertainty of given random variable (also called the equivocationof about ) is the average conditional entropy over :[9]

Because entropy can be conditioned on a random variable or on that random variable being a certain value, careshould be taken not to confuse these two definitions of conditional entropy, the former of which is in more commonuse. A basic property of this form of conditional entropy is that:

Mutual information (transinformation)Mutual information measures the amount of information that can be obtained about one random variable byobserving another. It is important in communication where it can be used to maximize the amount of informationshared between sent and received signals. The mutual information of relative to is given by:

where (Specific mutual Information) is the pointwise mutual information.A basic property of the mutual information is that

That is, knowing Y, we can save an average of bits in encoding X compared to not knowing Y.Mutual information is symmetric:

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) of the posteriorprobability distribution of X given the value of Y to the prior distribution on X:

In other words, this is a measure of how much, on the average, the probability distribution on X will change if we aregiven the value of Y. This is often recalculated as the divergence from the product of the marginal distributions to theactual joint distribution:

Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and themultinomial distribution and to Pearson's χ2 test: mutual information can be considered a statistic for assessingindependence between a pair of variables, and has a well-specified asymptotic distribution.

Information theory 136

Kullback–Leibler divergence (information gain)The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way ofcomparing two distributions: a "true" probability distribution p(X), and an arbitrary probability distribution q(X). Ifwe compress data in a manner that assumes q(X) is the distribution underlying some data, when, in reality, p(X) is thecorrect distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessaryfor compression. It is thus defined

Although it is sometimes used as a 'distance metric', it is not a true metric since it is not symmetric and does notsatisfy the triangle inequality (making it a semi-quasimetric).

Other quantitiesOther important information theoretic quantities include Rényi entropy, (a generalization of entropy,) differentialentropy, (a generalization of quantities of information to continuous distributions,) and the conditional mutualinformation.

Coding theory

A picture showing scratches on the readablesurface of a CD-R. Music and data CDs are codedusing error correcting codes and thus can still be

read even if they have minor scratches using errordetection and correction.

Coding theory is one of the most important and direct applications ofinformation theory. It can be subdivided into source coding theory andchannel coding theory. Using a statistical description for data,information theory quantifies the number of bits needed to describe thedata, which is the information entropy of the source.

• Data compression (source coding): There are two formulations forthe compression problem:

1. lossless data compression: the data must be reconstructed exactly;2. lossy data compression: allocates bits needed to reconstruct the

data, within a specified fidelity level measured by a distortionfunction. This subset of Information theory is called rate–distortion theory.

• Error-correcting codes (channel coding): While data compression removes as much redundancy as possible, anerror correcting code adds just the right kind of redundancy (i.e., error correction) needed to transmit the dataefficiently and faithfully across a noisy channel.

This division of coding theory into compression and transmission is justified by the information transmissiontheorems, or source–channel separation theorems that justify the use of bits as the universal currency for informationin many contexts. However, these theorems only hold in the situation where one transmitting user wishes tocommunicate to one receiving user. In scenarios with more than one transmitter (the multiple-access channel), morethan one receiver (the broadcast channel) or intermediary "helpers" (the relay channel), or more general networks,compression followed by transmission may no longer be optimal. Network information theory refers to thesemulti-agent communication models.

Information theory 137

Source theoryAny process that generates successive messages can be considered a source of information. A memoryless source isone in which each message is an independent identically-distributed random variable, whereas the properties ofergodicity and stationarity impose more general constraints. All such sources are stochastic. These terms are wellstudied in their own right outside information theory.

Rate

Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of eachsymbol, while, in the case of a stationary stochastic process, it is

that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of aprocess that is not necessarily stationary, the average rate is

that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the sameresult.[10]

It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, forexample, when the source of information is English prose. The rate of a source of information is related to itsredundancy and how well it can be compressed, the subject of source coding.

Channel capacityCommunications over a channel—such as an ethernet cable—is the primary motivation of information theory. Asanyone who's ever used a telephone (mobile or landline) knows, however, such channels often fail to produce exactreconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality. Howmuch information can one hope to communicate over a noisy (or otherwise imperfect) channel?Consider the communications process over a discrete channel. A simple model of the process is shown below:

Here X represents the space of messages transmitted, and Y the space of messages received during a unit time overour channel. Let be the conditional probability distribution function of Y given X. We will consider to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel).Then the joint distribution of X and Y is completely determined by our channel and by our choice of , themarginal distribution of messages we choose to send over the channel. Under these constraints, we would like tomaximize the rate of information, or the signal, we can communicate over the channel. The appropriate measure forthis is the mutual information, and this maximum mutual information is called the channel capacity and is given by:

This capacity has the following property related to communicating at information rate R (where R is usually bits per symbol). For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. In addition, for any rate R > C, it is impossible to transmit

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with arbitrarily small block error.Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisychannel with a small coding error at a rate near the channel capacity.

Capacity of particular channel models

• A continuous-time analog communications channel subject to Gaussian noise — see Shannon–Hartley theorem.• A binary symmetric channel (BSC) with crossover probability p is a binary input, binary output channel that flips

the input bit with probability p. The BSC has a capacity of bits per channel use, where is thebinary entropy function to the base 2 logarithm:

• A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The possiblechannel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss ofinformation about an input bit. The capacity of the BEC is 1 - p bits per channel use.

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Applications to other fields

Intelligence uses and secrecy applicationsInformation theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was usedin the Ultra project, breaking the German Enigma machine code and hastening the end of WWII in Europe. Shannonhimself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, itattempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A bruteforce attack can break systems based on asymmetric key algorithms or on most commonly used methods ofsymmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all suchmethods currently comes from the assumption that no known attack can break them in a practical amount of time.Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute forceattacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditionedon the key) can ensure proper transmission, while the unconditional mutual information between the plaintext andciphertext remains zero, resulting in absolutely secure communications. In other words, an eavesdropper would notbe able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key.However, as in any other cryptographic system, care must be used to correctly apply even information-theoreticallysecure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improperreuse of key material.

Information theory 140

Pseudorandom number generationPseudorandom number generators are widely available in computer language libraries and application programs.They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of moderncomputer equipment and software. A class of improved random number generators is termed cryptographicallysecure pseudorandom number generators, but even they require external to the software random seeds to work asintended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractorsis min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluatingrandomness in cryptographic systems. Although related, the distinctions among these measures mean that a randomvariable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptographyuses.

Seismic explorationOne early commercial application of information theory was in the field seismic oil exploration. Work in this fieldmade it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory anddigital signal processing offer a major improvement of resolution and image clarity over previous analogmethods.[11]

Miscellaneous applicationsInformation theory also has applications in gambling and investing, black holes, bioinformatics, and music.

References[1] F. Rieke, D. Warland, R Ruyter van Steveninck, W Bialek, Spikes: Exploring the Neural Code. The MIT press (1997).[2] cf. Huelsenbeck, J. P., F. Ronquist, R. Nielsen and J. P. Bollback (2001) Bayesian inference of phylogeny and its impact on evolutionary

biology, Science 294:2310-2314[3] Rando Allikmets, Wyeth W. Wasserman, Amy Hutchinson, Philip Smallwood, Jeremy Nathans, Peter K. Rogan, Thomas D. Schneider (http:/

/ www. lecb. ncifcrf. gov/ ~toms/ ), Michael Dean (1998) Organization of the ABCR gene: analysis of promoter and splice junction sequences,Gene 215:1, 111-122

[4] Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, SecondEdition (Springer Science, New York) ISBN 978-0-387-95364-9.

[5] Jaynes, E. T. (1957) Information Theory and Statistical Mechanics (http:/ / bayes. wustl. edu/ ), Phys. Rev. 106:620[6] Charles H. Bennett, Ming Li, and Bin Ma (2003) Chain Letters and Evolutionary Histories (http:/ / sciamdigital. com/ index.

cfm?fa=Products. ViewIssuePreview& ARTICLEID_CHAR=08B64096-0772-4904-9D48227D5C9FAC75), Scientific American 288:6,76-81

[7] David R. Anderson (November 1, 2003). "Some background on why people in the empirical sciences may want to better understand theinformation-theoretic methods" (http:/ / aicanderson2. home. comcast. net/ ~aicanderson2/ home. pdf) (pdf). . Retrieved 2010-06-23.

[8] Fazlollah M. Reza (1961, 1994). An Introduction to Information Theory (http:/ / books. google. com/ books?id=RtzpRAiX6OgC& pg=PA8&dq=intitle:"An+ Introduction+ to+ Information+ Theory"+ + "entropy+ of+ a+ simple+ source"& as_brr=0&ei=zP79Ro7UBovqoQK4g_nCCw& sig=j3lPgyYrC3-bvn1Td42TZgTzj0Q). Dover Publications, Inc., New York. ISBN 0-486-68210-2. .

[9] Robert B. Ash (1965, 1990). Information Theory (http:/ / books. google. com/ books?id=ngZhvUfF0UIC& pg=PA16&dq=intitle:information+ intitle:theory+ inauthor:ash+ conditional+ uncertainty& as_brr=0& ei=kKwNR4rbH5mepgKB4d2zBg&sig=YAsiCEVISjJ484R3uGoXpi-a5rI). Dover Publications, Inc.. ISBN 0-486-66521-6. .

[10] Jerry D. Gibson (1998). Digital Compression for Multimedia: Principles and Standards (http:/ / books. google. com/books?id=aqQ2Ry6spu0C& pg=PA56& dq=entropy-rate+ conditional& as_brr=3& ei=YGDsRtzGGKjupQKa2L2xDw&sig=o0UCtf0xZOf11lPIexPrjOKPgNc#PPA57,M1). Morgan Kaufmann. ISBN 1558603697. .

[11] The Corporation and Innovation, Haggerty, Patrick, Strategic Management Journal, Vol. 2, 97-118 (1981)

Information theory 141

The classic work• Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27,

pp. 379–423 & 623–656, July & October, 1948. PDF. (http:/ / cm. bell-labs. com/ cm/ ms/ what/ shannonday/shannon1948. pdf)Notes and other formats. (http:/ / cm. bell-labs. com/ cm/ ms/ what/ shannonday/ paper. html)

• R.V.L. Hartley, "Transmission of Information" (http:/ / www. dotrose. com/ etext/ 90_Miscellaneous/transmission_of_information_1928b. pdf), Bell System Technical Journal, July 1928

• Andrey Kolmogorov (1968), "Three approaches to the quantitative definition of information" in InternationalJournal of Computer Mathematics.

Other journal articles• J. L. Kelly, Jr., Saratoga.ny.us (http:/ / www. racing. saratoga. ny. us/ kelly. pdf), "A New Interpretation of

Information Rate" Bell System Technical Journal, Vol. 35, July 1956, pp. 917–26.• R. Landauer, IEEE.org (http:/ / ieeexplore. ieee. org/ search/ wrapper. jsp?arnumber=615478), "Information is

Physical" Proc. Workshop on Physics and Computation PhysComp'92 (IEEE Comp. Sci.Press, Los Alamitos,1993) pp. 1–4.

• R. Landauer, IBM.com (http:/ / www. research. ibm. com/ journal/ rd/ 441/ landauerii. pdf), "Irreversibility andHeat Generation in the Computing Process" IBM J. Res. Develop. Vol. 5, No. 3, 1961

Textbooks on information theory• Claude E. Shannon, Warren Weaver. The Mathematical Theory of Communication. Univ of Illinois Press, 1949.

ISBN 0-252-72548-4• Robert Gallager. Information Theory and Reliable Communication. New York: John Wiley and Sons, 1968. ISBN

0-471-29048-3• Robert B. Ash. Information Theory. New York: Interscience, 1965. ISBN 0-470-03445-9. New York: Dover

1990. ISBN 0-486-66521-6• Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience,

1991. ISBN 0-471-06259-6.2nd Edition. New York: Wiley-Interscience, 2006. ISBN 0-471-24195-4.

• Imre Csiszar, Janos Korner. Information Theory: Coding Theorems for Discrete Memoryless Systems AkademiaiKiado: 2nd edition, 1997. ISBN 963-05-7440-3

• Raymond W. Yeung. A First Course in Information Theory (http:/ / iest2. ie. cuhk. edu. hk/ ~whyeung/ book/ )Kluwer Academic/Plenum Publishers, 2002. ISBN 0-306-46791-7

• David J. C. MacKay. Information Theory, Inference, and Learning Algorithms (http:/ / www. inference. phy. cam.ac. uk/ mackay/ itila/ book. html) Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1

• Raymond W. Yeung. Information Theory and Network Coding (http:/ / iest2. ie. cuhk. edu. hk/ ~whyeung/ book2/) Springer 2008, 2002. ISBN 978-0-387-79233-0

• Stanford Goldman. Information Theory. New York: Prentice Hall, 1953. New York: Dover 1968 ISBN0-486-62209-6, 2005 ISBN 0-486-44271-3

• Fazlollah Reza. An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994.ISBN 0-486-68210-2

• Masud Mansuripur. Introduction to Information Theory. New York: Prentice Hall, 1987. ISBN 0-13-484668-0• Christoph Arndt: Information Measures, Information and its Description in Science and Engineering (Springer

Series: Signals and Communication Technology), 2004, ISBN 978-3-540-40855-0

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Other books• Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, [1956, 1962] 2004. ISBN 0-486-43918-6• James Gleick, The Information: A History, a Theory, a Flood, New York: Pantheon, 2011. ISBN

978-0375423727• A. I. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. ISBN 0-486-60434-9• H. S. Leff and A. F. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University

Press, Princeton, NJ (1990). ISBN 0-691-08727-X• Tom Siegfried, The Bit and the Pendulum, Wiley, 2000. ISBN 0-471-32174-5• Charles Seife, Decoding The Universe, Viking, 2006. ISBN 0-670-03441-X• Jeremy Campbell, Grammatical Man, Touchstone/Simon & Schuster, 1982, ISBN 0-671-44062-4• Henri Theil, Economics and Information Theory, Rand McNally & Company - Chicago, 1967.

External links• alum.mit.edu (http:/ / alum. mit. edu/ www/ toms/ paper/ primer), Eprint, Schneider, T. D., "Information Theory

Primer"• ND.edu (http:/ / www. nd. edu/ ~jnl/ ee80653/ tutorials/ sunil. pdf), Srinivasa, S. "A Review on Multivariate

Mutual Information"• Chem.wisc.edu (http:/ / jchemed. chem. wisc. edu/ Journal/ Issues/ 1999/ Oct/ abs1385. html), Journal of

Chemical Education, Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of EntropyIncrease? Nonsense!

• ITsoc.org (http:/ / www. itsoc. org/ index. html), IEEE Information Theory Society and ITsoc.org (http:/ / www.itsoc. org/ review. html) review articles

• Cam.ac.uk (http:/ / www. inference. phy. cam. ac. uk/ mackay/ itila/ ), On-line textbook: "Information Theory,Inference, and Learning Algorithms" by David MacKay - giving an entertaining and thorough introduction toShannon theory, including state-of-the-art methods from coding theory, such as arithmetic coding, low-densityparity-check codes, and Turbo codes.

• UMBC.edu (http:/ / research. umbc. edu/ ~erill/ Documents/ Introduction_Information_Theory. pdf), Eprint, Erill,I., "A gentle introduction to information content in transcription factor binding sites"

Cybernetics 143

CyberneticsCybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related toinformation theory, control theory and systems theory, at least in its first-order form. (Second-order cybernetics hascrucial methodological and epistemological implications that are fundamental to the field as a whole.) Both in itsorigins and in its evolution in the second half of the 20th century, cybernetics is equally applicable to physical andsocial (that is, language-based) systems.

Example of cybernetic thinking. On the one hand a company is approached as a system inan environment. On the other hand cybernetic factory can be modeled as a control system.

Overview

Cybernetics is most applicable whenthe system being analysed is involvedin a closed signal loop; that is, whereaction by the system causes somechange in its environment and thatchange is fed to the system viainformation (feedback) that causes thesystem to adapt to these newconditions: the system's changes affectits behavior. This "circular causal"relationship is necessary and sufficientfor a cybernetic perspective. SystemDynamics, a related field, originatedwith applications of electricalengineering control theory to otherkinds of simulation models (especially business systems) by Jay Forrester at MIT in the 1950s.

Contemporary cybernetics began as an interdisciplinary study connecting the fields of control systems, electricalnetwork theory, mechanical engineering, logic modeling, evolutionary biology, neuroscience, anthropology, andpsychology in the 1940s, often attributed to the Macy Conferences.

Other fields of study which have influenced or been influenced by cybernetics include game theory, system theory (amathematical counterpart to cybernetics), perceptual control theory, sociology, psychology (especiallyneuropsychology, behavioral psychology, cognitive psychology), philosophy, and architecture and organizationaltheory.[1]

Definition

The term cybernetics stems from the Greek κυβερνήτης (kybernētēs, steersman, governor,pilot, or rudder — the same root as government). Cybernetics is a broad field of study, butthe essential goal of cybernetics is to understand and define the functions and processes ofsystems that have goals and that participate in circular, causal chains that move from actionto sensing to comparison with desired goal, and again to action. Studies in cyberneticsprovide a means for examining the design and function of any system, including socialsystems such as business management and organizational learning, including for the

purpose of making them more efficient and effective.

Cybernetics was defined by Norbert Wiener, in his book of that title, as the study of control and communication in the animal and the machine. Stafford Beer called it the science of effective organization and Gordon Pask extended it

Cybernetics 144

to include information flows "in all media" from stars to brains. It includes the study of feedback, black boxes andderived concepts such as communication and control in living organisms, machines and organizations includingself-organization. Its focus is how anything (digital, mechanical or biological) processes information, reacts toinformation, and changes or can be changed to better accomplish the first two tasks [2] . A more philosophicaldefinition, suggested in 1956 by Louis Couffignal, one of the pioneers of cybernetics, characterizes cybernetics as"the art of ensuring the efficacy of action" [3] . The most recent definition has been proposed by Louis Kauffman,President of the American Society for Cybernetics, "Cybernetics is the study of systems and processes that interactwith themselves and produce themselves from themselves" [4] .Concepts studied by cyberneticists (or, as some prefer, cyberneticians) include, but are not limited to: learning,cognition, adaption, social control, emergence, communication, efficiency, efficacy and interconnectivity. Theseconcepts are studied by other subjects such as engineering and biology, but in cybernetics these are removed fromthe context of the individual organism or device.Other fields of study which have influenced or been influenced by cybernetics include game theory; system theory (amathematical counterpart to cybernetics); psychology, especially neuropsychology, behavioral psychology andcognitive psychology; philosophy; anthropology; and even theology,[5] telematic art, and architecture.[6]

History

The roots of cybernetic theoryThe word cybernetics was first used in the context of "the study of self-governance" by Plato in The Laws to signifythe governance of people. The word 'cybernétique' was also used in 1834 by the physicist André-Marie Ampère(1775–1836) to denote the sciences of government in his classification system of human knowledge.

James Watt

The first artificial automatic regulatory system, a water clock, was inventedby the mechanician Ktesibios. In his water clocks, water flowed from a sourcesuch as a holding tank into a reservoir, then from the reservoir to themechanisms of the clock. Ktesibios's device used a cone-shaped float tomonitor the level of the water in its reservoir and adjust the rate of flow of thewater accordingly to maintain a constant level of water in the reservoir, sothat it neither overflowed nor was allowed to run dry. This was the firstartificial truly automatic self-regulatory device that required no outsideintervention between the feedback and the controls of the mechanism.Although they did not refer to this concept by the name of Cybernetics (theyconsidered it a field of engineering), Ktesibios and others such as Heron andSu Song are considered to be some of the first to study cybernetic principles.

The study of teleological mechanisms (from the Greek τέλος or telos for end,goal, or purpose) in machines with corrective feedback dates from as far backas the late 18th century when James Watt's steam engine was equipped with agovernor, a centrifugal feedback valve for controlling the speed of the engine. Alfred Russel Wallace identified thisas the principle of evolution in his famous 1858 paper. In 1868 James Clerk Maxwell published a theoretical articleon governors, one of the first to discuss and refine the principles of self-regulating devices. Jakob von Uexküllapplied the feedback mechanism via his model of functional cycle (Funktionskreis) in order to explain animalbehaviour and the origins of meaning in general.

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The early 20th centuryContemporary cybernetics began as an interdisciplinary study connecting the fields of control systems, electricalnetwork theory, mechanical engineering, logic modeling, evolutionary biology and neuroscience in the 1940s.Electronic control systems originated with the 1927 work of Bell Telephone Laboratories engineer Harold S. Blackon using negative feedback to control amplifiers. The ideas are also related to the biological work of Ludwig vonBertalanffy in General Systems Theory.Early applications of negative feedback in electronic circuits included the control of gun mounts and radar antennaduring World War II. Jay Forrester, a graduate student at the Servomechanisms Laboratory at MIT during WWIIworking with Gordon S. Brown to develop electronic control systems for the U.S. Navy, later applied these ideas tosocial organizations such as corporations and cities as an original organizer of the MIT School of IndustrialManagement at the MIT Sloan School of Management. Forrester is known as the founder of System Dynamics.W. Edwards Deming, the Total Quality Management guru for whom Japan named its top post-WWII industrial prize,was an intern at Bell Telephone Labs in 1927 and may have been influenced by network theory. Deming made"Understanding Systems" one of the four pillars of what he described as "Profound Knowledge" in his book "TheNew Economics."Numerous papers spearheaded the coalescing of the field. In 1935 Russian physiologist P.K. Anokhin published abook in which the concept of feedback ("back afferentation") was studied. The study and mathematical modelling ofregulatory processes became a continuing research effort and two key articles were published in 1943. These paperswere "Behavior, Purpose and Teleology" by Arturo Rosenblueth, Norbert Wiener, and Julian Bigelow; and the paper"A Logical Calculus of the Ideas Immanent in Nervous Activity" by Warren McCulloch and Walter Pitts.Cybernetics as a discipline was firmly established by Wiener, McCulloch and others, such as W. Ross Ashby and W.Grey Walter.Walter was one of the first to build autonomous robots as an aid to the study of animal behaviour. Together with theUS and UK, an important geographical locus of early cybernetics was France.In the spring of 1947, Wiener was invited to a congress on harmonic analysis, held in Nancy, France. The event wasorganized by the Bourbaki, a French scientific society, and mathematician Szolem Mandelbrojt (1899–1983), uncleof the world-famous mathematician Benoît Mandelbrot.

John von Neumann

During this stay in France, Wiener received the offer to write amanuscript on the unifying character of this part of appliedmathematics, which is found in the study of Brownian motion and intelecommunication engineering. The following summer, back in theUnited States, Wiener decided to introduce the neologism cyberneticsinto his scientific theory. The name cybernetics was coined to denotethe study of "teleological mechanisms" and was popularized throughhis book Cybernetics, or Control and Communication in the Animaland Machine (Hermann & Cie, Paris, 1948). In the UK this became thefocus for the Ratio Club.

