1 Chapter 1 Automata: the Methods & the Madness Angkor Wat, Cambodia

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Chapter 1 Automata: the Chapter 1 Automata: the Methods & the MadnessMethods & the Madness

Angkor Wat, Cambodia

Chapter 1 - Chapter 1 - 22

OutlineOutline

1.1 Why Study Automata Theory?1.1 Why Study Automata Theory?1.2 Introduction to Formal Proof1.2 Introduction to Formal Proof1.3 Additional Forms of Proof1.3 Additional Forms of Proof1.4 Inductive Proofs1.4 Inductive Proofs1.5 Central Concepts of Automata 1.5 Central Concepts of Automata

TheoryTheory


1.1 Why Study Automata 1.1 Why Study Automata Theory?Theory?

1.1.1 Introduction to Automata1.1.1 Introduction to Automata

Properties of finite-state systemsProperties of finite-state systems

Having a Having a finite finite number of statesnumber of states

Entire history Entire history cannotcannot be remembered. be remembered.

Good to implement with a fixed set of resourcesGood to implement with a fixed set of resources

Can be modeled by Can be modeled by finite automata (FA)finite automata (FA)



Example 1.1 --- an FA modeling an on/off switchExample 1.1 --- an FA modeling an on/off switch

Example 1.2 --- an FA modeling recognition of the Example 1.2 --- an FA modeling recognition of the keyword “then” in a lexical analyzerkeyword “then” in a lexical analyzer

The double circles specify the final or “accepting state.”The double circles specify the final or “accepting state.”

startstart t

t h

th e

the n

then

offstartstart on

push

push



1.1.2 Structural Representation1.1.2 Structural RepresentationTwo other models (not automaton-like)Two other models (not automaton-like)

Grammar Grammar --- processing data with a recursive structure--- processing data with a recursive structureE.g., grammatical rule E E.g., grammatical rule E E + E for generating arithmetic e E + E for generating arithmetic e

xpressionsxpressions

Regular expressionRegular expression --- describing text strings --- describing text stringsE.g., UNIX-style regular expression E.g., UNIX-style regular expression

’’[A-Z][a-z]*[ ][A-Z][A-Z]’ describes [A-Z][a-z]*[ ][A-Z][A-Z]’ describes Ithaca NYIthaca NY, , Lafayette INLafayette IN……



1.1.3 Automata and Complexity1.1.3 Automata and Complexity

What can a computer do? --- What can a computer do? ---

Computability:Computability:

Studying problems which can be solved by Studying problems which can be solved by

computer, called computer, called decidable problems.decidable problems.

Decidability Decidability is the main topic in computability.is the main topic in computability.



1.1.3 Automata and Complexity1.1.3 Automata and ComplexityWhat can a computer do efficiently? --- What can a computer do efficiently? ---

Computational ComplexityComputational Complexity::Studying Studying tractable problemstractable problems solvable with some solvable with some

slowly growing function (like polynomial) of input slowly growing function (like polynomial) of input

size, & size, & intractableintractable problemsproblems solvable with fast solvable with fast

growing function (like exponential).growing function (like exponential).

Intractability Intractability is the main topic of computational is the main topic of computational

complexity.complexity.


1.2 Introduction to Formal 1.2 Introduction to Formal ProofProof

1.2.0 Concepts ---1.2.0 Concepts ---

Formal proof techniques are indispensable Formal proof techniques are indispensable

for proving theorems in theory of for proving theorems in theory of

computation.computation.

Understanding workings of correct Understanding workings of correct

programs is equivalent to proving theorems programs is equivalent to proving theorems

by induction.by induction.



1.2.1 Deductive Proofs (1.2.1 Deductive Proofs ( 演繹法演繹法 ))

Given facts (e.g. axioms), previous statementsGiven facts (e.g. axioms), previous statements

HypothesisHypothesis deduced statementsdeduced statements … … conclusion conclusion Logic principlesLogic principles

Logic principles: modus ponens (Logic principles: modus ponens ( 正前律、離斷規則、肯定正前律、離斷規則、肯定前件前件 )), syllogism (, syllogism ( 三段論三段論 )…)…

1.2.2~1.2.4 --- read by yourself1.2.2~1.2.4 --- read by yourself



Modus ponensModus ponens ( (正前律、離斷規則、肯定前正前律、離斷規則、肯定前件件 ))An example:An example:

If it rains, the sidewalk will be wet.If it rains, the sidewalk will be wet.It rains now.It rains now.So the sidewalk is wet.So the sidewalk is wet.

SyllogismSyllogism ( ( 三段論三段論 ) ) (supplemental)(supplemental)An example:An example:

All humans will die. All humans will die. Socrates is a human. Socrates is a human. So, Socrates will die.So, Socrates will die.


1.3 Additional Forms of Proof1.3 Additional Forms of Proof

1.3.0 Some more proof techniques ---1.3.0 Some more proof techniques ---Proofs about setsProofs about setsProofs by contradictionProofs by contradictionProofs by counterexamplesProofs by counterexamples



1.4 Inductive Proofs (1.4 Inductive Proofs ( 歸納法歸納法 ))

1.4.1 Induction on Integers1.4.1 Induction on IntegersTo prove a statement To prove a statement SS((nn) about an integer ) about an integer n n

by induction, we do:by induction, we do:

Basis Basis --- --- show show SS((ii) true for a particular ) true for a particular basis intebasis integerger ii (0 or 1 usually) (0 or 1 usually)

Inductive stepInductive step --- assume --- assume nn i (basis integer)i (basis integer), sh, show that the statement “if S(ow that the statement “if S(nn), then S(), then S(nn + 1)” is tr + 1)” is true.ue.