In the early 1940s John von Neumann, although better known for hiswork in mathematics and computer science, did contribute a uniqueand unusual addition to the world of cybernetics: Von Neumanncellular automata, and their logical follow up the Von NeumannUniversal Constructor. The result of these deceptively simplethought-experiments was the concept of self replication which

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cybernetics adopted as a core concept. The concept that the same properties of genetic reproduction applied to socialmemes, living cells, and even computer viruses is further proof of the somewhat surprising universality of cyberneticstudy.Wiener popularized the social implications of cybernetics, drawing analogies between automatic systems (such as aregulated steam engine) and human institutions in his best-selling The Human Use of Human Beings : Cyberneticsand Society (Houghton-Mifflin, 1950).While not the only instance of a research organization focused on cybernetics, the Biological Computer Lab [7] at theUniversity of Illinois, Urbana/Champaign, under the direction of Heinz von Foerster, was a major center ofcybernetic research [8] for almost 20 years, beginning in 1958.

The rebirth of cyberneticsIn the 1970s new cyberneticians emerged in multiple fields, but especially in biology. The ideas of Maturana, Varelaand Atlan, according to Dupuy (1986) "realized that the cybernetic metaphors of the program upon which molecularbiology had been based rendered a conception of the autonomy of the living being impossible. Consequently, thesethinkers were led to invent a new cybernetics, one more suited to the organizations which mankind discovers innature - organizations he has not himself invented"[9] . However, during the 1980s the question of whether thefeatures of this new cybernetics could be applied to social forms of organization remained open to debate.[9]

In political science, Project Cybersyn attempted to introduce a cybernetically controlled economy during the early1970s. In the 1980s, according to Harries-Jones (1988) "unlike its predecessor, the new cybernetics concerns itselfwith the interaction of autonomous political actors and subgroups, and the practical and reflexive consciousness ofthe subjects who produce and reproduce the structure of a political community. A dominant consideration is that ofrecursiveness, or self-reference of political action both with regards to the expression of political consciousness andwith the ways in which systems build upon themselves".[10]

One characteristic of the emerging new cybernetics considered in that time by Geyer and van der Zouwen, accordingto Bailey (1994), was "that it views information as constructed and reconstructed by an individual interacting withthe environment. This provides an epistemological foundation of science, by viewing it as observer-dependent.Another characteristic of the new cybernetics is its contribution towards bridging the "micro-macro gap". That is, itlinks the individual with the society"[11] Another characteristic noted was the "transition from classical cybernetics tothe new cybernetics [that] involves a transition from classical problems to new problems. These shifts in thinkinginvolve, among others, (a) a change from emphasis on the system being steered to the system doing the steering, andthe factor which guides the steering decisions.; and (b) new emphasis on communication between several systemswhich are trying to steer each other"[11] . The work of Gregory Bateson was also strongly influenced by cybernetics.Recent endeavors into the true focus of cybernetics, systems of control and emergent behavior, by such related fieldsas game theory (the analysis of group interaction), systems of feedback in evolution, and metamaterials (the study ofmaterials with properties beyond the Newtonian properties of their constituent atoms), have led to a revived interestin this increasingly relevant field.[2]

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Subdivisions of the fieldCybernetics is an earlier but still-used generic term for many types of subject matter. These subjects also extend intomany others areas of science, but are united in their study of control of systems.

Basic cyberneticsCybernetics studies systems of control as a concept, attempting to discover the basic principles underlying suchthings as

ASIMO uses sensors and intelligent algorithms toavoid obstacles and navigate stairs.

• Artificial intelligence• Robotics• Computer Vision• Control systems• Emergence• Learning organization• New Cybernetics• Second-order cybernetics• Interactions of Actors Theory• Conversation Theory• Self-organization in cybernetics

In biologyCybernetics in biology is the study of cybernetic systems present in biological organisms, primarily focusing on howanimals adapt to their environment, and how information in the form of genes is passed from generation togeneration[12] . There is also a secondary focus on combining artificial systems with biological systems.• Bioengineering• Biocybernetics• Bionics• Homeostasis• Medical cybernetics• Synthetic Biology• Systems Biology

In computer scienceComputer science directly applies the concepts of cybernetics to the control of devices and the analysis ofinformation.• Robotics• Decision support system• Cellular automaton• Simulation• Technology

Cybernetics 148

In engineeringCybernetics in engineering is used to analyze cascading failures and System Accidents, in which the small errors andimperfections in a system can generate disasters. Other topics studied include:

An artificial heart, a product of biomedicalengineering.

• Adaptive systems• Engineering cybernetics• Ergonomics• Biomedical engineering• Systems engineering

In management

• Entrepreneurial cybernetics• Management cybernetics• Organizational cybernetics

• Operations research• Systems engineering

In mathematicsMathematical Cybernetics focuses on the factors of information, interaction of parts in systems, and the structure ofsystems.• Dynamical system• Information theory• Systems theory

In psychology• Homunculus• Psycho-Cybernetics• Systems psychology• Perceptual Control Theory• Psychovector Analysis

In sociologyBy examining group behavior through the lens of cybernetics, sociology seeks the reasons for such spontaneousevents as smart mobs and riots, as well as how communities develop rules, such as etiquette, by consensus withoutformal discussion. Affect Control Theory explains role behavior, emotions, and labeling theory in terms ofhomeostatic maintenance of sentiments associated with cultural categories. The most comprehensive attempt evermade in the social sciences to increase cybernetics in a generalized theory of society was made by Talcott Parsons.In this way, cybernetics establish the basic hierarchy in Parsons' AGIL paradigm, which is the ordingsystem-dimension of his action theory. These and other cybernetic models in sociology are reviewed in a book editedby McClelland and Fararo[13] .• Affect Control Theory• Memetics• Sociocybernetics

Cybernetics 149

In artThe artist Roy Ascott theorised the cybernetics of art in "Behaviourist Art and the Cybernetic Vision". Cybernetica,Journal of the International Association for Cybernetics (Namur), 1967.• Telematic art• Interactive Art• Systems art

Related fields

Complexity scienceComplexity science attempts to understand the nature of complex systems.• Complex Adaptive System• Complex systems• Complexity theory

References[1] Tange, Kenzo (1966) "Function, Structure and Symbol".[2] Kelly, Kevin (1994). Out of control: The new biology of machines, social systems and the economic world. Boston: Addison-Wesley.

ISBN 0-201-48340-8. OCLC 221860672 32208523 40868076 56082721 57396750.[3] Couffignal, Louis, "Essai d’une définition générale de la cybernétique", The First International Congress on Cybernetics, Namur, Belgium,

June 26–29, 1956, Gauthier-Villars, Paris, 1958, pp. 46-54[4] CYBCON discusstion group 20 September 2007 18:15[5] Granfield, Patrick (1973). Ecclesial Cybernetics: A Study of Democracy in the Church. New York: MacMillan. pp. 280.[6] Hight, Christopher (2007). Architectural Principles in the age of Cybernetics. Routledge. pp. 248. ISBN 978-0415384827.[7] http:/ / www. ece. uiuc. edu/ pubs/ bcl/ mueller/ index. htm[8] http:/ / www. ece. uiuc. edu/ pubs/ bcl/ hutchinson/ index. htm[9] Jean-Pierre Dupuy, "The autonomy of social reality: on the contribution of systems theory to the theory of society" in: Elias L. Khalil &

Kenneth E. Boulding eds., Evolution, Order and Complexity, 1986.[10] Peter Harries-Jones (1988), "The Self-Organizing Polity: An Epistemological Analysis of Political Life by Laurent Dobuzinskis" in:

Canadian Journal of Political Science (Revue canadienne de science politique), Vol. 21, No. 2 (Jun., 1988), pp. 431-433.[11] Kenneth D. Bailey (1994), Sociology and the New Systems Theory: Toward a Theoretical Synthesis, p.163.[12] Note: this does not refer to the concept of Racial Memory but to the concept of cumulative adaptation to a particular niche, such as the case

of the pepper moth having genes for both light and dark environments.[13] McClelland, Kent A., and Thomas J. Fararo (Eds.). 2006. Purpose, Meaning, and Action: Control Systems Theories in Sociology. New

York: Palgrave Macmillan.

Further reading• Andrew Pickering (2010) The Cybernetic Brain: Sketches of Another Future (http:/ / www. amazon. com/

Cybernetic-Brain-Sketches-Another-Future/ dp/ 0226667898) University Of Chicago Press.• Slava Gerovitch (2002) From Newspeak to Cyberspeak: A History of Soviet Cybernetics (http:/ / web. mit. edu/

slava/ homepage/ newspeak. htm) MIT Press.• John Johnston, (2008) "The Allure of Machinic Life: Cybernetics, Artificial Life, and the New AI", MIT Press• Heikki Hyötyniemi (2006). Neocybernetics in Biological Systems (http:/ / neocybernetics. com/ report151/ ).

Espoo: Helsinki University of Technology, Control Engineering Laboratory.• Eden Medina, "Designing Freedom, Regulating a Nation: Socialist Cybernetics in Allende's Chile." Journal of

Latin American Studies 38 (2006):571-606.• Lars Bluma, (2005), Norbert Wiener und die Entstehung der Kybernetik im Zweiten Weltkrieg, Münster.• Francis Heylighen, and Cliff Joslyn (2001). " Cybernetics and Second Order Cybernetics (http:/ / pespmc1. vub.

ac. be/ Papers/ Cybernetics-EPST. pdf)", in: R.A. Meyers (ed.), Encyclopedia of Physical Science & Technology

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(3rd ed.), Vol. 4, (Academic Press, New York), p. 155-170.• Charles François (1999). " Systemics and cybernetics in a historical perspective (http:/ / www. uni-klu. ac. at/

~gossimit/ ifsr/ francois/ papers/ systemics_and_cybernetics_in_a_historical_perspective. pdf)". In: SystemsResearch and Behavioral Science. Vol 16, pp. 203–219 (1999)

• Heinz von Foerster, (1995), Ethics and Second-Order Cybernetics (http:/ / www. stanford. edu/ group/ SHR/ 4-2/text/ foerster. html).

• Steve J. Heims (1993), Constructing a Social Science for Postwar America. The Cybernetics Group, 1946-1953,Cambridge University Press, London, UK.

• Paul Pangaro (1990), "Cybernetics — A Definition", Eprint (http:/ / pangaro. com/ published/ cyber-macmillan.html).

• Stuart Umpleby (1989), "The science of cybernetics and the cybernetics of science" (ftp:/ / ftp. vub. ac. be/ pub/projects/ Principia_Cybernetica/ Papers_Umpleby/ Science-Cybernetics. txt), in: Cybernetics and Systems", Vol.21, No. 1, (1990), pp. 109–121.

• Michael A. Arbib (1987, 1964) Brains, Machines, and Mathematics (http:/ / www. amazon. com/Brains-Machines-Mathematics-Michael-Arbib/ dp/ 0387965394) Springer.

• B.C. Patten, and E.P. Odum (1981), "The Cybernetic Nature of Ecosystems", The American Naturalist 118,886-895.

• Hans Joachim Ilgauds (1980), Norbert Wiener, Leipzig.• Steve J. Heims (1980), John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life

and Death, 3. Aufl., Cambridge.• Stafford Beer (1974), Designing Freedom, John Wiley, London and New York, 1975.• Gordon Pask (1972), " Cybernetics (http:/ / www. cybsoc. org/ gcyb. htm)", entry in Encyclopædia Britannica

1972.• Helvey, T.C. The Age of Information: An Interdisciplinary Survey of Cybernetics. Englewood Cliffs, N.J.:

Educational Technology Publications, 1971.• Roy Ascott (1967). Behaviourist Art and the Cybernetic Vision. Cybernetica, Journal of the International

Association for Cybernetics (Namur), 10, pp. 25–56• W. Ross Ashby (1956), Introduction to Cybernetics. Methuen, London, UK. PDF text (http:/ / pespmc1. vub. ac.

be/ books/ IntroCyb. pdf).• Norbert Wiener (1948), Cybernetics or Control and Communication in the Animal and the Machine, (Hermann &

Cie Editeurs, Paris, The Technology Press, Cambridge, Mass., John Wiley & Sons Inc., New York, 1948).

External linksGeneral• Norbert Wiener and Stefan Odobleja - A Comparative Analysis (http:/ / www. bu. edu/ wcp/ Papers/ Comp/

CompJurc. htm)• Reading List for Cybernetics (http:/ / www. cscs. umich. edu/ ~crshalizi/ notabene/ cybernetics. html)• Principia Cybernetica Web (http:/ / pespmc1. vub. ac. be/ DEFAULT. html)• Web Dictionary of Cybernetics and Systems (http:/ / pespmc1. vub. ac. be/ ASC/ indexASC. html)• Glossary Slideshow (136 slides) (http:/ / www. gwu. edu/ ~asc/ slide/ s1. html)• Basics of Cybernetics (http:/ / www. smithsrisca. demon. co. uk/ cybernetics. html)• What is Cybernetics? (http:/ / www. youtube. com/ watch?v=_hjAXkNbPfk) Livas short introductory videos on

YouTube• A History of Systemic and Cybernetic Thought. From Homeostasis to the Teardrop (http:/ / www. pclibya. com/

cybernetic_teardrop. pdf)Societies

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• American Society for Cybernetics (http:/ / www. asc-cybernetics. org/ )• IEEE Systems, Man, & Cybernetics Society (http:/ / www. ieeesmc. org/ )• The Cybernetics Society (http:/ / www. cybsoc. org)

Systems theorySystems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that canbe applied to all types of systems in all fields of research. The term does not yet have a well-established, precisemeaning, but systems theory can reasonably be considered a specialization of systems thinking and a generalizationof systems science. The term originates from Bertalanffy's General System Theory (GST) and is used in later effortsin other fields, such as the action theory of Talcott Parsons and the system-theory of Niklas Luhmann.In this context the word systems is used to refer specifically to self-regulating systems, i.e. that are self-correctingthrough feedback. Self-regulating systems are found in nature, including the physiological systems of our body, inlocal and global ecosystems, and in climate.

Overview

Margaret Mead was an influential figure in systemstheory.

Contemporary ideas from systems theory have grown withdiversified areas, exemplified by the work of Béla H. Bánáthy,ecological systems with Howard T. Odum, Eugene Odum and FritjofCapra, organizational theory and management with individuals suchas Peter Senge, interdisciplinary study with areas like HumanResource Development from the work of Richard A. Swanson, andinsights from educators such as Debora Hammond and AlfonsoMontuori. As a transdisciplinary, interdisciplinary andmultiperspectival domain, the area brings together principles andconcepts from ontology, philosophy of science, physics, computerscience, biology, and engineering as well as geography, sociology,political science, psychotherapy (within family systems therapy) andeconomics among others. Systems theory thus serves as a bridge for interdisciplinary dialogue between autonomousareas of study as well as within the area of systems science itself.

In this respect, with the possibility of misinterpretations, von Bertalanffy[1] believed a general theory of systems"should be an important regulative device in science," to guard against superficial analogies that "are useless inscience and harmful in their practical consequences." Others remain closer to the direct systems concepts developedby the original theorists. For example, Ilya Prigogine, of the Center for Complex Quantum Systems at the Universityof Texas, Austin, has studied emergent properties, suggesting that they offer analogues for living systems. Thetheories of autopoiesis of Francisco Varela and Humberto Maturana are a further development in this field.Important names in contemporary systems science include Russell Ackoff, Béla H. Bánáthy, Anthony Stafford Beer,Peter Checkland, Robert L. Flood, Fritjof Capra, Michael C. Jackson, Edgar Morin and Werner Ulrich, amongothers.With the modern foundations for a general theory of systems following the World Wars, Ervin Laszlo, in the preface for Bertalanffy's book Perspectives on General System Theory, maintains that the translation of "general system theory" from German into English has "wrought a certain amount of havoc".[2] The preface explains that the original concept of a general system theory was "Allgemeine Systemtheorie (or Lehre)", pointing out the fact that "Theorie" (or "Lehre") just as "Wissenschaft" (translated Scholarship), "has a much broader meaning in German than the closest English words ‘theory’ and ‘science'".[2] With these ideas referring to an organized body of knowledge and

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"any systematically presented set of concepts, whether they are empirical, axiomatic, or philosophical, "Lehre" isassociated with theory and science in the etymology of general systems, but also does not translate from the Germanvery well; "teaching" is the "closest equivalent", but "sounds dogmatic and off the mark".[2] While many of the rootmeanings for the idea of a "general systems theory" might have been lost in the translation and many were led tobelieve that the systems theorists had articulated nothing but a pseudoscience, systems theory became anomenclature that early investigators used to describe the interdependence of relationships in organization bydefining a new way of thinking about science and scientific paradigms.A system from this frame of reference is composed of regularly interacting or interrelating groups of activities. Forexample, in noting the influence in organizational psychology as the field evolved from "an individually orientedindustrial psychology to a systems and developmentally oriented organizational psychology," it was recognized thatorganizations are complex social systems; reducing the parts from the whole reduces the overall effectiveness oforganizations.[3] This is different from conventional models that center on individuals, structures, departments andunits separate in part from the whole instead of recognizing the interdependence between groups of individuals,structures and processes that enable an organization to function. Laszlo[4] explains that the new systems view oforganized complexity went "one step beyond the Newtonian view of organized simplicity" in reducing the parts fromthe whole, or in understanding the whole without relation to the parts. The relationship between organizations andtheir environments became recognized as the foremost source of complexity and interdependence. In most cases thewhole has properties that cannot be known from analysis of the constituent elements in isolation. Béla H. Bánáthy,who argued—along with the founders of the systems society—that "the benefit of humankind" is the purpose ofscience, has made significant and far-reaching contributions to the area of systems theory. For the Primer Group atISSS, Bánáthy defines a perspective that iterates this view:

The systems view is a world-view that is based on the discipline of SYSTEM INQUIRY. Central to systemsinquiry is the concept of SYSTEM. In the most general sense, system means a configuration of partsconnected and joined together by a web of relationships. The Primer group defines system as a family ofrelationships among the members acting as a whole. Von Bertalanffy defined system as "elements in standingrelationship.—[5]

Similar ideas are found in learning theories that developed from the same fundamental concepts, emphasizing howunderstanding results from knowing concepts both in part and as a whole. In fact, Bertalanffy’s organismicpsychology paralleled the learning theory of Jean Piaget.[6] Interdisciplinary perspectives are critical in breakingaway from industrial age models and thinking where history is history and math is math, the arts and sciencesspecialized and separate, and where teaching is treated as behaviorist conditioning.[7] The influential contemporarywork of Peter Senge[8] provides detailed discussion of the commonplace critique of educational systems grounded inconventional assumptions about learning, including the problems with fragmented knowledge and lack of holisticlearning from the "machine-age thinking" that became a "model of school separated from daily life." It is in this waythat systems theorists attempted to provide alternatives and an evolved ideation from orthodox theories withindividuals such as Max Weber, Émile Durkheim in sociology and Frederick Winslow Taylor in scientificmanagement, which were grounded in classical assumptions.[9] The theorists sought holistic methods by developingsystems concepts that could be integrated with different areas.The contradiction of reductionism in conventional theory (which has as its subject a single part) is simply an example of changing assumptions. The emphasis with systems theory shifts from parts to the organization of parts, recognizing interactions of the parts are not "static" and constant but "dynamic" processes. Conventional closed systems were questioned with the development of open systems perspectives. The shift was from absolute and universal authoritative principles and knowledge to relative and general conceptual and perceptual knowledge,[10]

still in the tradition of theorists that sought to provide means in organizing human life. Meaning, the history of ideas that preceded were rethought not lost. Mechanistic thinking was particularly critiqued, especially the industrial-age

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mechanistic metaphor of the mind from interpretations of Newtonian mechanics by Enlightenment philosophers andlater psychologists that laid the foundations of modern organizational theory and management by the late 19thcentury.[11] Classical science had not been overthrown, but questions arose over core assumptions that historicallyinfluenced organized systems, within both social and technical sciences.

History

Timeline

Precursors

• Saint-Simon (1760–1825), Karl Marx (1817–1883), Herbert Spencer (1820–1903), Rudolf Clausius (1822–1888), Vilfredo Pareto(1848–1923), Émile Durkheim (1858–1917), Alexander Bogdanov (1873–1928), Nicolai Hartmann (1882–1950), Stafford Beer (1926–2002),Robert Maynard Hutchins (1929–1951), among others

Pioneers

• 1946-1953 Macy conferences• 1948 Norbert Wiener publishes Cybernetics or Control and Communication in the Animal and the Machine• 1954 Ludwig von Bertalanffy, Anatol Rapoport, Ralph W. Gerard, Kenneth Boulding establish Society for the Advancement of General

Systems Theory, in 1956 renamed to Society for General Systems Research.• 1955 W. Ross Ashby publishes Introduction to Cybernetics• 1968 Ludwig von Bertalanffy publishes General System theory: Foundations, Development, Applications

Developments

• 1970-1980s Second-order cybernetics developed by Heinz von Foerster, Gregory Bateson, Humberto Maturana and others• 1971-1973 Cybersyn, rudimentary internet and cybernetic system for democratic economic planning developed in Chile under Allende

government by Stafford Beer• 1970s Catastrophe theory (René Thom, E.C. Zeeman) Dynamical systems in mathematics.• 1977 Ilya Prigogine received the Nobel Prize for his works on self-organization, conciliating important systems theory concepts with system

thermodynamics.• 1980s Chaos theory David Ruelle, Edward Lorenz, Mitchell Feigenbaum, Steve Smale, James A. Yorke• 1986 Context theory, Anthony Wilden• 1988 International Society for Systems Science• 1990 Complex adaptive systems (CAS), John H. Holland, Murray Gell-Mann, W. Brian Arthur

Whether considering the first systems of written communication with Sumerian cuneiform to Mayan numerals, orthe feats of engineering with the Egyptian pyramids, systems thinking in essence dates back to antiquity.Differentiated from Western rationalist traditions of philosophy, C. West Churchman often identified with the IChing as a systems approach sharing a frame of reference similar to pre-Socratic philosophy and Heraclitus.[12] VonBertalanffy traced systems concepts to the philosophy of G.W. von Leibniz and Nicholas of Cusa's coincidentiaoppositorum. While modern systems are considerably more complicated, today's systems are embedded in history.An important step to introduce the systems approach, into (rationalist) hard sciences of the 19th century, was theenergy transformation, by figures like James Joule and Sadi Carnot. Then, the Thermodynamic of this century, withRudolf Clausius, Josiah Gibbs and others, built the system reference model, as a formal scientific object.Systems theory as an area of study specifically developed following the World Wars from the work of Ludwig vonBertalanffy, Anatol Rapoport, Kenneth E. Boulding, William Ross Ashby, Margaret Mead, Gregory Bateson, C.West Churchman and others in the 1950s, specifically catalyzed by the cooperation in the Society for GeneralSystems Research. Cognizant of advances in science that questioned classical assumptions in the organizationalsciences, Bertalanffy's idea to develop a theory of systems began as early as the interwar period, publishing "AnOutline for General Systems Theory" in the British Journal for the Philosophy of Science, Vol 1, No. 2, by 1950.Where assumptions in Western science from Greek thought with Plato and Aristotle to Newton's Principia havehistorically influenced all areas from the hard to social sciences (see David Easton's seminal development of the"political system" as an analytical construct), the original theorists explored the implications of twentieth centuryadvances in terms of systems.

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Subjects like complexity, self-organization, connectionism and adaptive systems had already been studied in the1940s and 1950s. In fields like cybernetics, researchers like Norbert Wiener, William Ross Ashby, John vonNeumann and Heinz von Foerster examined complex systems using mathematics. John von Neumann discoveredcellular automata and self-reproducing systems, again with only pencil and paper. Aleksandr Lyapunov and JulesHenri Poincaré worked on the foundations of chaos theory without any computer at all. At the same time Howard T.Odum, the radiation ecologist, recognised that the study of general systems required a language that could depictenergetics, thermodynamic and kinetics at any system scale. Odum developed a general systems, or Universallanguage, based on the circuit language of electronics to fulfill this role, known as the Energy Systems Language.Between 1929-1951, Robert Maynard Hutchins at the University of Chicago had undertaken efforts to encourageinnovation and interdisciplinary research in the social sciences, aided by the Ford Foundation with theinterdisciplinary Division of the Social Sciences established in 1931.[13] Numerous scholars had been activelyengaged in ideas before (Tectology of Alexander Bogdanov published in 1912-1917 is a remarkable example), but in1937 von Bertalanffy presented the general theory of systems for a conference at the University of Chicago.The systems view was based on several fundamental ideas. First, all phenomena can be viewed as a web ofrelationships among elements, or a system. Second, all systems, whether electrical, biological, or social, havecommon patterns, behaviors, and properties that can be understood and used to develop greater insight into thebehavior of complex phenomena and to move closer toward a unity of science. System philosophy, methodology andapplication are complementary to this science.[2] By 1956, the Society for General Systems Research wasestablished, renamed the International Society for Systems Science in 1988. The Cold War affected the researchproject for systems theory in ways that sorely disappointed many of the seminal theorists. Some began to recognizetheories defined in association with systems theory had deviated from the initial General Systems Theory (GST)view.[14] The economist Kenneth Boulding, an early researcher in systems theory, had concerns over themanipulation of systems concepts. Boulding concluded from the effects of the Cold War that abuses of power alwaysprove consequential and that systems theory might address such issues.[15] Since the end of the Cold War, there hasbeen a renewed interest in systems theory with efforts to strengthen an ethical view.

Developments in system theories

General systems research and systems inquiryMany early systems theorists aimed at finding a general systems theory that could explain all systems in all fields ofscience. The term goes back to Bertalanffy's book titled "General System theory: Foundations, Development,Applications" from 1968.[6] According to Von Bertalanffy, he developed the "allgemeine Systemlehre" (generalsystems teachings) first via lectures beginning in 1937 and then via publications beginning in 1946.[16]

Von Bertalanffy's objective was to bring together under one heading the organismic science that he had observed inhis work as a biologist. His desire was to use the word "system" to describe those principles which are common tosystems in general. In GST, he writes:

...there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective oftheir particular kind, the nature of their component elements, and the relationships or "forces" between them. Itseems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principlesapplying to systems in general.[17]

Ervin Laszlo[18] in the preface of von Bertalanffy's book Perspectives on General System Theory:[19]

Thus when von Bertalanffy spoke of Allgemeine Systemtheorie it was consistent with his view that he wasproposing a new perspective, a new way of doing science. It was not directly consistent with an interpretationoften put on "general system theory", to wit, that it is a (scientific) "theory of general systems." To criticize itas such is to shoot at straw men. Von Bertalanffy opened up something much broader and of much greatersignificance than a single theory (which, as we now know, can always be falsified and has usually an

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ephemeral existence): he created a new paradigm for the development of theories.Ludwig von Bertalanffy outlines systems inquiry into three major domains: Philosophy, Science, and Technology. Inhis work with the Primer Group, Béla H. Bánáthy generalized the domains into four integratable domains ofsystemic inquiry:

Domain Description

Philosophy the ontology, epistemology, and axiology of systems;

Theory a set of interrelated concepts and principles applying to all systems

Methodology the set of models, strategies, methods, and tools that instrumentalize systems theory and philosophy

Application the application and interaction of the domains

These operate in a recursive relationship, he explained. Integrating Philosophy and Theory as Knowledge, andMethod and Application as action, Systems Inquiry then is knowledgeable action.[20]

CyberneticsThe term cybernetics derives from a Greek word which meant steersman, and which is the origin of English wordssuch as "govern". Cybernetics is the study of feedback and derived concepts such as communication and control inliving organisms, machines and organisations. Its focus is how anything (digital, mechanical or biological) processesinformation, reacts to information, and changes or can be changed to better accomplish the first two tasks.The terms "systems theory" and "cybernetics" have been widely used as synonyms. Some authors use the termcybernetic systems to denote a proper subset of the class of general systems, namely those systems that includefeedback loops. However Gordon Pask's differences of eternal interacting actor loops (that produce finite products)makes general systems a proper subset of cybernetics. According to Jackson (2000), von Bertalanffy promoted anembryonic form of general system theory (GST) as early as the 1920s and 1930s but it was not until the early 1950sit became more widely known in scientific circles.Threads of cybernetics began in the late 1800s that led toward the publishing of seminal works (e.g., Wiener'sCybernetics in 1948 and von Bertalanffy's General Systems Theory in 1968). Cybernetics arose more fromengineering fields and GST from biology. If anything it appears that although the two probably mutually influencedeach other, cybernetics had the greater influence. Von Bertalanffy (1969) specifically makes the point ofdistinguishing between the areas in noting the influence of cybernetics: "Systems theory is frequently identified withcybernetics and control theory. This again is incorrect. Cybernetics as the theory of control mechanisms intechnology and nature is founded on the concepts of information and feedback, but as part of a general theory ofsystems;" then reiterates: "the model is of wide application but should not be identified with 'systems theory' ingeneral", and that "warning is necessary against its incautious expansion to fields for which its concepts are notmade." (17-23). Jackson (2000) also claims von Bertalanffy was informed by Alexander Bogdanov's three volumeTectology that was published in Russia between 1912 and 1917, and was translated into German in 1928. He alsostates it is clear to Gorelik (1975) that the "conceptual part" of general system theory (GST) had first been put inplace by Bogdanov. The similar position is held by Mattessich (1978) and Capra (1996). Ludwig von Bertalanffynever even mentioned Bogdanov in his works, which Capra (1996) finds "surprising".Cybernetics, catastrophe theory, chaos theory and complexity theory have the common goal to explain complex systems that consist of a large number of mutually interacting and interrelated parts in terms of those interactions. Cellular automata (CA), neural networks (NN), artificial intelligence (AI), and artificial life (ALife) are related fields, but they do not try to describe general (universal) complex (singular) systems. The best context to compare the different "C"-Theories about complex systems is historical, which emphasizes different tools and methodologies, from pure mathematics in the beginning to pure computer science now. Since the beginning of chaos theory when Edward Lorenz accidentally discovered a strange attractor with his computer, computers have become an

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indispensable source of information. One could not imagine the study of complex systems without the use ofcomputers today.

Complex adaptive systemsComplex adaptive systems are special cases of complex systems. They are complex in that they are diverse and madeup of multiple interconnected elements and adaptive in that they have the capacity to change and learn fromexperience. The term complex adaptive systems was coined at the interdisciplinary Santa Fe Institute (SFI), by JohnH. Holland, Murray Gell-Mann and others. However, the approach of the complex adaptive systems does not takeinto account the adoption of information which enables people to use it.CAS ideas and models are essentially evolutionary. Accordingly, the theory of complex adaptive systems bridgesdevelopments of the system theory with the ideas of 'generalized Darwinism', which suggests that Darwinianprinciples of evolution help explain a wide range of phenomena.

Applications of system theories

Living systems theoryLiving systems theory is an offshoot of von Bertalanffy's general systems theory, created by James Grier Miller,which was intended to formalize the concept of "life". According to Miller's original conception as spelled out in hismagnum opus Living Systems, a "living system" must contain each of 20 "critical subsystems", which are defined bytheir functions and visible in numerous systems, from simple cells to organisms, countries, and societies. In LivingSystems Miller provides a detailed look at a number of systems in order of increasing size, and identifies hissubsystems in each.James Grier Miller (1978) wrote a 1,102 pages volume to present his living systems theory. He constructed a generaltheory of living systems by focusing on concrete systems—nonrandom accumulations of matter-energy in physicalspace-time organized into interacting, interrelated subsystems or components. Slightly revising the original model adozen years later, he distinguished eight "nested" hierarchical levels in such complex structures. Each level is"nested" in the sense that each higher level contains the next lower level in a nested fashion.

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Organizational theory

Kurt Lewin attended the Macyconferences and is commonlyidentified as the founder of the

movement to study groupsscientifically.

The systems framework is also fundamental to organizational theory asorganizations are complex dynamic goal-oriented processes. One of the earlythinkers in the field was Alexander Bogdanov, who developed his Tectology, atheory widely considered a precursor of von Bertalanffy's GST, aiming to modeland design human organizations (see Mattessich 1978, Capra 1996). Kurt Lewinwas particularly influential in developing the systems perspective withinorganizational theory and coined the term "systems of ideology", from hisfrustration with behavioral psychologies that became an obstacle to sustainablework in psychology.[21] Jay Forrester with his work in dynamics and managementalongside numerous theorists including Edgar Schein that followed in theirtradition since the Civil Rights Era have also been influential.

The systems to organizations relies heavily upon achieving negative entropythrough openness and feedback. A systemic view on organizations istransdisciplinary and integrative. In other words, it transcends the perspectives ofindividual disciplines, integrating them on the basis of a common "code", or moreexactly, on the basis of the formal apparatus provided by systems theory. Thesystems approach gives primacy to the interrelationships, not to the elements of thesystem. It is from these dynamic interrelationships that new properties of thesystem emerge. In recent years, systems thinking has been developed to provide techniques for studying systems inholistic ways to supplement traditional reductionistic methods. In this more recent tradition, systems theory inorganizational studies is considered by some as a humanistic extension of the natural sciences.

Software and computingIn the 1960s, systems theory was adopted by the post John Von Neumann computing and information technologyfield and, in fact, formed the basis of structured analysis and structured design (see also Larry Constantine, TomDeMarco and Ed Yourdon). It was also the basis for early software engineering and computer-aided softwareengineering principles.By the 1970s, General Systems Theory (GST) was the fundamental underpinning of most commercial softwaredesign techniques, and by the 1980, W. Vaughn Frick and Albert F. Case, Jr. had used GST to design the "missinglink" transformation from system analysis (defining what's needed in a system) to system design (what's actuallyimplemented) using the Yourdon/DeMarco notation. These principles were incorporated into computer-aidedsoftware engineering tools delivered by Nastec Corporation, Transform Logic, Inc., KnowledgeWare (see FranTarkenton and James Martin), Texas Instruments, Arthur Andersen and ultimately IBM Corporation.

Sociology and SociocyberneticsSystems theory has also been developed within sociology. An important figure in the sociological systems perspective as developed from GST is Walter Buckley (who from Bertalanffy's theory). Niklas Luhmann (see Luhmann 1994) is also predominant in the literatures for sociology and systems theory. Miller's living systems theory was particularly influential in sociology from the time of the early systems movement. Models for dynamic equilibrium in systems analysis that contrasted classical views from Talcott Parsons and George Homans were influential in integrating concepts with the general movement. With the renewed interest in systems theory on the rise since the 1990s, Bailey (1994) notes the concept of systems in sociology dates back to Auguste Comte in the 19th century, Herbert Spencer and Vilfredo Pareto, and that sociology was readying into its centennial as the new systems theory was emerging following the World Wars. To explore the current inroads of systems theory into

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sociology (primarily in the form of complexity science) see sociology and complexity science.In sociology, members of Research Committee 51 of the International Sociological Association (which focuses onsociocybernetics), have sought to identify the sociocybernetic feedback loops which, it is argued, primarily controlthe operation of society. On the basis of research largely conducted in the area of education, Raven (1995) has, forexample, argued that it is these sociocybernetic processes which consistently undermine well intentioned publicaction and are currently heading our species, at an exponentially increasing rate, toward extinction. Seesustainability. He suggests that an understanding of these systems processes will allow us to generate the kind of(non "common-sense") targeted interventions that are required for things to be otherwise - i.e. to halt the destructionof the planet.

Systems biologySystems biology is a term used to describe a number of trends in bioscience research, and a movement which drawson those trends. Proponents describe systems biology as a biology-based inter-disciplinary study field that focuses oncomplex interactions in biological systems, claiming that it uses a new perspective (holism instead of reduction).Particularly from year 2000 onwards, the term is used widely in the biosciences, and in a variety of contexts. Anoften stated ambition of systems biology is the modeling and discovery of emergent properties, properties of asystem whose theoretical description is only possible using techniques which fall under the remit of systems biology.The term systems biology is thought to have been created by Ludwig von Bertalanffy in 1928.[22]

System dynamicsSystem Dynamics was founded in the late 1950s by Jay W. Forrester of the MIT Sloan School of Management withthe establishment of the MIT System Dynamics Group. At that time, he began applying what he had learned aboutsystems during his work in electrical engineering to everyday kinds of systems. Determining the exact date of thefounding of the field of system dynamics is difficult and involves a certain degree of arbitrariness. Jay W. Forresterjoined the faculty of the Sloan School at MIT in 1956, where he then developed what is now System Dynamics. Thefirst published article by Jay W. Forrester in the Harvard Business Review on "Industrial Dynamics", was publishedin 1958. The members of the System Dynamics Society have chosen 1957 to mark the occasion as it is the year inwhich the work leading to that article, which described the dynamics of a manufacturing supply chain, was done.As an aspect of systems theory, system dynamics is a method for understanding the dynamic behavior of complexsystems. The basis of the method is the recognition that the structure of any system — the many circular,interlocking, sometimes time-delayed relationships among its components — is often just as important indetermining its behavior as the individual components themselves. Examples are chaos theory and social dynamics.It is also claimed that, because there are often properties-of-the-whole which cannot be found among theproperties-of-the-elements, in some cases the behavior of the whole cannot be explained in terms of the behavior ofthe parts. An example is the properties of these letters which when considered together can give rise to meaningwhich does not exist in the letters by themselves. This further explains the integration of tools, like language, as amore parsimonious process in the human application of easiest path adaptability through interconnected systems.

Systems engineeringSystems engineering is an interdisciplinary approach and means for enabling the realization and deployment ofsuccessful systems. It can be viewed as the application of engineering techniques to the engineering of systems, aswell as the application of a systems approach to engineering efforts.[23] Systems engineering integrates otherdisciplines and specialty groups into a team effort, forming a structured development process that proceeds fromconcept to production to operation and disposal. Systems engineering considers both the business and the technicalneeds of all customers, with the goal of providing a quality product that meets the user needs.[24]

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Systems psychologySystems psychology is a branch of psychology that studies human behaviour and experience in complex systems. Itis inspired by systems theory and systems thinking, and based on the theoretical work of Roger Barker, GregoryBateson, Humberto Maturana and others. It is an approach in psychology, in which groups and individuals, areconsidered as systems in homeostasis. Systems psychology "includes the domain of engineering psychology, but inaddition is more concerned with societal systems and with the study of motivational, affective, cognitive and groupbehavior than is engineering psychology."[25] In systems psychology "characteristics of organizational behaviour forexample individual needs, rewards, expectations, and attributes of the people interacting with the systems areconsidered in the process in order to create an effective system".[26] The Systems psychology includes an illusion ofhomeostatic systems, although most of the living systems are in a continuous disequilibrium of various degrees.

References[1] Bertalanffy (1950: 142)[2] (Laszlo 1974)[3] (Schein 1980: 4-11)[4] Laslo (1972: 14-15)[5] (Banathy 1997: ¶ 22)[6] 1968, General System theory: Foundations, Development, Applications, New York: George Braziller, revised edition 1976: ISBN

0-8076-0453-4[7] (see Steiss 1967; Buckley, 1967)[8] Peter Senge (2000: 27-49)[9] (Bailey 1994: 3-8; see also Owens 2004)[10] (Bailey 1994: 3-8)[11] (Bailey 1994; Flood 1997; Checkland 1999; Laszlo 1972)[12] (Hammond 2003: 12-13)[13] Hammond 2003: 5-9[14] Hull 1970[15] (Hammond 2003: 229-233)[16] Karl Ludwig von Bertalanffy: ... aber vom Menschen wissen wir nichts, (English title: Robots, Men and Minds), translated by Dr.

Hans-Joachim Flechtner. page 115. Econ Verlag GmbH (1970), Düsseldorf, Wien. 1st edition.[17] (GST p.32)[18] perspectives_on_general_system_theory [ProjectsISSS] (http:/ / projects. isss. org/ perspectives_on_general_system_theory)[19] von Bertalanffy, Ludwig, (1974) Perspectives on General System Theory Edited by Edgar Taschdjian. George Braziller, New York[20] main_systemsinquiry [ProjectsISSS] (http:/ / projects. isss. org/ Main/ SystemsInquiry)[21] (see Ash 1992: 198-207)[22] 1928, Kritische Theorie der Formbildung, Borntraeger. In English: Modern Theories of Development: An Introduction to Theoretical

Biology, Oxford University Press, New York: Harper, 1933[23] Thomé, Bernhard (1993). Systems Engineering: Principles and Practice of Computer-based Systems Engineering. Chichester: John Wiley &

Sons. ISBN 0-471-93552-2.[24] INCOSE. "What is Systems Engineering" (http:/ / www. incose. org/ practice/ whatissystemseng. aspx). . Retrieved 2006-11-26.[25] Lester R. Bittel and Muriel Albers Bittel (1978), Encyclopedia of Professional Management, McGraw-Hill, ISBN 0-07-005478-9, p.498.[26] Michael M. Behrmann (1984), Handbook of Microcomputers in Special Education. College Hill Press. ISBN 0-933014-35-X. Page 212.

Further reading• Ackoff, R. (1978). The art of problem solving. New York: Wiley.• Ash, M.G. (1992). "Cultural Contexts and Scientific Change in Psychology: Kurt Lewin in Iowa." American

Psychologist, Vol. 47, No. 2, pp. 198–207.• Bailey, K.D. (1994). Sociology and the New Systems Theory: Toward a Theoretical Synthesis. New York: State

of New York Press.• Bánáthy, B (1996) Designing Social Systems in a Changing World New York Plenum• Bánáthy, B. (1991) Systems Design of Education. Englewood Cliffs: Educational Technology Publications

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• Bánáthy, B. (1992) A Systems View of Education. Englewood Cliffs: Educational Technology Publications.ISBN 0-87778-245-8

• Bánáthy, B.H. (1997). "A Taste of Systemics" (http:/ / www. newciv. org/ ISSS_Primer/ asem04bb. html), ThePrimer Project, Retrieved May 14, (2007)

• Bateson, G. (1979). Mind and nature: A necessary unity. New York: Ballantine• Bausch, Kenneth C. (2001) The Emerging Consensus in Social Systems Theory, Kluwer Academic New York

ISBN 0-306-46539-6• Ludwig von Bertalanffy (1968). General System Theory: Foundations, Development, Applications New York:

George Braziller• Bertalanffy, L. von (1950), "An Outline of General System Theory" (http:/ / www. isnature. org/ events/ 2009/

Summer/ r/ Bertalanffy1950-GST_Outline_SELECT. pdf), British Journal for the Philosophy of Science Vol. 1(No. 2), retrieved 24 October 2010

• Bertalanffy, L. von. (1955). "An Essay on the Relativity of Categories." Philosophy of Science, Vol. 22, No. 4,pp. 243–263.

• Bertalanffy, Ludwig von. (1968). Organismic Psychology and Systems Theory. Worchester: Clark UniversityPress.

• Bertalanffy, Ludwig Von. (1974). Perspectives on General System Theory Edited by Edgar Taschdjian. GeorgeBraziller, New York.

• Buckley, W. (1967). Sociology and Modern Systems Theory. New Jersey: Englewood Cliffs.• Mario Bunge (1979) Treatise on Basic Philosophy, Volume 4. Ontology II A World of Systems. Dordrecht,

Netherlands: D. Reidel.• Capra, F. (1997). The Web of Life-A New Scientific Understanding of Living Systems, Anchor ISBN

978-0-385-47676-8• Checkland, P. (1981). Systems thinking, Systems practice. New York: Wiley.• Checkland, P. 1997. Systems Thinking, Systems Practice. Chichester: John Wiley & Sons, Ltd.• Churchman, C.W. (1968). The systems approach. New York: Laurel.• Churchman, C.W. (1971). The design of inquiring systems. New York: Basic Books.• Corning, P. (1983) The Synergism Hupothesis: A Theory of Progressive Evolution. New York: McGraw Hill• Davidson, Mark. (1983). Uncommon Sense: The Life and Thought of Ludwig von Bertalanffy, Father of General

Systems Theory. Los Angeles: J.P. Tarcher, Inc.• Durand, D. La systémique, Presses Universitaires de France• Flood, R.L. 1999. Rethinking the Fifth Discipline: Learning within the unknowable." London: Routledge.• Charles François. (2004). Encyclopedia of Systems and Cybernetics, Introducing the 2nd Volume (http:/ /

benking. de/ systems/ encyclopedia/ concepts-and-models. htm) and further links to the ENCYCLOPEDIA, K GSaur, Munich (http:/ / benking. de/ encyclopedia/ ) see also (http:/ / wwwu. uni-klu. ac. at/ gossimit/ ifsr/ francois/encyclopedia. htm)

• Kahn, Herman. (1956). Techniques of System Analysis, Rand Corporation• Laszlo, E. (1995). The Interconnected Universe. New Jersey, World Scientific. ISBN 981-02-2202-5• François, C. (1999). Systemics and Cybernetics in a Historical Perspective (http:/ / www. uni-klu. ac. at/

~gossimit/ ifsr/ francois/ papers/ systemics_and_cybernetics_in_a_historical_perspective. pdf)• Jantsch, E. (1980). The Self Organizing Universe. New York: Pergamon.• Gorelik, G. (1975) Reemergence of Bogdanov's Tektology in. Soviet Studies of Organization, Academy of

Management Journal. 18/2, pp. 345–357• Hammond, D. 2003. The Science of Synthesis. Colorado: University of Colorado Press.• Hinrichsen, D. and Pritchard, A.J. (2005) Mathematical Systems Theory. New York: Springer. ISBN

978-3-540-44125-0• Hull, D.L. 1970. "Systemic Dynamic Social Theory." Sociological Quarterly, Vol. 11, Issue 3, pp. 351–363.

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• Hyötyniemi, H. (2006). Neocybernetics in Biological Systems (http:/ / neocybernetics. com/ report151/ ). Espoo:Helsinki University of Technology, Control Engineering Laboratory.

• Jackson, M.C. 2000. Systems Approaches to Management. London: Springer.• Klir, G.J. 1969. An Approach to General Systems Theory. New York: Van Nostrand Reinhold Company.• Ervin László 1972. The Systems View of the World. New York: George Brazilier.• Laszlo, E. (1972a). The systems view of the world. The natural philosophy of the new developments in the

sciences. New York: George Brazillier. ISBN 0-8076-0636-7• Laszlo, E. (1972b). Introduction to systems philosophy. Toward a new paradigm of contemporary thought. San

Francisco: Harper.• Laszlo, Ervin. 1996. The Systems View of the World. Hampton Press, NJ. (ISBN 1-57273-053-6).• Lemkow, A. (1995) The Wholeness Principle: Dynamics of Unity Within Science, Religion & Society. Quest

Books, Wheaton.• Niklas Luhmann (1996),"Social Systems",Stanford University Press, Palo Alto, CA• Mattessich, R. (1978) Instrumental Reasoning and Systems Methodology: An Epistemology of the Applied and

Social Sciences. Reidel, Boston• Minati, Gianfranco. Collen, Arne. (1997) Introduction to Systemics Eagleye books. ISBN 0-924025-06-9• Montuori, A. (1989). Evolutionary Competence. Creating the Future. Amsterdam: Gieben.• Morin, E. (2008). On Complexity. Cresskill, NJ: Hampton Press.• Odum, H. (1994) Ecological and General Systems: An introduction to systems ecology, Colorado University

Press, Colorado.• Olmeda, Christopher J. (1998). Health Informatics: Concepts of Information Technology in Health and Human

Services. Delfin Press. ISBN 0-9821442-1-0• Owens, R.G. (2004). Organizational Behavior in Education: Adaptive Leadership and School Reform, Eighth

Edition. Boston: Pearson Education, Inc.• Pharaoh, M.C. (online). Looking to systems theory for a reductive explanation of phenomenal experience and

evolutionary foundations for higher order thought (http:/ / homepage. ntlworld. com/ m. pharoah/ ) RetrievedDec.14 2007.

• Science as Paradigmatic Complexity by Wallace H. Provost Jr. (http:/ / philosophy. freeopenu. org/ mod/resource/ view. php?id=8721) 1984 in the International Journal of General Systems

• Schein, E.H. (1980). Organizational Psychology, Third Edition. New Jersey: Prentice-Hall.• Peter Senge (1990). The Fifth Discipline. The art and practice of the learning organization. New York:

Doubleday.• Senge, P., Ed. (2000). Schools That Learn: A Fifth Discipline Fieldbook for Educators, Parents, and Everyone

Who Cares About Education. New York: Doubleday Dell Publishing Group.• Snooks, G.D. (2008). "A general theory of complex living systems: Exploring the demand side of dynamics",

Complexity,13: 12-20.• Steiss, A.W. (1967). Urban Systems Dynamics. Toronto: Lexington Books.• Gerald Weinberg. (1975). An Introduction to General Systems Thinking (1975 ed., Wiley-Interscience) (2001 ed.

Dorset House).• Wiener, N. (1967). The human use of human beings. Cybernetics and Society. New York: Avon.• Young, O. R., “A Survey of General Systems Theory”, General Systems, vol. 9 (1964), pages 61–80. (overview

about different trends and tendencies, with bibliography)

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External links• Systems theory (http:/ / pespmc1. vub. ac. be/ SYSTHEOR. html) at Principia Cybernetica WebOrganizations• International Society for the System Sciences (http:/ / projects. isss. org/ Main/ Primer)• New England Complex Systems Institute (http:/ / www. necsi. edu/ )• System Dynamics Society (http:/ / www. systemdynamics. org/ )

Queueing theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis ofseveral related processes, including arriving at the (back of the) queue, waiting in the queue (essentially a storageprocess), and being served at the front of the queue. The theory permits the derivation and calculation of severalperformance measures including the average waiting time in the queue or the system, the expected number waitingor receiving service, and the probability of encountering the system in certain states, such as empty, full, having anavailable server or having to wait a certain time to be served.Queueing theory has applications in diverse fields,[1] including telecommunications,[2] traffic engineering,computing[3] and the design of factories, shops, offices and hospitals.[4]

OverviewThe word queue comes, via French, from the Latin cauda, meaning tail. The spelling "queueing" over "queuing" istypically encountered in the academic research field. In fact, one of the flagship journals of the profession is named"Queueing Systems".Queueing theory is generally considered a branch of operations research because the results are often used whenmaking business decisions about the resources needed to provide service. It is applicable in a wide variety ofsituations that may be encountered in business, commerce, industry, healthcare,[5] public service and engineering.Applications are frequently encountered in customer service situations as well as transport and telecommunication.Queueing theory is directly applicable to intelligent transportation systems, call centers, PABXs, networks,telecommunications, server queueing, mainframe computer of telecommunications terminals, advancedtelecommunications systems, and traffic flow.Notation for describing the characteristics of a queueing model was first suggested by David G. Kendall in 1953.Kendall's notation introduced an A/B/C queueing notation that can be found in all standard modern works onqueueing theory, for example, Tijms.[6]

The A/B/C notation designates a queueing system having A as interarrival time distribution, B as service timedistribution, and C as number of servers. For example, "G/D/1" would indicate a General (may be anything) arrivalprocess, a Deterministic (constant time) service process and a single server. More details on this notation are given inthe article about queueing models.

Queueing theory 163

HistoryAgner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the firstpaper on queueing theory in 1909.[7]

David G. Kendall introduced an A/B/C queueing notation in 1953. Important work on queueing theory used inmodern packet switching networks was performed in the early 1960s by Leonard Kleinrock.

Application to telephonyThe public switched telephone network (PSTN) is designed to accommodate the offered traffic intensity with only asmall loss. The performance of loss systems is quantified by their grade of service, driven by the assumption that ifsufficient capacity is not available, the call is refused and lost.[8] Alternatively, overflow systems make use ofalternative routes to divert calls via different paths — even these systems have a finite traffic carrying capacity.[8]

However, the use of queueing in PSTNs allows the systems to queue their customers' requests until free resourcesbecome available. This means that if traffic intensity levels exceed available capacity, customer's calls are not lost;customers instead wait until they can be served.[9] This method is used in queueing customers for the next availableoperator.A queueing discipline determines the manner in which the exchange handles calls from customers.[9] It defines theway they will be served, the order in which they are served, and the way in which resources are divided among thecustomers.[9] [10] Here are details of four queueing disciplines:First in first out

This principle states that customers are served one at a time and that the customer that has been waiting thelongest is served first.[10]

Last in first outThis principle also serves customers one at a time, however the customer with the shortest waiting time will beserved first.[10] Also known as a stack.

Processor sharingCustomers are served equally. Network capacity is shared between customers and they all effectivelyexperience the same delay.[10]

PriorityCustomers with high priority are served first.[10]

Queueing is handled by control processes within exchanges, which can be modelled using state equations.[9] [10]

Queueing systems use a particular form of state equations known as a Markov chain that models the system in eachstate.[9] Incoming traffic to these systems is modelled via a Poisson distribution and is subject to Erlang’s queueingtheory assumptions viz.[8]

• Pure-chance traffic – Call arrivals and departures are random and independent events.[8]

• Statistical equilibrium – Probabilities within the system do not change.[8]

• Full availability – All incoming traffic can be routed to any other customer within the network.[8]

• Congestion is cleared as soon as servers are free.[8]

Classic queueing theory involves complex calculations to determine waiting time, service time, server utilization andother metrics that are used to measure queueing performance.[9] [10]

Queueing theory 164

Queueing networksNetworks of queues are systems which contain an arbitrary, but finite, number m of queues. Customers, sometimesof different classes,[11] travel through the network and are served at the nodes. The state of a network can bedescribed by a vector , where ki is the number of customers at queue i. In open networks, customers canjoin and leave the system, whereas in closed networks the total number of customers within the system remainsfixed.The first significant result in the area was Jackson networks, for which an efficient product form equilibriumdistribution exists.

Role of Poisson process, exponential distributionsA useful queueing model represents a real-life system with sufficient accuracy and is analytically tractable. Aqueueing model based on the Poisson process and its companion exponential probability distribution often meetsthese two requirements. A Poisson process models random events (such as a customer arrival, a request for actionfrom a web server, or the completion of the actions requested of a web server) as emanating from a memorylessprocess. That is, the length of the time interval from the current time to the occurrence of the next event does notdepend upon the time of occurrence of the last event. In the Poisson probability distribution, the observer records thenumber of events that occur in a time interval of fixed length. In the (negative) exponential probability distribution,the observer records the length of the time interval between consecutive events. In both, the underlying physicalprocess is memoryless.Models based on the Poisson process often respond to inputs from the environment in a manner that mimics theresponse of the system being modeled to those same inputs. The analytically tractable models that result yield bothinformation about the system being modeled and the form of their solution. Even a queueing model based on thePoisson process that does a relatively poor job of mimicking detailed system performance can be useful. The factthat such models often give "worst-case" scenario evaluations appeals to system designers who prefer to include asafety factor in their designs. Also, the form of the solution of models based on the Poisson process often providesinsight into the form of the solution to a queueing problem whose detailed behavior is poorly mimicked. As a result,queueing models are frequently modeled as Poisson processes through the use of the exponential distribution.

Limitations of queueing theoryThe assumptions of classical queueing theory may be too restrictive to be able to model real-world situations exactly.The complexity of production lines with product-specific characteristics cannot be handled with those models.Therefore specialized tools have been developed to simulate, analyze, visualize and optimize time dynamic queueingline behavior.For example; the mathematical models often assume infinite numbers of customers, infinite queue capacity, or nobounds on inter-arrival or service times, when it is quite apparent that these bounds must exist in reality. Often,although the bounds do exist, they can be safely ignored because the differences between the real-world and theory isnot statistically significant, as the probability that such boundary situations might occur is remote compared to theexpected normal situation. Furthermore, several studies show the robustness of queueing models outside theirassumptions. In other cases the theoretical solution may either prove intractable or insufficiently informative to beuseful.Alternative means of analysis have thus been devised in order to provide some insight into problems that do not fallunder the scope of queueing theory, although they are often scenario-specific because they generally consist ofcomputer simulations or analysis of experimental data. See network traffic simulation.

Queueing theory 165

References[1] Andrewferrier.com (http:/ / www. andrewferrier. com/ oldpages/ queueing_theory/ Henry/ index. html)[2] TU Berlin: Technische Universität Berlin (http:/ / www. tkn. tu-berlin. de/ curricula/ ws0203/ ue-kn/ qt. pdf)[3] Lawrence W. Dowdy, Virgilio A.F. Almeida, Daniel A. Menasce (Thursday Janery 15, 2004). "Performance by Design: Computer Capacity

Planning By Example" (http:/ / www. cs. gmu. edu/ ~menasce/ perfbyd/ ). pp. 480.[4] Schlechter, Kira (Monday March 02, 2009). "Hershey Medical Center to open redesigned emergency room" (http:/ / www. pennlive. com/

midstate/ index. ssf/ 2009/ 03/ hershey_med_to_open_redesigned. html). The Patriot-News.[5] Mayhew, Les; Smith, David (December 2006). Using queuing theory to analyse completion times in accident and emergency departments in

the light of the Government 4-hour target (http:/ / www. cass. city. ac. uk/ media/ stories/ story_96_105659_69284. html). Cass BusinessSchool. ISBN 978-1-905752-06-5. . Retrieved 2008-05-20.

[6] Tijms, H.C, Algorithmic Analysis of Queues", Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003[7] http:/ / pass. maths. org. uk/ issue2/ erlang/ index. html[8] Flood, J.E. Telecommunications Switching, Traffic and Networks, Chapter 4: Telecommunications Traffic, New York: Prentice-Hall, 1998.[9] Bose S.J., Chapter 1 - An Introduction to Queueing Systems, Kluwer/Plenum Publishers, 2002.[10] Penttinen A., Chapter 8 – Queueing Systems, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory.[11] F. P. Kelly Networks of Queues with Customers of Different Types Journal of Applied Probability, Vol. 12, No. 3 (Sep., 1975), pp. 542-554

Further reading• Gross, Donald; Carl M. Harris (1998). Fundamentals of Queueing Theory. Wiley. ISBN 047132812X.• Deitel, Harvey M. (1984) [1982]. An introduction to operating systems (http:/ / portal. acm. org/ citation.

cfm?id=79046& dl=GUIDE& coll=GUIDE) (revisited first ed.). Addison-Wesley. pp. 673. ISBN 0-201-14502-2.chap.15, pp. 380–412

• Lazowska, Edward D.; John Zahorjan, G. Scott Graham, Kenneth C. Sevcik (1984). Quantitative SystemPerformance: Computer System Analysis Using Queueing Network Models (http:/ / www. cs. washington. edu/homes/ lazowska/ qsp/ ). Prentice-Hall, Inc. ISBN 0137469756.

• Zukerman, Moshe. Introduction to Queueing Theory and Stochastic Teletraffic Models (http:/ / www. ee. cityu.edu. hk/ ~zukerman/ classnotes. pdf).

External links• Queueing theory calculator (http:/ / www. supositorio. com/ rcalc/ rcalclite. htm)• Virtamo's Queueing Theory Course (http:/ / www. netlab. tkk. fi/ opetus/ s383143/ kalvot/ english. shtml)• Shmula's Queueing Theory Page (http:/ / www. shmula. com/ queueing-theory)• Myron Hlynka's Queueing Theory Page (http:/ / web2. uwindsor. ca/ math/ hlynka/ queue. html)• Queueing Theory Basics (http:/ / www. eventhelix. com/ RealtimeMantra/ CongestionControl/ queueing_theory.

htm)• Java Modelling Tools - A GPL suite of queueing theory tools (http:/ / jmt. sourceforge. net)

Markov chain 166

Markov chain

A simple two-state Markov chain.

A Markov chain, named for Andrey Markov, is a mathematicalsystem that undergoes transitions from one state to another (from afinite or countable number of possible states) in a chainlike manner. Itis a random process characterized as memoryless (i.e., exhibiting theMarkov property): the next state depends only on the current state andnot on the entire past. Markov chains have many applications asstatistical models of real-world processes.

Introduction

Formally, a Markov chain is a discrete (discrete-time) random processwith the Markov property. Often, the term "Markov chain" is used tomean a Markov process which has a discrete (finite or countable)state-space. Usually a Markov chain would be defined for a discrete setof times (i.e. a discrete-time Markov chain)[1] although some authors use the same terminology where "time" cantake continuous values.[2] [3] Also see continuous-time Markov process. The use of the term in Markov chain MonteCarlo methodology covers cases where the process is in discrete-time (discrete algorithm steps) with a continuousstate space. The following concentrates on the discrete-time discrete-state-space case.

A "discrete-time" random process involves a system which is in a certain state at each "step", with the state changingrandomly between steps. The steps are often thought of as time, but they can equally well refer to physical distanceor any other discrete measurement; formally, the steps are just the integers or natural numbers, and the randomprocess is a mapping of these to states. The Markov property states that the conditional probability distribution forthe system at the next step (and in fact at all future steps) given its current state depends only on the current state ofthe system, and not additionally on the state of the system at previous steps.Since the system changes randomly, it is generally impossible to predict the exact state of the system in the future.However, the statistical properties of the system's future can be predicted. In many applications it is these statisticalproperties that are important.The changes of state of the system are called transitions, and the probabilities associated with various state-changesare called transition probabilities. The set of all states and transition probabilities completely characterizes a Markovchain. By convention, we assume all possible states and transitions have been included in the definition of theprocesses, so there is always a next-state and the process goes on forever.A famous Markov chain is the so-called "drunkard's walk", a random walk on the number line where, at each step,the position may change by +1 or −1 with equal probability. From any position there are two possible transitions, tothe next or previous integer. The transition probabilities depend only on the current position, not on the way theposition was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all othertransition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4or 6.Another example is the dietary habits of a creature who eats only grapes, cheese or lettuce, and whose dietary habitsconform to the following rules:• It eats exactly once a day.• If it ate cheese today, tomorrow it will eat lettuce or grapes with equal probability.• If it ate grapes today, tomorrow it will eat grapes with probability 1/10, cheese with probability 4/10 and lettuce

with probability 5/10.

Markov chain 167

• If it ate lettuce today, it will not eat lettuce again tomorrow but will eat grapes with probability 4/10 or cheesewith probability 6/10.

This creature's eating habits can be modeled with a Markov chain since its choice depends solely on what it atetoday, not what it ate yesterday or even farther in the past. One statistical property that could be calculated is theexpected percentage, over a long period, of the days on which the creature will eat grapes.A series of independent events—for example, a series of coin flips—does satisfy the formal definition of a Markovchain. However, the theory is usually applied only when the probability distribution of the next step dependsnon-trivially on the current state.Many other examples of Markov chains exist.

Formal definitionA Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov property, namely that, given thepresent state, the future and past states are independent. Formally,

The possible values of Xi form a countable set S called the state space of the chain.Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of goingfrom one state to the other states.

Variations• Continuous-time Markov processes have a continuous index.• Time-homogeneous Markov chains (or stationary Markov chains) are processes where

for all n. The probability of the transition is independent of n.• A Markov chain of order m (or a Markov chain with memory m) where m is finite, is a process satisfying

In other words, the future state depends on the past m states. It is possible to construct a chain (Yn) from (Xn)which has the 'classical' Markov property as follows:Let Yn = (Xn, Xn−1, ..., Xn−m+1), the ordered m-tuple of X values. Then Yn is a Markov chain with state space Sm

and has the classical Markov property.• An additive Markov chain of order m where m is finite, is where

for n > m.

Markov chain 168

Example

A simple example is shown in the figure on the right, using adirected graph to picture the state transitions. The states representwhether the economy is in a bull market, a bear market, or arecession, during a given week. According to the figure, a bullweek is followed by another bull week 90% of the time, a bearmarket 7.5% of the time, and a recession the other 2.5%. From thisfigure it is possible to calculate, for example, the long-termfraction of time during which the economy is in a recession, or onaverage how long it will take to go from a recession to a bullmarket.A thorough development and many examples can be found in theon-line monograph Meyn & Tweedie 2005.[4] The appendix ofMeyn 2007,[5] also available on-line, contains an abridged Meyn& Tweedie.

A finite state machine can be used as a representation of a Markovchain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from abinary alphabet chosen by coin tosses), if the machine is in state y at time n, then the probability that it moves to statex at time n + 1 depends only on the current state.

Markov chainsThe probability of going from state i to state j in n time steps is

and the single-step transition is

For a time-homogeneous Markov chain:

and

The n-step transition probabilities satisfy the Chapman–Kolmogorov equation, that for any k such that 0 < k < n,

where S is the state space of the Markov chain.The marginal distribution Pr(Xn = x) is the distribution over states at time n. The initial distribution is Pr(X0 = x). Theevolution of the process through one time step is described by

Note: The superscript (n) is an index and not an exponent.

Markov chain 169

ReducibilityA state j is said to be accessible from a state i (written i → j) if a system started in state i has a non-zero probabilityof transitioning into state j at some point. Formally, state j is accessible from state i if there exists an integer n ≥ 0such that

Allowing n to be zero means that every state is defined to be accessible from itself.A state i is said to communicate with state j (written i ↔ j) if both i → j and j → i. A set of states C is acommunicating class if every pair of states in C communicates with each other, and no state in C communicateswith any state not in C. It can be shown that communication in this sense is an equivalence relation and thus thatcommunicating classes are the equivalence classes of this relation. A communicating class is closed if the probabilityof leaving the class is zero, namely that if i is in C but j is not, then j is not accessible from i.That said, communicating classes need not be commutative, in that classes achieving greater periodic frequenciesthat encompass 100% of the phases of smaller periodic frequencies, may still be communicating classes provided aform of either diminished, downgraded, or multiplexed cooperation exists within the higher frequency class.A state i is said to be essential if for all j such that i → j it is also true that j → i. A state i is inessential if it is notessential.[6]

Finally, a Markov chain is said to be irreducible if its state space is a single communicating class; in other words, ifit is possible to get to any state from any state.

PeriodicityA state i has period k if any return to state i must occur in multiples of k time steps. Formally, the period of a state isdefined as

(where "gcd" is the greatest common divisor). Note that even though a state has period k, it may not be possible toreach the state in k steps. For example, suppose it is possible to return to the state in {6, 8, 10, 12, ...} time steps; kwould be 2, even though 2 does not appear in this list.If k = 1, then the state is said to be aperiodic: returns to state i can occur at irregular times. Otherwise (k > 1), thestate is said to be periodic with period k.It can be shown that every state in a communicating class must have overlapping periods with allequivalent-or-larger occurring sample(s).It can be also shown that every state of a bipartite graph has an even period.

RecurrenceA state i is said to be transient if, given that we start in state i, there is a non-zero probability that we will neverreturn to i. Formally, let the random variable Ti be the first return time to state i (the "hitting time"):

The number

is the probability that we return to state i for the first time after n steps. Therefore, state i is transient if

State i is recurrent (or persistent) if it is not transient. Recurrent states have finite hitting time with probability 1.

Markov chain 170

Mean recurrence time

Even if the hitting time is finite with probability 1, it need not have a finite expectation. The mean recurrence timeat state i is the expected return time Mi:

State i is positive recurrent (or non-null persistent) if Mi is finite; otherwise, state i is null recurrent (or nullpersistent).

Expected number of visits

It can be shown that a state i is recurrent if and only if the expected number of visits to this state is infinite, i.e.,

Absorbing states

A state i is called absorbing if it is impossible to leave this state. Therefore, the state i is absorbing if and only if

ErgodicityA state i is said to be ergodic if it is aperiodic and positive recurrent. If all states in an irreducible Markov chain areergodic, then the chain is said to be ergodic.It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. A model has theergodic property if there's a finite number N such that any state can be reached from any other state in exactly Nsteps. In case of a fully-connected transition matrix where all transitions have a non-zero probability, this conditionis fulfilled with N=1. A model with just one out-going transition per state cannot be ergodic.

Steady-state analysis and limiting distributionsIf the Markov chain is a time-homogeneous Markov chain, so that the process is described by a single,time-independent matrix , then the vector is called a stationary distribution (or invariant measure) if itsentries are non-negative and sum to 1 and if it satisfies

An irreducible chain has a stationary distribution if and only if all of its states are positive recurrent. In that case, π isunique and is related to the expected return time:

Further, if the chain is both irreducible and aperiodic, then for any i and j,

Note that there is no assumption on the starting distribution; the chain converges to the stationary distributionregardless of where it begins. Such π is called the equilibrium distribution of the chain.If a chain has more than one closed communicating class, its stationary distributions will not be unique (consider anyclosed communicating class in the chain; each one will have its own unique stationary distribution. Any of these willextend to a stationary distribution for the overall chain, where the probability outside the class is set to zero).However, if a state j is aperiodic, then

Markov chain 171

and for any other state i, let fij be the probability that the chain ever visits state j if it starts at i,

If a state i is periodic with period k > 1 then the limit

does not exist, although the limit

does exist for every integer r.

Steady-state analysis and the time-inhomogeneous Markov chain

A Markov chain need not necessarily be time-homogeneous to have an equilibrium distribution. If there is aprobability distribution over states such that

for every state j and every time n then is an equilibrium distribution of the Markov chain. Such can occur inMarkov chain Monte Carlo (MCMC) methods in situations where a number of different transition matrices are used,because each is efficient for a particular kind of mixing, but each matrix respects a shared equilibrium distribution.

Finite state spaceIf the state space is finite, the transition probability distribution can be represented by a matrix, called the transitionmatrix, with the (i, j)th element of P equal to

Since each row of P sums to one and all elements are non-negative, P is a right stochastic matrix.

Time-homogeneous Markov chain with a finite state spaceIf the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-steptransition probability can be computed as the k-th power of the transition matrix, Pk.The stationary distribution π is a (row) vector, whose entries are non-negative and sum to 1, that satisfies theequation

In other words, the stationary distribution π is a normalized (meaning that the sum of its entries is 1) left eigenvectorof the transition matrix associated with the eigenvalue 1.Alternatively, π can be viewed as a fixed point of the linear (hence continuous) transformation on the unit simplexassociated to the matrix P. As any continuous transformation in the unit simplex has a fixed point, a stationarydistribution always exists, but is not guaranteed to be unique, in general. However, if the Markov chain is irreducibleand aperiodic, then there is a unique stationary distribution π. Additionally, in this case Pk converges to a rank-onematrix in which each row is the stationary distribution π, that is,

where 1 is the column vector with all entries equal to 1. This is stated by the Perron–Frobenius theorem. If, by whatever means, is found, then the stationary distribution of the Markov chain in question can be easily

Markov chain 172

determined for any starting distribution, as will be explained below.

For some stochastic matrices P, the limit does not exist, as shown by this example:

Because there are a number of different special cases to consider, the process of finding this limit if it exists can be alengthy task. However, there are many techniques that can assist in finding this limit. Let P be an n×n matrix, anddefine It is always true that

Subtracting Q from both sides and factoring then yields

where In is the identity matrix of size n, and 0n,n is the zero matrix of size n×n. Multiplying together stochasticmatrices always yields another stochastic matrix, so Q must be a stochastic matrix. It is sometimes sufficient to usethe matrix equation above and the fact that Q is a stochastic matrix to solve for Q.Here is one method for doing so: first, define the function f(A) to return the matrix A with its right-most columnreplaced with all 1's. If [f(P − In)]−1 exists then

Explain: The original matrix equation is equivalent to a system of n×n linear equations in n×n variables. Andthere are n more linear equations from the fact that Q is a right stochastic matrix whose each row sums to 1. Soit needs any n×n independent linear equations of the (n×n+n) equations to solve for the n×n variables. In thisexample, the n equations from “Q multiplied by the right-most column of (P-In)” have been replaced by the nstochastic ones.

One thing to notice is that if P has an element Pi,i on its main diagonal that is equal to 1 and the ith row or column isotherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers Pk. Hence,the ith row or column of Q will have the 1 and the 0's in the same positions as in P.

Reversible Markov chainA Markov chain is said to be reversible if there is a probability distribution over states, π, such that

for all times n and all states i and j. This condition is also known as the detailed balance condition (some booksrefer the local balance equation). With a time-homogeneous Markov chain, Pr(Xn+1 = j | Xn = i) does not change withtime n and it can be written more simply as . In this case, the detailed balance equation can be written morecompactly as

Summing the original equation over i gives

so, for reversible Markov chains, π is always a steady-state distribution of Pr(Xn+1 = j | Xn = i) for every n.If the Markov chain begins in the steady-state distribution, i.e., if Pr(X0 = i) = πi, then Pr(Xn = i) = πi for all n and thedetailed balance equation can be written as

Markov chain 173

The left- and right-hand sides of this last equation are identical except for a reversing of the time indices n and n + 1.Reversible Markov chains are common in Markov chain Monte Carlo (MCMC) approaches because the detailedbalance equation for a desired distribution π necessarily implies that the Markov chain has been constructed so that πis a steady-state distribution. Even with time-inhomogeneous Markov chains, where multiple transition matrices areused, if each such transition matrix exhibits detailed balance with the desired π distribution, this necessarily impliesthat π is a steady-state distribution of the Markov chain.

Bernoulli schemeA Bernoulli scheme is a special case of a Markov chain where the transition probability matrix has identical rows,which means that the next state is even independent of the current state (in addition to being independent of the paststates). A Bernoulli scheme with only two possible states is known as a Bernoulli process.

General state spaceMany results for Markov chains with finite state space can be generalized to chains with uncountable state spacethrough Harris chains. The main idea is to see if there is a point in the state space that the chain hits with probabilityone. Generally, it is not true for continuous state space, however, we can define sets A and B along with a positivenumber ε and a probability measure ρ, such that

1.2.Then we could collapse the sets into an auxiliary point α, and a recurrent Harris chain can be modified to contain α.Lastly, the collection of Harris chains is a comfortable level of generality, which is broad enough to contain a largenumber of interesting examples, yet restrictive enough to allow for a rich theory.

ApplicationsMarkov chains are applied in a number of ways to many different fields. Often they are used as a mathematicalmodel from some random physical process; if the parameters of the chain are known, quantitative predictions can bemade. In other cases, they are used to model a more abstract process, and are the theoretical underpinning of analgorithm.

PhysicsMarkovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are usedto represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant,and that no relevant history need be considered which is not already included in the state description.

Chemistry

Michaelis-Menten kinetics. The enzyme (E) binds asubstrate (S) and produces a product (P). Each reaction

is a state transition in a Markov chain.

Chemistry is often a place where Markov chains andcontinuous-time Markov processes are especially useful becausethese simple physical systems tend to satisfy the Markov propertyquite well. The classical model of enzyme activity,Michaelis-Menten kinetics, can be viewed as a Markov chain,where at each time step the reaction proceeds in some direction.While Michaelis-Menten is fairly straightforward, far morecomplicated reaction networks can also be modeled with Markovchains.

Markov chain 174

An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals in silicotowards a desired class of compounds such as drugs or natural products.[7] As a molecule is grown, a fragment isselected from the nascent molecule as the "current" state. It is not aware of its past (i.e., it is not aware of what isalready bonded to it). It then transitions to the next state when a fragment is attached to it. The transitionprobabilities are trained on databases of authentic classes of compounds.Also, the growth (and composition) of copolymers may be modeled using Markov chains. Based on the reactivityratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (e.g.,whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due to steric effects,second-order Markov effects may also play a role in the growth of some polymer chains.

TestingSeveral theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoiningMarkov chains to form a "Markov blanket", arranging these chains in several recursive layers ("wafering") andproducing more efficient test sets—samples—as a replacement for exhaustive testing. MCSTs also have uses intemporal state-based networks; Chilukuri et al.'s paper entitled "Temporal Uncertainty Reasoning Networks forEvidence Fusion with Applications to Object Detection and Tracking" (ScienceDirect) gives a background and casestudy for applying MCSTs to a wider range of applications.

Information sciencesMarkov chains are used throughout information processing. Claude Shannon's famous 1948 paper A mathematicaltheory of communication, which in a single step created the field of information theory, opens by introducing theconcept of entropy through Markov modeling of the English language. Such idealized models can capture many ofthe statistical regularities of systems. Even without describing the full structure of the system perfectly, such signalmodels can make possible very effective data compression through entropy encoding techniques such as arithmeticcoding. They also allow effective state estimation and pattern recognition.Markov chains are also the basis for Hidden Markov Models, which are an important tool in such diverse fields astelephone networks (for error correction), speech recognition and bioinformatics. The world's mobile telephonesystems depend on the Viterbi algorithm for error-correction, while hidden Markov models are extensively used inspeech recognition and also in bioinformatics, for instance for coding region/gene prediction. Markov chains alsoplay an important role in reinforcement learning.

Queueing theoryMarkov chains are the basis for the analytical treatment of queues (queueing theory). This makes them critical foroptimizing the performance of telecommunications networks, where messages must often compete for limitedresources (such as bandwidth).[5]

Internet applicationsThe PageRank of a webpage as used by Google is defined by a Markov chain.[8] It is the probability to be at page in the stationary distribution on the following Markov chain on all (known) webpages. If is the number of known

webpages, and a page has links then it has transition probability for all pages that are linked to

and for all pages that are not linked to. The parameter is taken to be about 0.85.[9]

Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on aparticular website can be modeled using first- or second-order Markov models and can be used to make predictionsregarding future navigation and to personalize the web page for an individual user.

Markov chain 175

StatisticsMarkov chain methods have also become very important for generating sequences of random numbers to accuratelyreflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC).In recent years this has revolutionized the practicability of Bayesian inference methods, allowing a wide range ofposterior distributions to be simulated and their parameters found numerically.

Economics and financeMarkov chains are used in Finance and Economics to model a variety of different phenomena, including asset pricesand market crashes. The first financial model to use a Markov chain was from Prasad et al. in 1974.[10] Another wasthe regime-switching model of James D. Hamilton (1989), in which a Markov chain is used to model switchesbetween periods of high volatility and low volatility of asset returns.[11] A more recent example is the MarkovSwitching Multifractal asset pricing model, which builds upon the convenience of earlier regime-switchingmodels.[12] It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.Dynamic macroeconomics heavily uses Markov chains. An example is using Markov chains to exogenously modelprices of equity (stock) in a general equilibrium setting.[13]

Leontief's Input-output model is a Markov chain.

Social sciencesMarkov chains are generally used in describing path-dependent arguments, where current structural configurationscondition future outcomes. An example is the commonly argued link between economic development and the rise ofcapitalism. Once a country reaches a specific level of economic development, the configuration of structural factors,such as size of the commercial bourgeoisie, the ratio of urban to rural residence, the rate of political mobilization,etc., will generate a higher probability of transitioning from authoritarian to capitalist.

Mathematical biologyMarkov chains also have many applications in biological modelling, particularly population processes, which areuseful in modelling processes that are (at least) analogous to biological populations. The Leslie matrix is one suchexample, though some of its entries are not probabilities (they may be greater than 1). Another example is themodeling of cell shape in dividing sheets of epithelial cells.[14] Yet another example is the state of Ion channels incell membranes.Markov chains are also used in simulations of brain function, such as the simulation of the mammalian neocortex.[15]

GamesMarkov chains can be used to model many games of chance. The children's games Snakes and Ladders and "Hi Ho!Cherry-O", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (ona given square) and from there has fixed odds of moving to certain other states (squares).

MusicMarkov chains are employed in algorithmic music composition, particularly in software programs such as CSound orMax or SuperCollider. In a first-order chain, the states of the system become note or pitch values, and a probabilityvector for each note is constructed, completing a transition probability matrix (see below). An algorithm isconstructed to produce and output note values based on the transition matrix weightings, which could be MIDI notevalues, frequency (Hz), or any other desirable metric.[16]

Markov chain 176

1st-order matrix

Note A C♯ E♭

A 0.1 0.6 0.3

C♯ 0.25 0.05 0.7

E♭ 0.7 0.3 0

2nd-order matrix

Note A D G

AA 0.18 0.6 0.22

AD 0.5 0.5 0

AG 0.15 0.75 0.1

DD 0 0 1

DA 0.25 0 0.75

DG 0.9 0.1 0

GG 0.4 0.4 0.2

GA 0.5 0.25 0.25

GD 1 0 0

A second-order Markov chain can be introduced by considering the current state and also the previous state, asindicated in the second table. Higher, nth-order chains tend to "group" particular notes together, while 'breaking off'into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense ofphrasal structure, rather than the 'aimless wandering' produced by a first-order system.[17]

It should also be noted that Markov chains can be used structurally, as in Xenakis's Analogique A and B, asdescribed in his book 'Formalized Music: Mathematics and Thought in Composition'

BaseballMarkov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Eachhalf-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered.During any at-bat, there are 24 possible combinations of number of outs and position of the runners. Mark Pankinshows that Markov chain models can be used to evaluate runs created for both individual players as well as ateam.[18] The author also discusses various kinds of strategies and play conditions how Markov chain models havebeen used to analyze statistics for game situations such as bunting and base stealing and differences when playing ongrass vs. astroturf.[19]

Markov chain 177

Markov text generatorsMarkov processes can also be used to generate superficially "real-looking" text given a sample document: they areused in a variety of recreational "parody generator" software (see dissociated press, Jeff Harrison [20], Mark VShaney[21] [22] ).These processes are also used by spammers to inject real-looking hidden paragraphs into unsolicited email in anattempt to get these messages past spam filters.

HistoryAndrey Markov produced the first results (1906) for these processes, purely theoretically. A generalization tocountably infinite state spaces was given by Kolmogorov (1936). Markov chains are related to Brownian motion andthe ergodic hypothesis, two topics in physics which were important in the early years of the twentieth century, butMarkov appears to have pursued this out of a mathematical motivation, namely the extension of the law of largenumbers to dependent events. In 1913, he applied his findings for the first time to the first 20,000 letters of Pushkin'sEugene Onegin.Seneta provides an account of Markov's motivations and the theory's early development.[23] The term "chain" wasused by Markov (1906).[24]

Notes[1] Everitt,B.S. (2002) The Cambridge Dictionary of Statistics. CUP. ISBN 0-521-81099-x[2] Parzen, E. (1962) Stochastic Processes, Holden-Day. ISBN 0-8162-6664-6 (Table 6.1))[3] Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entry for "Markov chain")[4] S. P. Meyn and R.L. Tweedie, 2005. Markov Chains and Stochastic Stability (https:/ / netfiles. uiuc. edu/ meyn/ www/ spm_files/ book.

html). Second edition to appear, Cambridge University Press, 2008.[5] S. P. Meyn, 2007. Control Techniques for Complex Networks (http:/ / decision. csl. uiuc. edu/ ~meyn/ pages/ CTCN/ CTCN. html),

Cambridge University Press, 2007.[6] Asher Levin, David. Markov chains and mixing times (http:/ / books. google. co. uk/ books?id=6Cg5Nq5sSv4C& lpg=PA16&

ots=ATv27idyx7& dq=essential class markov& pg=PA16#v=onepage). pp. 16. .[7] Kutchukian, Peter; Lou, David; Shakhnovich, Eugene (2009). "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of

Molecules occupying Druglike Chemical". Journal of Chemical Information and Modeling 49 (7): 1630–1642. doi:10.1021/ci9000458.PMID 19527020.

[8] U.S. Patent 6285999 (http:/ / www. google. com/ patents?vid=6285999)[9] Page, Lawrence and Brin, Sergey and Motwani, Rajeev and Winograd, Terry (1999). The PageRank Citation Ranking: Bringing Order to the

Web (http:/ / citeseerx. ist. psu. edu/ viewdoc/ summary?doi=10. 1. 1. 31. 1768). Technical Report .[10] Prasad, NR; RC Ender,ST Reilly,G Nesgos (1974). "Allocation of resources on a minimized cost basis" (http:/ / ieeexplore. ieee. org/

Xplore/ defdeny. jsp?url=http:/ / ieeexplore. ieee. org/ stamp/ stamp. jsp?tp=& arnumber=4045263& userType=inst& denyReason=-133&arnumber=4045263& productsMatched=null& userType=inst). 1974 IEEE Conference on Decision and Control including the 13thSymposium on Adaptive Processes 13: 402–3. doi:10.1109/CDC.1974.270470. .

[11] Hamilton, James (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle" (http:/ / jstor. org/stable/ 1912559). Econometrica (Econometrica, Vol. 57, No. 2) 57 (2): 357–84. doi:10.2307/1912559. .

[12] Calvet, Laurent; Adlai Fisher (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes".Journal of Financial Econometrics 2: 49–83. doi:10.1093/jjfinec/nbh003.

[13] Brennan, Michael; Xiab, Yihong. "Stock Price Volatility and the Equity Premium" (http:/ / bbs. cenet. org. cn/ uploadImages/200352118122167693. pdf). Department of Finance, the Anderson School of Management, UCLA. .

[14] Emergence of geometric order in proliferating metazoan epithelia (http:/ / www. eecs. harvard. edu/ ~abpatel/ drosophila_wing_model. htm)by Ankit B. Patel, Radhika Nagpal

[15] George, Dileep; Hawkins, Jeff (2009). "Towards a Mathematical Theory of Cortical Micro-circuits". PLoS Comput Biol 5 (10): e1000532.doi:10.1371/journal.pcbi.1000532.

[16] K McAlpine, E Miranda, S Hoggar (1999). "Making Music with Algorithms: A Case-Study System" (http:/ / www. mitpressjournals. org/doi/ abs/ 10. 1162/ 014892699559733). Computer Music Journal 23 (2): 19. doi:10.1162/014892699559733. .

[17] Curtis Roads (ed.) (1996). The Computer Music Tutorial. MIT Press. ISBN 0262181584.[18] Pankin, Mark D.. "MARKOV CHAIN MODELS: THEORETICAL BACKGROUND" (http:/ / www. pankin. com/ markov/ theory. htm). .

Retrieved 2007-11-26.

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[19] Pankin, Mark D.. "BASEBALL AS A MARKOV CHAIN" (http:/ / www. pankin. com/ markov/ intro. htm). . Retrieved 2009-04-24.[20] http:/ / www. fieralingue. it/ modules. php?name=Content& pa=list_pages_categories& cid=111[21] Kenner, Hugh; O'Rourke, Joseph (November 1984). "A Travesty Generator for Micros". BYTE 9 (12): 129–131, 449–469[22] Hartman, Charles (1996). The Virtual Muse: Experiments in Computer Poetry. Hanover, NH: Wesleyan University Press. ISBN 0819522392[23] Seneta, E. (1996). "Markov and the Birth of Chain Dependence Theory". International Statistical Review 64 (3): 255–263. JSTOR 1403785.[24] Upton, G.; Cook, I. (2008). Oxford Dictionary of Statistics. OUP. ISBN 0199541454.

References• A.A. Markov. "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga". Izvestiya

Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya, tom 15, pp. 135–156, 1906.• A.A. Markov. "Extension of the limit theorems of probability theory to a sum of variables connected in a chain".

reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wileyand Sons, 1971.

• Classical Text in Translation: A. A. Markov, An Example of Statistical Investigation of the Text Eugene OneginConcerning the Connection of Samples in Chains, trans. David Link. Science in Context 19.4 (2006): 591–600.Online: http:/ / journals. cambridge. org/ production/ action/ cjoGetFulltext?fulltextid=637500

• Leo Breiman. Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society forIndustrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. (See Chapter 7.)

• J.L. Doob. Stochastic Processes. New York: John Wiley and Sons, 1953. ISBN 0-471-52369-0.• S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. London: Springer-Verlag, 1993. ISBN

0-387-19832-6. online: https:/ / netfiles. uiuc. edu/ meyn/ www/ spm_files/ book. html . Second edition to appear,Cambridge University Press, 2009.

• S. P. Meyn. Control Techniques for Complex Networks. Cambridge University Press, 2007. ISBN978-0-521-88441-9. Appendix contains abridged Meyn & Tweedie. online: https:/ / netfiles. uiuc. edu/ meyn/www/ spm_files/ CTCN/ CTCN. html

• Booth, Taylor L. (1967). Sequential Machines and Automata Theory (1st ed.). New York: John Wiley and Sons,Inc.. Library of Congress Card Catalog Number 67-25924. Extensive, wide-ranging book meant for specialists,written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of stateminimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment ofMarkov processes pp. 449ff. Discusses Z-transforms, D transforms in their context.

• Kemeny, John G.; Hazleton Mirkil, J. Laurie Snell, Gerald L. Thompson (1959). Finite Mathematical Structures(1st ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc.. Library of Congress Card Catalog Number 59-12841.Classical text. cf Chapter 6 Finite Markov Chains pp. 384ff.

• E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press,1984, 2004. ISBN 0-521-60494-X

• Seneta, E. Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Seriesin Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) ISBN 978-0-387-29765-1

• Kishor S. Trivedi, Probability and Statistics with Reliability, Queueing, and Computer Science Applications, JohnWiley & Sons, Inc. New York, 2002. ISBN 0-471-33341-7.

• K.S.Trivedi and R.A.Sahner, SHARPE at the age of twenty-two, vol. 36, no. 4, pp.-52-57, ACM SIGMETRICSPerformance Evaluation Review, 2009.

• R.A.Sahner, K.S.Trivedi and A. Puliafito, Performance and reliability analysis of computer systems: anexample-based approach using the SHARPE software package, Kluwer Academic Publishers, 1996. ISBN0-7923-9650-2.

• G.Bolch, S.Greiner, H.de Meer and K.S.Trivedi, Queueing Networks and Markov Chains, John Wiley, 2ndedition, 2006. ISBN 9780792396505.

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External links• Techniques to Understand Computer Simulations: Markov Chain Analysis (http:/ / jasss. soc. surrey. ac. uk/ 12/ 1/

6. html)• Markov Chains chapter in American Mathematical Society's introductory probability book (http:/ / www.

dartmouth. edu/ ~chance/ teaching_aids/ books_articles/ probability_book/ Chapter11. pdf)(pdf)• Markov Chains (http:/ / crypto. mat. sbg. ac. at/ ~ste/ diss/ node6. html)• Class structure (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=5765) on PlanetMath• Chapter 5: Markov Chain Models (http:/ / www. math. rutgers. edu/ courses/ 338/ coursenotes/ chapter5. pdf)• A JavaScript Markovian text compressor (http:/ / adriano. mitre. com. br/ markov/ ) (in Portuguese).• Application to Markov Chains (http:/ / aix1. uottawa. ca/ ~jkhoury/ markov. htm)• Generating Text:Section 15.3 of Programming Pearls (http:/ / www. cs. bell-labs. com/ cm/ cs/ pearls/ sec153.

html)• Generating Text Online: Markov Random Walk v1.1 (http:/ / antonalexeev. hop. ru/ markov/ index. html)• a list of recourses related to Markov chains (http:/ / www. compression-links. info/ Markov)

Graph theory

A drawing of a graph

In mathematics and computer science, graph theory isthe study of graphs, mathematical structures used tomodel pairwise relations between objects from a certaincollection. A "graph" in this context refers to acollection of vertices or 'nodes' and a collection ofedges that connect pairs of vertices. A graph may beundirected, meaning that there is no distinctionbetween the two vertices associated with each edge, orits edges may be directed from one vertex to another;see graph (mathematics) for more detailed definitionsand for other variations in the types of graphs that arecommonly considered. The graphs studied in graphtheory should not be confused with graphs of functionsor other kinds of graphs.

Graphs are one of the prime objects of study in discrete mathematics. Refer to glossary of graph theory for basicdefinitions in graph theory.

ApplicationsGraphs are among the most ubiquitous models of both natural and human-made structures. They can be used tomodel many types of relations and process dynamics in physical, biological[1] and social systems. Many problems ofpractical interest can be represented by graphs.In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. One practical example: The link structure of a website could be represented by a directed graph. The vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. A similar approach can be taken to problems in travel, biology, computer chip design, and many other fields. The development of algorithms to handle graphs is therefore of major interest in computer science. There, the transformation of graphs is often formalized and represented by graph rewrite systems.

Graph theory 180

They are either directly used or properties of the rewrite systems (e.g. confluence) are studied. Complementary tograph transformation systems focussing on rule-based in-memory manipulation of graphs are graph databases gearedtowards transaction-safe, persistent storing and querying of graph-structured data.Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural languageoften lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-basedstructures, whose expressive power lies in the Principle of Compositionality, modeled in a hierarchical graph. Withinlexical semantics, especially as applied to computers, modeling word meaning is easier when a given word isunderstood in terms of related words; semantic networks are therefore important in computational linguistics. Stillother methods in phonology (e.g. Optimality Theory, which uses lattice graphs) and morphology (e.g. finite-statemorphology, using finite-state transducers) are common in the analysis of language as a graph. Indeed, theusefulness of this area of mathematics to linguistics has borne organizations such as TextGraphs [2], as well asvarious 'Net' projects, such as WordNet, VerbNet, and others.Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the threedimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statisticson graph-theoretic properties related to the topology of the atoms. For example, Franzblau's shortest-path (SP) rings.In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. Thisapproach is especially used in computer processing of molecular structures, ranging from chemical editors todatabase searching. In statistical physics, graphs can represent local connections between interacting parts of asystem, as well as the dynamics of a physical process on such systems.Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explorediffusion mechanisms, notably through the use of social network analysis software.Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions wherecertain species exist (or habitats) and the edges represent migration paths, or movement between the regions. Thisinformation is important when looking at breeding patterns or tracking the spread of disease, parasites or howchanges to the movement can affect other species.In mathematics, graphs are useful in geometry and certain parts of topology, e.g. Knot Theory. Algebraic graphtheory has close links with group theory.A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, orweighted graphs, are used to represent structures in which pairwise connections have some numerical values. Forexample if a graph represents a road network, the weights could represent the length of each road.A digraph with weighted edges in the context of graph theory is called a network. Network analysis have manypractical applications, for example, to model and analyze traffic networks. Applications of network analysis splitbroadly into three categories:1. First, analysis to determine structural properties of a network, such as the distribution of vertex degrees and the

diameter of the graph. A vast number of graph measures exist, and the production of useful ones for variousdomains remains an active area of research.

2. Second, analysis to find a measurable quantity within the network, for example, for a transportation network, thelevel of vehicular flow within any portion of it.

3. Third, analysis of dynamical properties of networks.

Graph theory 181

History

The Königsberg Bridge problem

The paper written by Leonhard Euler on the Seven Bridges ofKönigsberg and published in 1736 is regarded as the first paper inthe history of graph theory.[3] This paper, as well as the onewritten by Vandermonde on the knight problem, carried on withthe analysis situs initiated by Leibniz. Euler's formula relating thenumber of edges, vertices, and faces of a convex polyhedron wasstudied and generalized by Cauchy[4] and L'Huillier,[5] and is atthe origin of topology.

More than one century after Euler's paper on the bridges ofKönigsberg and while Listing introduced topology, Cayley was ledby the study of particular analytical forms arising from differentialcalculus to study a particular class of graphs, the trees. This studyhad many implications in theoretical chemistry. The involved techniques mainly concerned the enumeration ofgraphs having particular properties. Enumerative graph theory then rose from the results of Cayley and thefundamental results published by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in1959. Cayley linked his results on trees with the contemporary studies of chemical composition.[6] The fusion of theideas coming from mathematics with those coming from chemistry is at the origin of a part of the standardterminology of graph theory.

In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature, where he drawsan analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:[7]

"[...] Every invariant and co-variant thus becomes expressible by a graph precisely identical with a Kekuléandiagram or chemicograph. [...] I give a rule for the geometrical multiplication of graphs, i.e. for constructing agraph to the product of in- or co-variants whose separate graphs are given. [...]" (italics as in the original).

One of the most famous and productive problems of graph theory is the four color problem: "Is it true that any mapdrawn in the plane may have its regions colored with four colors, in such a way that any two regions having acommon border have different colors?" This problem was first posed by Francis Guthrie in 1852 and its first writtenrecord is in a letter of De Morgan addressed to Hamilton the same year. Many incorrect proofs have been proposed,including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait, Heawood,Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus.Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersenand Kőnig. The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at theorigin of another branch of graph theory, extremal graph theory.The four color problem remained unsolved for more than a century. In 1969 Heinrich Heesch published a method forsolving the problem using computers.[8] A computer-aided proof produced in 1976 by Kenneth Appel and WolfgangHaken makes fundamental use of the notion of "discharging" developed by Heesch.[9] [10] The proof involvedchecking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to itscomplexity. A simpler proof considering only 633 configurations was given twenty years later by Robertson,Seymour, Sanders and Thomas.[11]

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works ofJordan, Kuratowski and Whitney. Another important factor of common development of graph theory and topologycame from the use of the techniques of modern algebra. The first example of such a use comes from the work of thephysicist Gustav Kirchhoff, who published in 1845 his Kirchhoff's circuit laws for calculating the voltage andcurrent in electric circuits.

Graph theory 182

The introduction of probabilistic methods in graph theory, especially in the study of Erdős and Rényi of theasymptotic probability of graph connectivity, gave rise to yet another branch, known as random graph theory, whichhas been a fruitful source of graph-theoretic results.

Drawing graphsGraphs are represented graphically by drawing a dot or circle for every vertex, and drawing an arc between twovertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow.A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are severalways to structure the graph drawing. All that matters is which vertices are connected to which others by how manyedges and not the exact layout. In practice it is often difficult to decide if two drawings represent the same graph.Depending on the problem domain some layouts may be better suited and easier to understand than others.

Graph-theoretic data structuresThere are different ways to store graphs in a computer system. The data structure used depends on both the graphstructure and the algorithm used for manipulating the graph. Theoretically one can distinguish between list andmatrix structures but in concrete applications the best structure is often a combination of both. List structures areoften preferred for sparse graphs as they have smaller memory requirements. Matrix structures on the other handprovide faster access for some applications but can consume huge amounts of memory.

List structuresIncidence list

The edges are represented by an array containing pairs (tuples if directed) of vertices (that the edge connects)and possibly weight and other data. Vertices connected by an edge are said to be adjacent.

Adjacency listMuch like the incidence list, each vertex has a list of which vertices it is adjacent to. This causes redundancyin an undirected graph: for example, if vertices A and B are adjacent, A's adjacency list contains B, while B'slist contains A. Adjacency queries are faster, at the cost of extra storage space.

Matrix structuresIncidence matrix

The graph is represented by a matrix of size |V | (number of vertices) by |E| (number of edges) where the entry[vertex, edge] contains the edge's endpoint data (simplest case: 1 - incident, 0 - not incident).

Adjacency matrixThis is an n by n matrix A, where n is the number of vertices in the graph. If there is an edge from a vertex x toa vertex y, then the element is 1 (or in general the number of xy edges), otherwise it is 0. In computing,this matrix makes it easy to find subgraphs, and to reverse a directed graph.

Laplacian matrix or Kirchhoff matrix or Admittance matrixThis is defined as D − A, where D is the diagonal degree matrix. It explicitly contains both adjacencyinformation and degree information. (However, there are other, similar matrices that are also called "Laplacianmatrices" of a graph.)

Distance matrix

A symmetric n by n matrix D, where n is the number of vertices in the graph. The element is the lengthof a shortest path between x and y; if there is no such path = infinity. It can be derived from powers of A

Graph theory 183

Problems in graph theory

EnumerationThere is a large literature on graphical enumeration: the problem of counting graphs meeting specified conditions.Some of this work is found in Harary and Palmer (1973).

Subgraphs, induced subgraphs, and minorsA common problem, called the subgraph isomorphism problem, is finding a fixed graph as a subgraph in a givengraph. One reason to be interested in such a question is that many graph properties are hereditary for subgraphs,which means that a graph has the property if and only if all subgraphs have it too. Unfortunately, finding maximalsubgraphs of a certain kind is often an NP-complete problem.• Finding the largest complete graph is called the clique problem (NP-complete).A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties arehereditary with respect to induced subgraphs, which means that a graph has a property if and only if all inducedsubgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. Forexample,• Finding the largest edgeless induced subgraph, or independent set, called the independent set problem

(NP-complete).Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. Aminor or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges.Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minorshave it too. A famous example:

• A graph is planar if it contains as a minor neither the complete bipartite graph (See the Three-cottageproblem) nor the complete graph .

Another class of problems has to do with the extent to which various species and generalizations of graphs aredetermined by their point-deleted subgraphs, for example:• The reconstruction conjecture.

Graph coloringMany problems have to do with various ways of coloring graphs, for example:• The four-color theorem• The strong perfect graph theorem• The Erdős–Faber–Lovász conjecture (unsolved)• The total coloring conjecture (unsolved)• The list coloring conjecture (unsolved)• The Hadwiger conjecture (graph theory) (unsolved).

Graph theory 184

Route problems• Hamiltonian path and cycle problems• Minimum spanning tree• Route inspection problem (also called the "Chinese Postman Problem")• Seven Bridges of Königsberg• Shortest path problem• Steiner tree• Three-cottage problem• Traveling salesman problem (NP-hard)

Network flowThere are numerous problems arising especially from applications that have to do with various notions of flows innetworks, for example:• Max flow min cut theorem

Visibility graph problems• Museum guard problem

Covering problemsCovering problems are specific instances of subgraph-finding problems, and they tend to be closely related to theclique problem or the independent set problem.• Set cover problem• Vertex cover problem

Graph classesMany problems involve characterizing the members of various classes of graphs. Overlapping significantly withother types in this list, this type of problem includes, for instance:• Enumerating the members of a class• Characterizing a class in terms of forbidden substructures• Ascertaining relationships among classes (e.g., does one property of graphs imply another)• Finding efficient algorithms to decide membership in a class• Finding representations for members of a class.

Interdependent NetworksInterdependent networks is a system of coupled networks where nodes of one or more networks depend on nodes inother networks. This dependency is enhanced by the developments in modern technology. Such dependencies maylead to cascading failures between the networks and a relatively small damage can lead to a catastrophic breakdownof the system. Blackouts are a fascinating demonstration of the important role played by the dependencies betweennetworks. A recent study developed a framework to study the cascading failures in an interdependent networkssystem.[12]

Graph theory 185

Dependency linksIn percolation theory, many systems are characterized by small subgroups in the network which depend on eachother. For example, consider a financial network: Each company has trading and sales connections with othercompanies (connectivity links). These connections enable the companies to interact with others and function togetheras a global financial market. In addition, companies that belong to the same owner strongly depend on one another(dependency links). If one company fails, the owner might not be able to finance the other companies, which will failtoo. Another example is an online social network (Facebook or Twitter): Each individual communicates with hisfriends (connectivity links), thus forming a social network through which information and rumors can spread.However, many individuals will only participate in a social network if other individuals with common interests alsoparticipate in that social network, thereby forming dependency links.[12] [13] [14]

Notes[1] Mashaghi A et al. Investigation of a protein complex network EUROPEAN PHYSICAL JOURNAL B 41(1) 113-121 (2004)[2] http:/ / www. textgraphs. org[3] Biggs, N.; Lloyd, E. and Wilson, R. (1986), Graph Theory, 1736-1936, Oxford University Press[4] Cauchy, A.L. (1813), "Recherche sur les polyèdres - premier mémoire", Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.[5] L'Huillier, S.-A.-J. (1861), "Mémoire sur la polyèdrométrie", Annales de Mathématiques 3: 169–189.[6] Cayley, A. (1875), "Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie

chemischer Verbindungen", Berichte der deutschen Chemischen Gesellschaft 8 (2): 1056–1059, doi:10.1002/cber.18750080252.[7] John Joseph Sylvester (1878), Chemistry and Algebra. Nature, volume 17, page 284. doi:10.1038/017284a0. Online version (http:/ / www.

archive. org/ stream/ nature15unkngoog#page/ n312/ mode/ 1up). Retrieved 2009-12-30.[8] Heinrich Heesch: Untersuchungen zum Vierfarbenproblem. Mannheim: Bibliographisches Institut 1969.[9] Appel, K. and Haken, W. (1977), "Every planar map is four colorable. Part I. Discharging", Illinois J. Math. 21: 429–490.[10] Appel, K. and Haken, W. (1977), "Every planar map is four colorable. Part II. Reducibility", Illinois J. Math. 21: 491–567.[11] Robertson, N.; Sanders, D.; Seymour, P. and Thomas, R. (1997), "The four color theorem", Journal of Combinatorial Theory Series B 70:

2–44, doi:10.1006/jctb.1997.1750.[12] S.V.Buldyrev, R.Parshani, G.Paul ,H.E.Stanley, S. Havlin (2010). "Catastrophic cascade of failures in interdependent networks" (http:/ /

havlin. biu. ac. il/ Publications. php?keyword=Catastrophic+ cascade+ of+ failures+ in+ interdependent+ networks& year=*& match=all).Nature 465: 08932. .

[13] R. Parshani, S. V. Buldyrev, and , S. Havlin (2011). "Critical effect of dependency groups on the function of networks" (http:/ / havlin. biu.ac. il/ Publications. php?keyword=Critical+ effect+ of+ dependency+ groups+ on+ the+ function+ of+ networks& year=*& match=all). ProcNatl. Acad. Sci 1008: 1007. .

[14] A. Bashan, R. Parshani, S. Havlin (2011). ", Percolation in networks composed of connectivity and dependency links" (http:/ / havlin. biu.ac. il/ Publications. php?keyword=Percolation+ in+ networks+ composed+ of+ connectivity+ and+ dependency+ links& year=*& match=all).Phys, Rev. E 83: 051127. .

References• Berge, Claude (1958), Théorie des graphes et ses applications, Collection Universitaire de Mathématiques, II,

Paris: Dunod. English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish,Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first Englishedition, Dover, New York 2001.

• Biggs, N.; Lloyd, E.; Wilson, R. (1986), Graph Theory, 1736–1936, Oxford University Press.• Bondy, J.A.; Murty, U.S.R. (2008), Graph Theory, Springer, ISBN 978-1-84628-969-9.• Chartrand, Gary (1985), Introductory Graph Theory, Dover, ISBN 0-486-24775-9.• Gibbons, Alan (1985), Algorithmic Graph Theory, Cambridge University Press.• Golumbic, Martin (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press.• Harary, Frank (1969), Graph Theory, Reading, MA: Addison-Wesley.• Harary, Frank; Palmer, Edgar M. (1973), Graphical Enumeration, New York, NY: Academic Press.• Mahadev, N.V.R.; Peled, Uri N. (1995), Threshold Graphs and Related Topics, North-Holland.

Graph theory 186

External links

Online textbooks• Graph Theory with Applications (http:/ / www. math. jussieu. fr/ ~jabondy/ books/ gtwa/ gtwa. html) (1976) by

Bondy and Murty• Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs (http:/ / arxiv. org/

pdf/ cond-mat/ 0602129) (2006) by Hartmann and Weigt• Digraphs: Theory Algorithms and Applications (http:/ / www. cs. rhul. ac. uk/ books/ dbook/ ) 2007 by Jorgen

Bang-Jensen and Gregory Gutin• Graph Theory, by Reinhard Diestel (http:/ / diestel-graph-theory. com/ index. html)

Other resources• Graph theory tutorial (http:/ / www. utm. edu/ departments/ math/ graph/ )• A searchable database of small connected graphs (http:/ / www. gfredericks. com/ main/ sandbox/ graphs)• Image gallery: graphs (http:/ / web. archive. org/ web/ 20060206155001/ http:/ / www. nd. edu/ ~networks/

gallery. htm)• Concise, annotated list of graph theory resources for researchers (http:/ / www. babelgraph. org/ links. html)

Article Sources and Contributors 187

Article Sources and ContributorsUser:Rajah2770  Source: http://en.wikipedia.org/w/index.php?oldid=416910398  Contributors: ArcAngel, JohnCD, MatthewVanitas, Rajah2770

Operations research  Source: http://en.wikipedia.org/w/index.php?oldid=440244737  Contributors: Abkeshvari, Ace Frahm, Afluegel, Aleenf1, Altenmann, Alterrabe, Amcfreely, Amckern,Andrewpmk, Andycjp, Angus Lepper, AnmaFinotera, Anwar saadat, Arthena, Arzel, Ask123, Axr15, BDE, Baa, BarryList, Bdwilliamscraig, Beetstra, BenFrantzDale, Benjaminevans82,Bentogoa, Bluezy, Boing! said Zebedee, Borgx, Brianhe, Bsimmons666, Btyner, CRGreathouse, Caliprincess, Canterbury Tail, Cheese Sandwich, Chrislk02, Clemwang, Czyl, D nath1, DXBari,David from Downunder, Deanlaw, Decora, Deor, Dominus, Dralbertomarquez, Dramaturgid, Dungodung, Dysprosia, Dúnadan, Earthandmoon, EdH, Eddie.willers, Eleusis, Elgrandragon,Encyclops, EnigmaMcmxc, Erianna, Erkan Yilmaz, Erxnmedia, FTGHSmith, Fastfission, Fieldday-sunday, Fintor, Formol, G Qian, Galoubet, Giftlite, Gracefool, Gracenotes, GraemeLeggett,Graham Sharp, Gwrede, Haham hanuka, HanPritcher, Henrygb, Hike395, Homarjun, Hu, Ian Dunster, Ibizzavic, Imajoebob, Isaac.holeman, Isheden, Iwnbap, Ixfd64, JJL, JaeDyWolf, JamieS93,Jay.Here, Jdpipe, Jimmaths, Jitse Niesen, Jon Awbrey, Joyous!, Jpo, Jredmond, Just Another Dan, Jyoti buet, KenKendall, Khendon, Kiefer.Wolfowitz, Knotwork, Knowledge Incarnate, Koczy,Krj373, Leonard G., Leszek Jańczuk, LittleOldMe old, Lucidish, Manop, Masoudsa, Matforddavid, Maurreen, Maury Markowitz, Maxsonbd, Mdd, Mdgarvey, Melcombe, Merveunuvar, MichaelHardy, Mintleaf, Mnh, Monkeyman, Msh210, Mtrick, Muness, Mydogategodshat, Myleslong, Nelson50, NerdyNSK, Nilmerg, Nnhsky, NorthernCounties, Notreadbyhumans, Novacatz,Nrlsouza, Oberiko, Obladi, Ohnoitsjamie, Open2universe, Oxymoron83, Panscient, Pcap, Penfold, Petrus, PhDinMS, Philip Baird Shearer, Philip ea, Piotrus, Pol098, Protonk, Qwfp, R'n'B, RJN,Requestion, Rich Janis, Rjpbi, Robin klein, Ronz, RoyBoy, S2000magician, Sadads, Samansouri, SandyGeorgia, Saxifrage, Seaphoto, Shuroo, Sidna, Slakr, Slambo, Snumath89, StopThat,SueHay, Surendra mohnot, Tabletop, Tbackstr, Tcody84, That Guy, From That Show!, The Anome, TheTechieGeek63, Thewellman, Toddy1, Togo, Trekphiler, Tribaal, Vader07d, Vgy7ujm,VictorK1965, Vignaux, Vocaro, Wally Tharg, Wayiran, WebsterRiver, Welsh, Will Beback Auto, Will Thomas, Williamsrus, Wshun, Wuhwuzdat, Wxm29, Wyckyd Sceptre, Xa4, Xn4, Yqwen,Zeno Gantner, Zvika, 310 anonymous edits

Mathematical optimization  Source: http://en.wikipedia.org/w/index.php?oldid=442990730  Contributors: AManWithNoPlan, APH, Aaronbrick, Ablevy, Ajgorhoe, Alphachimp, AnAj, Andris,Anonymous Dissident, Antonielly, Ap, Armehrabian, Arnab das, Arthur Rubin, Artur adib, Asadi1978, Ashburyjohn, Asm4, Auntof6, Awaterl, AxelBoldt, BenFrantzDale, BlakeJRiley, Bonnans,Boxplot, Bpadmakumar, Bradgib, Brianboonstra, Burgaz, CRGreathouse, Carbaholic, Carbo1200, Carlo.milanesi, Cfg1777, Chan siuman, Chaos, Charles Matthews, CharlesGillingham,Charlesreid1, Chester Markel, ConstantLearner, Crisgh, Ct529, Czenek, DRHagen, Damian Yerrick, Daniel Dickman, Daryakav, Dattorro, Daveagp, David Martland, David.Monniaux,Deeptrivia, Delaszk, Deuxoursendormis, Dianegarey, Diego Moya, Diracula, Discospinster, Dmitrey, Doobliebop, Dpbert, Dsz4, Duoduoduo, Dwassel, Dysprosia, Edesigner, Ekojnoekoj,EncMstr, Encyclops, Erkan Yilmaz, Fintor, Fred Bauder, G.de.Lange, Galoubet, Georg Stillfried, Gglockner, Ggpauly, Giftlite, H Padleckas, H.ehsaan, Harrycaterpillar, HiYoSilver01, Hike395,Hosseininassab, Hu12, Hua001, Iknowyourider, Ish ishwar, Isheden, Isnow, JFPuget, Jackzhp, Jason Quinn, Jasonb05, Jean-Charles.Gilbert, Jitse Niesen, Jmc200, John of Reading,Johngcarlsson, JonMcLoone, Jonnat, Jurgen, Justin W Smith, KaHa242, Kamitsaha, Karada, Katie O'Hare, Kiefer.Wolfowitz, Kiril Simeonovski, Klochkov.ivan, Knillinux, KrakatoaKatie,Krystofer, LSpring, LastChanceToBe, Lavaka, Lethe, Ltk, Lycurgus, Lylenorton, MIT Trekkie, MSchlueter, Mange01, Mangogirl2, Marcol, MarkSweep, MartinDK, Martynas Patasius, Matcross, MaxSem, MaximizeMinimize, Mcmlxxxi, Mdd, Mdwang, Metafun, Michael Hardy, Mikewax, Misfeldt, Moink, MrOllie, Mrbynum, Msh210, Mxn, Myleslong, Nacopt, Nageh, Nimur,NormDor, Nwbeeson, Obradovic Goran, Ojigiri, Oleg Alexandrov, Olegalexandrov, Oli Filth, Optimering, Optiy, Oğuz Ergin, Paolo.dL, Patrick, Pcap, Peterlin, Philip Trueman, PhotoBox,PimBeers, Polar Bear, Pontus, Pownuk, Procellarum, Pschaus, Psogeek, RKUrsem, Rade Kutil, Ravelite, Rbdevore, Retired username, Riedel, Rinconsoleao, Roleplayer, Rxnt, Ryguasu, Sabamo,Sahinidis, Salix alba, Sapphic, Saraedum, Schaber, Schlitz4U, Skifreak, Sliders06, Smartcat, Smmurphy, Srinnath, Stebulus, Stevan White, Struway, Suegerman, Syst analytic, TPlantenga,Tbbooher, TeaDrinker, The Anome, The Nut, Thiscomments voice, Thoughtfire, Tizio, Topbanana, Travis.a.buckingham, Truecobb, Tsirel, Twocs, Van helsing, Vermorel, VictorAnyakin,Voyevoda, Wamanning, Wikibuki, Wmahan, X7q, Xprime, YuriyMikhaylovskiy, Zundark, Zwgeem, Іванко1, Щегол, 311 anonymous edits

Mathematical model  Source: http://en.wikipedia.org/w/index.php?oldid=444014305  Contributors: 130.159.254.xxx, 213.253.39.xxx, Abdull, Abdullais4u, Abune, Agor153, Aiwing, AlLemos, Aliotra, Altenmann, AndrewDressel, Arvindn, Atlant, Audriusa, Awickert, AxelBoldt, Azuris, BD2412, BenBaker, Binarypower, Brianjd, CBM, CarlHewitt, Carlodn6, Cazort,Conversion script, Cst17, Cureden, Cybercobra, DARTH SIDIOUS 2, Dakart, Derek Ross, Dori, Dterp, Dvsphanindra, Dysprosia, Elapsed, EmersonLowry, Emperorbma, Eurosong, EyeSerene,Farazgiki, Filippof, Fioravante Patrone en, Frau Holle, G.de.Lange, GKantaris, GiantSloth, Giftlite, Globalsolidarity, Gogo Dodo, Goodralph, Graham87, Gregbard, Grubber, Hans Adler,Hede2000, Helix84, Henrygb, Hesperian, HisashiKobayashi, HolgerK, Howard.noble323, Hxxvxxy, Hydrargyrum, JRSpriggs, JWSchmidt, Jarash, Jdpipe, John Deas, JohnOwens, Jonnat,Josephfrodo, Kai.velten, Karnesky, Kbdank71, Kingturtle, Kwaku, Lambiam, Lexor, MCrawford, Maedin, Magister Mathematicae, Maksim-e, Manop, Marek69, Martin451,Math.geek3.1415926, MathMartin, Matqkks, Maurice Carbonaro, Mayooranathan, Mdd, Meemee878, Michael Hardy, Millancad, MrOllie, Msh210, Neilc, Nikai, Nshackles, Obradovic Goran,Oleg Alexandrov, Oliver Lineham, Olivier, Omicronpersei8, PMDrive1061, Paolo.dL, Passw0rd, Pdenapo, Ph7five, Phanerozoic, Philip Trueman, Piano non troppo, Porcher, Prazan, Ranilb5,Rbakker99, Rich Farmbrough, RoyBoy, Sam Douglas, Sangwinc, Seaphoto, Silverhelm, Sina92, Sjö, Skbkekas, Smmurphy, Soy ivan, Spacemika, Spiritia, Spradlig, Squidonius,Stephanhartmannde, Stricklandjs59, T.ogar, Tasudrty, Tbackstr, Tbsmith, The Anome, The enemies of god, Tilin, TittoAssini, Tom Fararo, Tomas e, Tparameter, Tpbradbury, Trebor, Trovatore,Vina, Vlado0605, Wavehunter, Wavelength, Wesley, WikiTome, Winhunter, Wireless friend, Woohookitty, Wsiegmund, Yotaloop, 182 anonymous edits

Game theory  Source: http://en.wikipedia.org/w/index.php?oldid=443738912  Contributors: 63.208.190.xxx, 7&6=thirteen, A F K When Needed, A bit iffy, AMuseo, APH, Aaker, AapoLaitinen, Abigail-II, Abiyoyo, Abu badali, Acalamari, Acebrock, Acitrano, Action Jackson IV, AdamSmithee, Adashiel, Adoniscik, Adrian.benko, AdultSwim, Ahoerstemeier, Aim Here, Alakon,Aliahmedraza, Aliekens, Amcfreely, Andonic, Andre Dahlke, Andrew Levine, Angus Lepper, AnneDrew55, Anonymous Dissident, Antandrus, Anthere, Anthon.Eff, AnthonyQBachler, Arbor,Arjayay, Arvindn, AxelBoldt, B7T, Bdesham, Behavioralethics, BenFrantzDale, Billymac00, Bjankuloski06en, Bloodshedder, Blow of Light, Bluemoose, Bobblewik, Bogsat, Bongwarrior,Bracton, Brendan Moody, Bret101x, Brighterorange, BrokenSegue, C mon, CRGreathouse, CSTAR, Calabraxthis, Calton, Camrn86, Can't sleep, clown will eat me, Canderson7, CardinalDan,Catgut, Cats and kittens, Cb6, Cdc, Chairboy, Charledl, Charles Matthews, Cheeseisafruit, Chesleya, Childhoodsend, Clay Juicer, CloudNine, Cometstyles, Common Man, Conversion script,Cphay, Cretog8, Curps, DVdm, Dameyawn, Damian Yerrick, Danger, Dank, Dave101, David Eppstein, David Haslam, David Levy, David Shay, DavidCary, DavidScotson, Davidbod,Davidcarfi, Davidmayberry, Dcabrilo, Delirium, DerHexer, Dexter inside, Dgscott4, Dicklyon, Digitalme, Dionyziz, Discospinster, Diuturno, Dlgiffen, DouglasGreen, DrDentz, DrJonesNelson,Dramaturgid, Drobinsonatlaur, Droll, Dromedary, Duncharris, Dureo, Dysprosia, EachyJ, Eags, Ebricca, EconoPhysicist, Economicprof, EdJohnston, Edward, Edward321, El C, Electricbassguy,Elwikipedista, EmilJ, Encephalon, Enchanter, Enochlau, EnumaElish, Erianna, Espetkov, EvanYares, Everyking, Everything counts, Ewlyahoocom, Faintly, Ferstel, FerzenR, Fioravante Patroneen, Firien, Fish and karate, Fixitrich, Flowerpotman, Fnielsen, Fortunecookie289, Freegoods, Furrykef, GHe, GabrielAPetrie, Gametheoryguy, Gametheoryme, Gary King, Gaurav1146, Gauss,Geometry guy, GeorgeLouis, Gfoley4, Giftlite, Gimboid13, Goarany, Gombang, Googl, Graham87, Gregalton, Grick, Gronky, Gtcs-student, Guyasleep, Hairy Dude, Hakperest, HappyCamper,Hardy Littlewood, Harryboyles, Haylon357, Hellkillz, Henrygb, Hoodam, Hooverbag, Hoziron, Hqb, Hubriscantilever, Huon, Hve, Hwansokcho, I203.129.46.242I, II MusLiM HyBRiD II,Ikanreed, Iluvcapra, Indego Prophecy, Infarom, Inkbacker, Isdhbcfj, Isfisk, Isomorphic, Ivemadeahugemistake, J heisenberg, J.delanoy, JDspeeder1, JForget, Jacob Finn, Jakob.scholbach,JamesGecko, JavOs, Jaymay, Jecar, Jeekc, Jfpierce, Jhanley, Jimmaths, Jitse Niesen, Jlairdpdx, Joe.Dirt, JoergenB, John Quiggin, John254, Jon Roland, Jonel, Jpers36, JuPitEer, JuanOso,Jumbuck, Justin W Smith, Jwdietrich2, Kallimina, Kanmalachoa, Kenneth M Burke, Kesten, Khendon, Kiefer.Wolfowitz, KingOfBurgers, Kinos, Kirtag Hratiba, Kizor, Kku, Knowsetfree, Kntg,Kntrabssi, Koczy, Kukuliik, Kungfuadam, Kymacpherson, Kzollman, La Pianista, Lakinekaki, Landroni, Larknoll, Larry Sanger, LeaveSleaves, LeiserJ, Leon math, Lesslame, Levineps, Lexor,Lgallindo, Liftarn, Lights, Lihaas, LizardJr8, Lotje, LukeNukem, Lupin, Luqui, MER-C, MIT Trekkie, MLCommons, Machine Elf 1735, Magmi, MarcoLittel, Marenio, Mark Renier, Marqueed,Martin Kozák, Marykateolson, Matgerke, Mathprog777, Mattisse, Matveims, Mav, Mayumashu, McMorph23, Mcculley, Mdd4696, Meelar, Mentisock, MercyBreeze, Mfandersen, MichaelHardy, Mike2vil, Mild Bill Hiccup, Mindmatrix, MisfitToys, Mitsuhirato, Mk270, Mk5384, Mkch, Monjur 99, Moonlight Poor, Morphh, Mousomer, Moxon, Mr. G. 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Simulation  Source: http://en.wikipedia.org/w/index.php?oldid=444263839  Contributors: 16@r, 6024kingedward1, A. Parrot, AManWithNoPlan, Aaron Solomon Adelman, Aavviof, Acdoyle2000, Addshore, Aeroman213, Aerotone, Aetheling, Agilemolecule, Agostinobruzzone, Ahanberg, Alan Fitzgerald, Albert ip, Aleenf1, Amorymeltzer, Andre Engels, AndrewDressel, Anomalocaris, Ap, Arasut, Arthena, Arthur Rubin, Asparagus, Atlant, Axlq, B7582, BIONICLE233, BMF81, BackwardsBoy, Bapho, BarryList, Barticus88, Bdalgarno, Bdel63, Behaafarid, Bel.Alena, Ben Ben, Bobbajobb, Bobblehead, Bobet, Brucevdk, C.Fred, CLandsberg85, COMPFUNK2, Caitlin Timms, Capricorn42, Cddejong, Celticsfan730, ChemGardener, Cleared as filed, Closedmouth, Commander Keane, Cortes IDS5717C, Cyclonenim, Dannyc77, Dastly75, Davehi1, David Wright, Davidcofer73, Dayorkmd, DeepAnim1, Dgibson, Dhchilies, Dirkbb, Dlacombe,

Article Sources and Contributors 188

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Artificial intelligence  Source: http://en.wikipedia.org/w/index.php?oldid=444478767  Contributors: 100110100, 132.204.25.xxx, 172.141.188.xxx, 17Drew, 1dragon, 200.191.188.xxx,A157247, AAAAA, APH, APL, ARC Gritt, AVBAI, AVGavrilov, Abdullahazzam, Abeg92, Academic Challenger, Acerperi, Acroterion, AdSR, Adam Roush, Adamreiswig, AdjustShift,AdultSwim, AgadaUrbanit, Agemoi, AgnosticPreachersKid, Ahoerstemeier, Ai24081983, Aitias, AlGreen00, Alan Rockefeller, Alansohn, Aldux, Alex naish, Alexf, AlistairMcMillan,Allstarecho, Aloisdimpflmoser, Alsandro, Amareshjoshi, AmiDaniel, Anandcv, Andre Engels, AndrewHZ, Andreworkney, Andy Dingley, Andy Marchbanks, Angr, Aniu, Ankank, Anniepoo,Annonnimus, Antandrus, Anthony Appleyard, Anville, ArielGold, Arne Heise, Arthena, Arthur Rubin, Arthur Smart, ArthurWeasley, ArtificioSapiens, Arvindn, Asbestos, Ashish krazzy,Astrobase36, Atamyrat, Avenged Eightfold, AxelBoldt, Axl, Axon, Aykantspel, AzaToth, Azlan Iqbal, B3virq3b, Baby16, Banes, Banus, 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Probability theory  Source: http://en.wikipedia.org/w/index.php?oldid=443357270  Contributors: 129.116.226.xxx, 1ForTheMoney, APH, Aastrup, Abce2, Ali Obeid, Andeggs, AnonymousDissident, Arcfrk, Arjay369, AvidDismantler, Beland, Betterusername, Bjankuloski06en, Bjcairns, Bobo192, Boleslav Bobcik, Booniesyeo, Borislav, Bryan Derksen, Bsmath802, Btyner,CRGreathouse, Calabraxthis, Capricorn42, Charles Matthews, ChicXulub, Christopher Connor, Conversion script, Coyets, Cretog8, Cyfal, Cyrillic, DARTH SIDIOUS 2, DarkAudit, DavidEppstein, Dbtfz, Debresser, Den fjättrade ankan, Drizzd, DutchDevil, Dyaa, Dylan Lake, Dysprosia, Eeekster, El C, Elassint, Ensign beedrill, Fantastic4boy, Fastfission, Flammifer, Frankman,FutureNJGov, Gala.martin, Gbr3, Geometry guy, Gerhardvalentin, Giftlite, Gill110951, Givegains, Goatasaur, Goldfinger 93, Graham87, Gutsul, Hadal, Hayabusa future, Hirak 99, INic, JasonPatton, Jauhienij, JayJasper, Jheald, Johannes Hüsing, Jonik, Josang, Josephbrophy, Justin W Smith, KingTT, Knutux, KoyaanisQatsi, Krun, Kungfuadam, Kurtan, Lambiam, Larry_Sanger, LeeDaniel Crocker, Lenthe, Leroytirebiter, Lethe, Levineps, Liko81, MER-C, MH, Magmi, Malhonen, MathMartin, Mathstat, Maximaximax, Mayumashu, McSly, Mdd, Melcombe, Michael Hardy,Michael Slone, Miguel, MisterSheik, Msh210, Myasuda, Ncmathsadist, Nguyen Thanh Quang, Niteowlneils, Numbo3, Obradovic Goran, Oda Mari, Oleg Alexandrov, Omicronpersei8, PAR,PJTraill, Patrick, Paul August, Pax:Vobiscum, Pb30, Pinethicket, Porcher, PrisonerOfIce, Progicnet, Quiet photon, Qwfp, RJHall, RainbowOfLight, Raymond Meredith, Raymondwinn,Reidbaileyrocks, RexNL, Rgclegg, Rich Farmbrough, Rjwilmsi, Robertetaylor, Roman V. Odaisky, Salix alba, Sceptre, ShaunMacPherson, SiobhanHansa, Sleeping123, Snielsen, Some jerk onthe Internet, Spebudmak, Spudbeach, Srinivasasha, Suruena, TMLutas, Tayste, The Anome, TheMandarin, Tiddly Tom, Tide rolls, Tizio, Tosha, Treisijs, Trovatore, Tsirel, Unionhawk, Urdutext,User27091, Utcursch, Vivacissamamente, Wavelength, Weialawaga, Wyatt915, Wynand.winterbach, Ynh, Zaharous, Zenohockey, Zundark, Zwilson, Борис Пряха, 241 anonymous edits

Article Sources and Contributors 189

Linear programming  Source: http://en.wikipedia.org/w/index.php?oldid=442827614  Contributors: .:Ajvol:., AJim, Abdullahadnan, Abovechief, Ajgorhoe, Akasha27, Ale jrb, Almi, Alotau,Andre Engels, Andreas Fabri, Andris, Apdevries, Arie ten Cate, Arjuna-vallabha, Artier, Ash211, Asm4, Barak, BarryList, Batpox, Bender2k14, Bengalftbl55, Bird of paradox, Bobblewik,Bobo192, Borgx, CRGreathouse, Caelumluna, Caesura, Charvest, ChaudhryZafar, Chemuser, Chenopodiaceous, Chris the speller, Chrislk02, Cjohnson, Ckhung, Cointyro, Cremepuff222, Crisgh,Ctbolt, CyberShadow, Cybercobra, Czyl, DSP-user, DSachan, Dakee, Daveagp, Davekpeck, David.Monniaux, Dcoetzee, Ddechene, Dekart, Delirium, Dianegarey, Diego Moya, Dmitrey,Doobliebop, Doyley, Dpv, Dspoerl, Dysprosia, EdJohnston, Edadk, El C, EmmetCaulfield, Endersdouble, Erniepan, Falcon8765, Faturita, Fox, Fredrik, Freedomlinux, Freeman77, Fschoenm,Func, Fyyer, G.de.Lange, Gcm, Gergely.Szekely, Ghettoblaster, Giftlite, Gilliam, Gms, Gnasher729, Gotozeus, Greco-German, Gshaham, Gwernol, Hairy Dude, Hgkamath, Hgreenbe,HiYoSilver01, Hike395, Hjjalal, Hu12, Ike9898, Inductiveload, Intellec7, Isaacto, It Is Me Here, JFB80, JRSP, JYolkowski, Jacj, Jakob.scholbach, Jhausauer, Jitse Niesen, Jlyonsmith, John ofReading, Johnflux, Johngcarlsson, Johnuniq, JonMcLoone, Jonkerz, Joriki, JoshuaZ, Jpkotta, Jugander, Junglecat, Justforasecond, Justin W Smith, Jwestbrook, Kbdank71, Keilana,Kiefer.Wolfowitz, KnowledgeOfSelf, LOL, LachlanA, Ladislav the Posthumous, Lancedgang, Lavaka, Liechtensteiner, LilHelpa, Lissajous, Lobizón, LorisRomito, Lradrama, Lschrage,Macintosh User, Marco Krohn, Marcol, Matforddavid, Matthijs, Mcmlxxxi, Mdd, Medvall, Michael Hardy, Michal Jurosz, Misfeldt, Misof, Miym, Mlpkr, Mmortal03, MrOllie, Msh210,Myleslong, Natebrix, NathanHurst, Nbarth, Nelson50, NeoChaosX, New Thought, Nick, Nigholith, Nimur, Nkhanna, Numsgil, Obradovic Goran, Oleg Alexandrov, Oliphaunt, OlivierMiR,Optimalon, Orderud, Oskar Sigvardsson, PMajer, PapLorinc, Patrick, Pfortuny, Piano non troppo, PimBeers, Pkbmx, Qonnec, Quinnculver, RainbowOfLight, Rgrimson, Ric.s.cabral, Riedel,Rinconsoleao, RobinK, Robinh, Rsandvik, Saf37, Salman3037, Schmausschmaus, Schutz, Sdr, Senarclens, Sergnov, Shahab, SimpleKFC, Smokedsalmon crispyduck, Soliloquial, Spacepotato,Spalding, Splintercellguy, Steeldragon24680, StitchProgramming, Stu Mitchell, Tamfang, That Guy, From That Show!, The Anome, The Thing That Should Not Be, Thiseye, Thrufir,Thunderfish24, Thv, Tijfo098, Tmkly3, Tobias Bergemann, Tohd8BohaithuGh1, Trubadur, Twocs, Twri, Ulric1313, Undsoweiter, Urdutext, Vanderbei, Viridian, Vkuncak, Vocaro, Voltteri,Wastingmytime, Wenga66, Wikipelli, WuTheFWasThat, Ylloh, Zheric, Zido, Ɯ, 497 anonymous edits

Nonlinear programming  Source: http://en.wikipedia.org/w/index.php?oldid=440365847  Contributors: Ajgorhoe, Alexander.mitsos, BarryList, Broom eater, Brunner7, Charles Matthews,Dmitrey, Dto, EconoPhysicist, EdJohnston, EncMstr, Frau Holle, FrenchIsAwesome, G.de.Lange, Giftlite, Hike395, Hu12, Isheden, Jamelan, Jaredwf, Jean-Charles.Gilbert, Jitse Niesen,Kiefer.Wolfowitz, KrakatoaKatie, Leonard G., McSush, Mcmlxxxi, Mdd, Metiscus, Miaow Miaow, Michael Hardy, Mike40033, Monkeyman, MrOllie, Myleslong, Nacopt, Oleg Alexandrov,Olegalexandrov, PimBeers, Psvarbanov, RekishiEJ, Sabamo, Stevenj, Tgdwyer, User A1, Vgmddg, 56 anonymous edits

Integer programming  Source: http://en.wikipedia.org/w/index.php?oldid=436746449  Contributors: Amontjoy, Bender2k14, CBM, CRGreathouse, Daveagp, Delirium, Isheden,Kiefer.Wolfowitz, Matforddavid, Michael Hardy, Miym, Shahab, Tijfo098, X7q, Іванко1, 8 anonymous edits

Dynamic programming  Source: http://en.wikipedia.org/w/index.php?oldid=444536782  Contributors: 1baumann, A. Pichler, AHMartin, Abi79, Aceituno, Aeons, Agreppin, Alex.altmaster,Altenmann, Ancheta Wis, AshtonBenson, Atif.hussain, Babbage, Beefman, Beland, Bluebusy, Brentsmith101, Bunyk, Cancan101, Cannin, Chan siuman, Chipuni, Conskeptical, Crystallina,Cybercobra, D h benson, Damian.frank, David Eppstein, Dbroadwell, Dcoetzee, Drilnoth, Edschofield, Erxnmedia, Ethan, Eupedia, Freakofnurture, Fredrik, Furrykef, FvdP, Gaius Cornelius,Gene.arboit, Giftlite, Gnorthup, Grafen, Guahnala, Guslacerda, Hgranqvist, Hike395, HorsePunchKid, Huggie, Humanengr, Hyad, Imran, Intgr, Isheden, JFB80, JMatthews, JaGa, Jackzhp,Jaredwf, Jeff Edmonds, Jirislaby, Jiuguang Wang, Jmeppley, JonH, Julesd, Justin W Smith, Kaeslin, Karl-Henner, Ketil, Kiefer.Wolfowitz, Kku, LX, LachlanA, Leonard G., LiDaobing, Mahlon,Mark T, Matforddavid, Mdd, Meonkeys, Miaow Miaow, Mikeblas, Miss Madeline, Miym, Mlpkr, MrGBug, Mwj, Nils Grimsmo, Nowhere man, Npansare, Oleg Alexandrov, Orange Suede Sofa,Oskar Sigvardsson, Paddu, Patrick O'Leary, Pekrau, Phatsphere, Philip Trueman, Pixiefeet, Pjrm, Pm215, Popnose, Pozix604, Prunesqualer, Qz, R'n'B, Rajkumar.p84, Richienumnum,Rinconsoleao, RustyWP, SSJemmett, Sam Hocevar, SamIAmNot, SavantEdge, SeldonPlan, Shantavira, Shuroo, Signalhead, Smmurphy, Sniedo, Spinmeister, Spireguy, Spmeyn, Sriganeshs,Stannered, Sydneyfong, Szarka, Tamfang, Terrifictriffid, The Thing That Should Not Be, TheMandarin, Tom Duff, Tommy2010, Tonya49, TripleF, Utcursch, VKokielov, Vanished user47736712, Vecter, Vegpuff, Vssun, Waxmop, Wonglijie, Zahlentheorie, Zarei.h, Zhouhowe, Zirconscot, Ztobor, Zzyzx11, 347 anonymous edits

Goal programming  Source: http://en.wikipedia.org/w/index.php?oldid=442539774  Contributors: Abigail Clarke, Bryan Derksen, Charles Matthews, DCDuring, Dfj, FrankTobia, Lycurgus,Mdd, MuthuKutty, Nddstudent, Ronz, Shadowjams, The Thing That Should Not Be, War, 37 anonymous edits

Information theory  Source: http://en.wikipedia.org/w/index.php?oldid=441051589  Contributors: (, 213.122.18.xxx, APH, Alejo2083, AlexanderMalmberg, Algorithms,AllGloryToTheHypnotoad, Allchopin, Almkglor, Ammarsakaji, Ancheta Wis, Andres, Andrewbadr, Andycjp, Angela, Ann O'nyme, Annacoder, AnonMoos, Ap, Arthur Rubin, Arunkumar,AxelBoldt, Bemba, Bemoeial, BenjaminGittins, Bethnim, Bidabadi, Bobby D. Bryant, Bobo192, Brion VIBBER, BryanD, Buettcher, C9900, CALR, COMPATT, CRGreathouse, Calbaer,CapitalR, Cassandra Cathcart, Cazort, Cburnett, Chancemill, Charles Matthews, Chris Pressey, Chungc, Cihan, Commander Nemet, Constant314, Conversion script, Creidieki, Crunchy Frog, D,D6, DV8 2XL, Daggerstab, DanBri, Dani.gomezdp, David Eppstein, Deepmath, Dicklyon, Djcmackay, Dysprosia, Dziewa, E-Kartoffel, EPM, Eatsaq, Edchi, El C, Elektron, Ericamick,Eweinber, Expooz, Eyreland, Fleem, FrancisTyers, FrankTobia, Fredrik, FreezBee, Giftlite, Gnomehacker, Graham Chapman, Graham87, Grgarza, Grubber, Guaka, HSRT, Haham hanuka,Hannes Hirzel, HappyCamper, Harryboyles, Heidijane, Henri de Solages, Het, Hpalaiya, HumphreyW, Hyacinth, Iluvcapra, Imz, Informationtheory, Informationtricks, Isheden, Isomorphic,Isvish, Ivan Štambuk, Jahiegel, Jheald, Jimmaths, Jingluolaodao, Joeoettinger, Johnuniq, Jon Awbrey, Josh Parris, Jthomp4338, Jwdietrich2, Kanenas, Karada, Kevin Baas, Kjells, KuniShiro,Kusma, L.exsteens, L353a1, Lachico, Lee J Haywood, Light current, Linas, LittleDan, Loom91, Lordspaz, Lotje, LouScheffer, Lupo, Lyrl, MH, ML, Magmi, ManuelGR, Maria Vargas,Masgatotkaca, Masrudin, Materialscientist, Matthew Verey, Maurreen, Mayumashu, Mceliece, Mct mht, Mdd, Metacomet, Michael Hardy, Michael Ross, Michael Slone, Mindmatrix,Minesweeper, MisterSheik, Mitch Ames, Moe Epsilon, Momergil, Mpeisenbr, MrJosiahT, Msh210, Munford, MuthuKutty, Nabarry, Nageh, Nearfar, Netalarm, Neuromancien,NeuronExMachina, Nick Green, Noisy, Nothingmuch, Novum, Oldrubbie, OverlordQ, PHansen, Pax:Vobiscum, Pcarbonn, Pcontrop, Pearle, Peerc, Picapica, Poor Yorick, PoulyM, Pouya,Pulkitgrover, Pzrq, Radagast3, RainbowCrane, Rakesh kumar, Rbj, Rdsmith4, Reedy, Rich Farmbrough, Roman Cheplyaka, Ruud Koot, Rvollmert, SC, ScottHolden, Securiger, Seglea,SeventyThree, Siddhant, Sigmundg, Simon South, Sina2, Sir Nicholas de Mimsy-Porpington, SkyMachine, Smalljim, SoWhy, Spellchecker, Spoon!, Srleffler, Staffwaterboy, Starrymessenger,StephanWehner, SudoMonas, Tamer ih, Taxisfolder, Tedunning, Terra Novus, Thermochap, Thomas Keyes, TimeOfDei, Timo Honkasalo, Tiramisoo, Toby Bartels, Twohoos, Tyrson, UncleBill, Unnikrishnan.am, Useight, Varunrebel, Vegetator, Velho, Vinodmp, WikiIT, Wile E. Heresiarch, Wizard191, Woohookitty, Ww, XJamRastafire, Xezbeth, Xlasne, Yahya Abdal-Aziz, Ynh,514 anonymous edits

Cybernetics  Source: http://en.wikipedia.org/w/index.php?oldid=442898147  Contributors: .:Ajvol:., 1exec1, AbsolutDan, Acroterion, AdiJapan, Aeternus, Alan Liefting, AllanG,AndrewHowse, AndriuZ, Andy Denis, Angela, Apeiron07, Argumzio, Arthena, Asukara, Beland, Bhadani, Bobby D. Bryant, Bookuser, Boris Krassi, Brandelf, Breinbaas, Brona, BrotherGeorge,Bryan Derksen, BryanD, Cbdorsett, Chaosdruid, Chriscf, Ciphergoth, CommonsDelinker, ConceptExp, Conversion script, Daileychwaliboy, Davehi1, Deadclever23, Deathphoenix, DenisDiderot, Deodar, Dhcsoul, Djdoobwah, Doctahdrey, Donreed, Doradus, Drjheise, Duke Ganote, EWS23, Ed Poor, Ercolev, Erkan Yilmaz, Estranom, Fastfission, Fenice, FrancisTyers, FredBauder, Fynsnman, Gary Cziko, Giftlite, Goldorak, GraemeL, Grivalta, Guyonthesubway, Haaqfun, Harriv, Hawdon15, HenkvD, Hithereimdan, Hmains, Howardjp, Hu, Hu12, IMSoP,IanManka, Ike9898, Impact49, J04n, JDspeeder1, Jayjg, Jefffire, Jheald, Jimmaths, Jleedev, Jm34harvey, Joel Russ, Joffeloff, Jon Awbrey, Jondel, JuJube, Junkfather, Jwdietrich2, KarolLangner, Kelisi, Kenneth M Burke, Kesaloma, Kevin Baas, LSmok3, Larry laptop, Lexor, Lightmouse, LilHelpa, Lolzolzolz, Loremaster, Lowellian, Luctor IV, M.e, Magnusr, ManuelGR,Marcdelm, Martarius, Martian, Mato, Matt B., Maurice Carbonaro, MaxHund, MayaSimFan, Mdd, Metrax, Michael Hardy, MichaelWattam, Mmortal03, Mogadorien, Morenosastoque, Motters,Mr3641, MrOllie, Mschel, Mxn, NZUlysses, Nam H T Rae, Nave.notnilc, NawlinWiki, NeilN, Nejtan, Neuromancien, Nick Green, Nihiltres, Nile, NoiselessJoe, Normxxx,NotQuiteEXPComplete, Oldekop, Omicron18, Orderud, Ortolan88, Ott, Paliku, Panoramix, Paul A, Paulpangaro, Pcontrop, Peterdjones, Pgk, Plasticup, Poccil, Pyrsmis, Quiddity, Raiph,Raphaele7, Raven4x4x, Rbudegb, Rejimissac, Rememberway, Ricky81682, Robin klein, Ronz, Rsanz, Sah111-666, Sam Hocevar, Sanguis Sanies, Schakelaar, Scorpion451, SeventyThree,Shizhao, Sholto Maud, Sietse Snel, Sigmundg, Slicky, Sm8900, Snowded, Spawn Man, Sten, Stephenchou0722, Stratocracy, Suryadas, Swedenborg, Tabor, TastyPoutine, Taxisfolder, Tha Masta,The Cunctator, The Noosphere, The Transhumanist, Theresa knott, Tomsega, Trclark9, Verbum Veritas, Vlad, Vuo, Warrickball, Wikiborg, Will Beback, Wotnow, Xgeom, Zoniedude,Александър, 284 anonymous edits

Systems theory  Source: http://en.wikipedia.org/w/index.php?oldid=443170291  Contributors: 7195Prof, APH, Academic Challenger, Action potential, Afbcasejr, Ai24, Alan Liefting,Anikasavage, Aoxiang, Aqualung, Arialboundaries123, Asrghasrhiojadrhr, Astharoth1, Athkalani, Attilios, Baumjay, Bcastel3, Bender235, Benking, BertSeghers, Blazotron, Blood sliver,Borisgloger, Brighterorange, Btphelps, Campjacob, Captain-tucker, Centrx, Cgs, Charles Matthews, ChrisHodgesUK, Chrisdel, Chzz, Colonies Chris, Conversion script, Cretog8, Cyde, Cícero,D.h, DNewhall, DashaKat, Davidkinnen, Deipnosophista, Dggreen, Dialectic, Dougwalton, Dr. Gabriel Gojon, Dranorter, Dukey, El C, Epolk, Erkan Yilmaz, Etherclear, Evercat, FT2,Fatherjeromeusa, Feeeshboy, Fences and windows, Fenice, Ffangs, FidelDrumbo, Fixaller, Flewis, Fplay, Fratrep, Fred Bauder, Fuhghettaboutit, GarOgar, Gemmink2, GeoW, George100,Giftlite, Gilliam, Gintautasm, Graham87, Greudin, Hadal, Hannes Hirzel, Headbomb, Helvetius, HenkvD, Hetar, IOLJeff, IPSOS, Inwind, Iridescent, IsabellaElhasan, JForget, JFromm, JHunterJ,JKeck, JS Nelson, JaGa, JenLouise, Jessica72, Jfusion, Jncraton, John Abbe, John D. Croft, Jon Awbrey, JorgeGG, Jose Icaza, JoseREMY, Jpaulm, Jpbowen, JuanR, Jwdietrich2, Kate, KellyMartin, Kenneth M Burke, Kenyon, Knowledgeum, Krauss, LaosLos, Lenov, Leolaursen, Letranova, Lexor, Lid, Limetom, Ljasie, M Payne, Magntuve, Mange01, Margaret9mary, Martg76,Mathieugp, Mathmanta, Maunus, MaxHund, McSly, Mdd, Michael Hardy, Michaelbusch, Moncrief, Mporter, N5iln, Nbaig, Nectarflowed, Neilbeach, Nick Green, NickRichmond, Nigholith,Nihiltres, Niki K, Ntsimp, Obradovic Goran, Ombudsman, Ordermaven, OrgasGirl, Palapa, Pan BMP, Pargeter1, Paul August, Pcontrop, Penfold, Peterdjones, Phanerozoic, Philosotox, Piotrus,Pitlane02, Pivi11, Pjrich, Plasticup, Pmod, Poliorcetes, Psychohistorian, RDF-SAS, RJBurkhart3, Ragesoss, Ramir, Ray Van De Walker, Razimantv, Rbudegb, RedHouse18, Refsworldlee,Remipeds, Reyk, Rich Farmbrough, Richard D. LeCour, Ritafelgate, Rjwilmsi, Rl, RobertG, Roesser, Ronewolf, Ronz, Rory Torrens, Rrburke, Rursus, ST47, Schmorgluck, Scottprovost,Sevenoaks, Shadowjams, Sholto Maud, Skipsievert, Slark, Smmurphy, Soph2, Sr388frau, Steven Zhang, Stirling Newberry, Subtractive, SummerPhD, Sunray, Svadhisthana, Tarquin,TheMadBaron, Theo10011, Tide rolls, Tim Starling, Togo, Toh, Tom Mandel, Tomisti, Tommysun, Tomsega, Tothebarricades.tk, Trautw, Trisapeace, Trusilver, Ty580, Useight, Vald, Vegetator,Viriditas, VirtualDelight, Washingtoncool, WikiDao, Wl219, Wotnow, Zvar, 达伟, 348 anonymous edits

Queueing theory  Source: http://en.wikipedia.org/w/index.php?oldid=439460148  Contributors: AbsolutDan, Alison22, Ancheta Wis, Apoorv020, Arnaudf, Atomskninja, Avatar, Axl, BMF81, Bcat, Bertoli, Bezenek, BillNace, Blaisorblade, Bobo192, Brianhe, Cameron Dewe, Charles Matthews, Chrike, Cjrs 79, Clicketyclack, Cyko, Daf, Damian Yerrick, Danikolt, David Cooke, DavidLevinson, Deewiant, Diberri, Everything counts, EyeSerene, Gaius Cornelius, Gardar Rurak, Gareth Jones, Giftlite, Hgfernan, HisashiKobayashi, Ian Geoffrey Kennedy, Ingolfson, Isnow,

Article Sources and Contributors 190

J.delanoy, Jasper Chua, Jedibob5, Jemimsa, Jimmaths, Jnc, John Quiggin, John Reed Riley, JonHarder, Joshdick, Joy77, Jrdioko, Justsail, Kevin Saff, Kiscica, Kitsonk, Kku, Kmweber, Kvdveer,KyraVixen, LachlanA, LoopZilla, Lotje, Lupin, Mange01, MartinHarper, Mdd, Melcombe, Michael Hardy, Moonsun1981, MrZeebo, Mydogategodshat, Myleslong, Nick.follows, Ohnoitsjamie,Oleg Alexandrov, OrgasGirl, PhysPhD, Pleckaitis, Pointillist, PsyberS, Qbmessiah, Quest for Truth, Qwertyus, R3m0t, RHaworth, RedWolf, Reetep, Rgclegg, Rjwilmsi, Ronz, Rrburke, Seapwc,SimonD, Skittleys, Spmeyn, Stephen Turner, StephenJennings, Stephenb, Swampfox, Telemakh0s, Think outside the box, Thisisbossi, Tinctorius, Tomi, U89djt, Umar420e, Vrenator, Wcdriscoll,Wikibarista, Yqwen, Zigger, Zzuuzz, 153 anonymous edits

Markov chain  Source: http://en.wikipedia.org/w/index.php?oldid=443655755  Contributors: 1ForTheMoney, A5, Aardvark23a, AdjustShift, AlanUS, Aliekens, Altarego, Andrei Stroe, Andres,Andreyf, Andytango, Angela, Ankitpatel715, Anthonywong, Archelon, ArglebargleIV, Arthena, Aurelius173, Autotomic, AxelBoldt, Begoon, BenFrantzDale, Bender235, Bjcairns, Boing! saidZebedee, Boyd Steere, BrotherE, Brusegadi, Bryanjohnston, Bubba hotep, BusError, C S, Capodistria, Centrx, Chadloder, Charles Matthews, ChrisKennedy, Christopher Parham, Cihan, Cosfly,CosineKitty, Cuaxdon, Cww, David Eppstein, DavidCBryant, DavidCary, DavidFHoughton, Deb, Denis.arnaud, Digfarenough, DmitriyV, Doctahdrey, DoctorW, Doczilla, Drae, Dreadstar,Drilnoth, Droll, Drseudo, Dyaa, Dysprosia, Déjà Vu, Eassin, Ed Poor, Eed3si9n, Eeera, Eggstone, Egkauston, Ehudshapira, Ejrh, Ekotkie, El C, Elmindreda, Epbr123, Ericamick, Ezrakilty,F.Shelley, Falcon8765, Faridani, Fcady2007, Fij, FocalPoint, Forwardmeasure, Fratrep, Frymaster, Furrykef, Gaius Cornelius, Gausseliminering, Giftlite, Gintautasm, Gioto, Greender, Gregsearle, Grubber, Guanhomer, Gwalla, Gzuckier, Hadal, Henawy 1, Henning Makholm, Hirak 99, Hl, Ht686rg90, HyDeckar, Indy90, Ino5hiro, IradBG, Iridescent, Isnow, JHunterJ, JMK,JYOuyang, JaGa, Jafusmaximus, Jdlh, Jdthood, Jheald, Jim 14159, Jim.belk, Jj1236, Jmchen, Joke137, JonMcLoone, Josephorourke, Jtsch, Julienbarlan, Julovi, K-squire, Karl Stroetmann, KarolLangner, Kazantsev, Kdcoo, Kenyon, Khim1, Kiefer.Wolfowitz, Kjoonlee, Kku, Kokoklems, L Kensington, LOL, LachlanA, Leandro79, Leonard G., Lightmouse, Linas, Ljaun, Lnummeli,Loodog, LouScheffer, Lovibond, Luqui, MK8, Mack2, Maghnus, Magic Window, Marasmusine, MartinGugino, Matthew Meta, Maurizio.polito, Maxlittle2007, Mbloore, Melcombe, Mellum,Merriam, Michael Hardy, Mikeeg555, Mindmatrix, Miserlou, MisterSheik, Mistercupcake, Mmortal03, ModernMajor, MorganGreen, Mpaa, MrOllie, Msh210, Mythobeast, Nanmus, Ncauthor,Negrulio, Neilc, Nicolaennio, Nijdam, O18, OdedSchramm, OlEnglish, Oleg Alexandrov, Olivier, Paulthree, PeterWT, Ph0t0phobic, Phil Boswell, Planetmarshall, Pseudomonas, Qlmatrix,Quantling, Quibik, Quintote, Qwfp, RDBury, RainbowOfLight, Randomblue, Rdbrady, Requestion, Rjwilmsi, Rkrish67, Rotring, Ruakh, SSZ, Samuelsen, SanderEvers, Sarma.bhs, Seth Ilys,SgtThroat, Shabda, Shanes, Shepard, Shepelyansky, Shiftchange, Shorespirit, Sigma0 1, Skittleys, Slakr, Sligocki, So nazzer it's pav, Someguy1221, Soundray, Spmeyn, Steve98052, Stoni,Stpasha, Stunetii, Stypex, Svein Olav Nyberg, TWiStErRob, Tarotcards, Tayste, Telemakh0s, Terra Xin, The Anome, TheKMan, Thomas Bliem, Timboe, Timwi, Tobias Bergemann, TomLougheed, Tomi, Tregoweth, Trevor.tombe, Tyrenius, Undsoweiter, Urdutext, Urhixidur, Vaahteramäen Eemeli, Voice In The Wilderness, Vvarkey, W0lfie, Wikiklrsc, Wile E. Heresiarch,WriterHound, Wvbailey, Xicer9, Yayavar, Ylloh, Yoderj, Zalgo, Zoonfafer, 408 anonymous edits

Graph theory  Source: http://en.wikipedia.org/w/index.php?oldid=443935412  Contributors: APH, Aaronzat, Abeg92, Agro1986, Ajweinstein, Aknxy, Akutagawa10, Alan Au, Alansohn,Alcidesfonseca, Alex Bakharev, Almi, Altenmann, Ams80, Andre Engels, Andrea105, Andreas Kaufmann, Andris, Ankitbhatt, Anna Lincoln, Anonymous Dissident, Anthonynow12, Aquae,Arbor, Armoreno10, Arvindn, Astrophil, AxelBoldt, Ayda D, Bact, Beda42, Bereziny, Berteun, Bkell, Blair Sutton, BlckKnght, Boleslav Bobcik, Booyabazooka, Brent Gulanowski, BrickThrower, Brona, Bumbulski, C S, CRGreathouse, Camembert, CanadianLinuxUser, Caramdir, Cerber, CesarB, Chalst, Charles Matthews, Chas zzz brown, Chmod007, Conversion script, Corpx,Csl77, D75304, DFRussia, Damien Karras, Daniel Quinlan, David Eppstein, Davidfstr, Dbenbenn, Delaszk, DerHexer, Dgrant, Dicklyon, Diego UFCG, Dina, Disavian, Discospinster, Dittaeva,Doctorbozzball, Dr.S.Ramachandran, Duncharris, Dycedarg, Dysprosia, Dze27, ElBenevolente, Eleuther, Elf, Eric.weigle, Eugeneiiim, Favonian, FayssalF, Fchristo, Feeeshboy,Fivelittlemonkeys, Fred Bradstadt, Fredrik, FvdP, GGordonWorleyIII, GTBacchus, Gaius Cornelius, Gandalf61, Garyzx, Geometry guy, George Burgess, Gianfranco, Giftlite, GiveAFishABone,Glinos, Gmelli, Goochelaar, Gragragra, Graham87, GraphTheoryPwns, GregorB, Grog, Gutza, Gyan, Hannes Eder, Hbruhn, Headbomb, Henning Makholm, Hirzel, Idiosyncratic-bumblebee,Ignatzmice, Igodard, Ino5hiro, Jakob Voss, Jalpar75, Jaxl, JeLuF, Jeronimo, Jheuristic, Jinma, Joel B. Lewis, Joespiff, Jojit fb, Jon Awbrey, Jon har, Jorge Stolfi, Jwissick, Kalogeropoulos, KarlE. V. Palmen, KellyCoinGuy, King Bee, Knutux, Kope, Kpjas, Kruusamägi, LC, LOL, Lanem, Ldecola, Liao, Linas, Looxix, Low-frequency internal, Lpgeffen, Lupin, Madewokherd, MagisterMathematicae, Magmi, Mahlerite, Mailer diablo, Mani1, Marek69, Marianocecowski, Mark Foskey, Mark Renier, MathMartin, Matusz, Maurobio, Maxime.Debosschere, McKay, Meekohi,Melchoir, Michael Hardy, Michael Slone, Miguel, Mild Bill Hiccup, Miym, Mlpkr, Morphh, Msh210, Mxn, Myanw, MynameisJayden, Naerbnic, Nanshu, Nasnema, Nichtich, Ntsimp,Obankston, Obradovic Goran, Ohnoitsjamie, Oleg Alexandrov, Oli Filth, Onorem, Optim, OrangeDog, Oroso, Oskar Flordal, Outraged duck, Pashute, Patrick, Paul August, Paul Murray,PaulHoadley, PaulTanenbaum, Pcb21, Peter Kwok, Piano non troppo, PierreAbbat, Poor Yorick, Populus, Powerthirst123, Protonk, Pstanton, Puckly, Pugget, Quaeler, Qutezuce, RUVARD,Radagast3, Ratiocinate, Renice, Requestion, Restname, Rev3nant, RexNL, Rmiesen, Roachmeister, Robert Merkel, Robin klein, RobinK, Ronz, Ruud Koot, SCEhardt, Sacredmint, Sangak,Shanes, Shd, Shepazu, Shikhar1986, Shizhao, Sibian, SlumdogAramis, Smoke73, Solitude, Sonia, Spacefarer, Stochata, Sundar, Sverdrup, TakuyaMurata, Tangi-tamma, Tarotcards, Taw,Taxipom, Tckma, Template namespace initialisation script, The Cave Troll, The Isiah, Thesilverbail, Thv, Titanic4000, Tomo, Tompw, Trinitrix, Tyir, Tyler McHenry, Uncle Dick, Vonkje,Watersmeetfreak, Whiteknox, Whyfish, Womiller99, Wshun, Wsu-dm-jb, Wsu-f, XJamRastafire, Xdenizen, Xiong, XxjwuxX, Yecril, Ylloh, Youngster68, Zaslav, Zero0000, Zoicon5, Zundark,Александър, Канеюку, 411 anonymous edits

Image Sources, Licenses and Contributors 191

Image Sources, Licenses and ContributorsImage:Dr.A.B.Rajib Hazarika & his kids.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Dr.A.B.Rajib_Hazarika_&_his_kids.jpg  License: Public Domain  Contributors:Rajah2770File:Kammhuber Line Map - Agent Tegal.png  Source: http://en.wikipedia.org/w/index.php?title=File:Kammhuber_Line_Map_-_Agent_Tegal.png  License: Public Domain  Contributors:UnknownFile:MaximumParaboloid.png  Source: http://en.wikipedia.org/w/index.php?title=File:MaximumParaboloid.png  License: GNU Free Documentation License  Contributors: Original uploaderwas Sam Derbyshire at en.wikipediaImage:Ultimatum Game Extensive Form.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ultimatum_Game_Extensive_Form.svg  License: Creative CommonsAttribution-ShareAlike 3.0 Unported  Contributors: Kevin Zollman --KzollmanImage:Centipede game.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Centipede_game.svg  License: Creative Commons Attribution-ShareAlike 3.0 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http://en.wikipedia.org/w/index.php?title=File:Hierarchical-control-system.svg  License: Creative Commons Attribution 3.0  Contributors:user:Pbroks13File:ParseTree.svg  Source: http://en.wikipedia.org/w/index.php?title=File:ParseTree.svg  License: Public Domain  Contributors: Traced by User:StanneredFile:Kismet robot at MIT Museum.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Kismet_robot_at_MIT_Museum.jpg  License: GNU Free Documentation License  Contributors:User:Tsca.bot, User:Waldir, user:PolimerekFile:ArtificialFictionBrain.png  Source: http://en.wikipedia.org/w/index.php?title=File:ArtificialFictionBrain.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Gengiskanhg, Jtneill, Scientist94, 1 anonymous editsFile:Artificial neural network.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Artificial_neural_network.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: en:User:CburnettFile:Automated online 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