1.5 Central Concepts of 1.5 Central Concepts of Automata TheoryAutomata Theory

1.5.0 Three basic concepts1.5.0 Three basic concepts

Alphabet Alphabet --- a set of symbols--- a set of symbols

Strings Strings --- a sequence of symbols from an --- a sequence of symbols from an

alphabetalphabet

Language Language --- a set of strings from the same --- a set of strings from the same

alphabetalphabet



1.5.1 Alphabets1.5.1 Alphabets

Definition --- Definition ---

An An alphabetalphabet is a is a finitefinite, , nonempty nonempty set set

of symbols.of symbols.

Conventional notation --- Conventional notation ---



1.5.1 Alphabets1.5.1 Alphabets

The term “symbolsymbol” is usually undefined.

Examples ---

Binary alphabet

a, b, …, z} …



1.5.2 Strings1.5.2 StringsDefinition --- Definition ---

A A stringstring (or word) is a (or word) is a finite sequencefinite sequence of of

symbols from an alphabet.symbols from an alphabet.

Example --- Example --- 1011 is a string from alphabet 1011 is a string from alphabet



1.5.2 Strings1.5.2 Strings

Empty string Empty string --- a string with zero --- a string with zero

occurrences of symbolsoccurrences of symbols

Length |Length |ww| | of string of string ww --- the number of --- the number of

positions positions for symbols in for symbols in ww

Examples --- |0111|=4, |Examples --- |0111|=4, ||=0, …|=0, …




PowerPower of an alphabet of an alphabet kk --- ---

a set of all strings of length a set of all strings of length kk

Examples --- Examples ---

givengiven we havewe have

= {= {22 = {00, 01, 10, 11} = {00, 01, 10, 11}

Supplemental --- Supplemental --- 1100 = = (01)(01)00 = = … …




Set of all strings over Set of all strings over --- denoted as --- denoted as

It is not difficult to know that It is not difficult to know that

∪∪ ∪∪ ∪∪ ……



1.5.2 Strings1.5.2 Stringsset of nonempty strings fromset of nonempty strings from

Therefore, we haveTherefore, we have

∪∪ ∪∪ ∪∪……

∪∪



1.5.2 Strings1.5.2 StringsConcatenationConcatenation of two strings of two strings xx and y --- and y --- xyxy

Example --- Example ---

ifif x x = 01101= 01101, y, y = = 110110, then, then

xyxy = 01101 = 01101110, 110, xx xx = x= x= = 0110101101, …0110101101, …

is the is the identity identity for concatenation for concatenation

since since ww = = www.w.



1.5.2 Strings 1.5.2 Strings (supplemental)(supplemental)PowerPower of a string --- of a string ---

Defined by concatenation ---Defined by concatenation ---

xxii = = xxxx……xx ( (xx concatenated concatenated ii times) times)

Defined by recursion ---Defined by recursion ---

xx00 = = (by definition) (by definition)

xxii = = xxxxii-1-1



1.5.3 Languages1.5.3 LanguagesDefinition --- Definition ---

a a languagelanguage is a set of strings all chosen is a set of strings all chosen from some from some **

If If is an alphabet, and is an alphabet, and LL**, then , then LL is a is a language over language over ..



1.5.3 Languages1.5.3 LanguagesExamples ---Examples ---

The set of all legal English The set of all legal English words words is a is a language. Why? What is the alphabet language. Why? What is the alphabet here? here?

Answer:Answer: the set of all letters the set of all letters

A legal program of C is a language. Why? A legal program of C is a language. Why? What is the alphabet here? What is the alphabet here?

Answer:Answer: a subset of the ASCII characters. a subset of the ASCII characters.



1.5.3 Languages1.5.3 Languages

More examples of languages ---More examples of languages ---

The set of all strings of The set of all strings of nn 0’s followed by 0’s followed by nn 1’s 1’s

for for nn 0 0::

{{, 01, 0011, 000111, …}, 01, 0011, 000111, …}

* * is anis an infiniteinfinite language for any alphabet language for any alphabet ..




More examples of languages (cont’d) ---More examples of languages (cont’d) ---

the the empty languageempty language (not the empty string (not the empty string

is a language over any alphabet.is a language over any alphabet.

{{}} is a language over is a language over anyany alphabet (consisting alphabet (consisting

of only one string, the empty string of only one string, the empty string ).).



1.5.3 Languages1.5.3 Languages Ways to describe languages (1/3) ---Ways to describe languages (1/3) ---

Description byDescription by exhaustive listing exhaustive listing --- ---

LL1 = {1 = {aa, , abab, , abcabc} (finite language; listed one by one)} (finite language; listed one by one)

LL2 = {2 = {aa, , abab, , abbabb, , abbb, abbb, ...} (infinite language; listed pa...} (infinite language; listed pa

rtially)rtially)

LL3 = 3 = LL((abab*)*) (infinite language; expressed by a regular (infinite language; expressed by a regular

expression) expression)




Ways to define languages (2/3) ---Ways to define languages (2/3) ---

Description byDescription by generic elements generic elements --- ---

LL4 = {4 = {xx | | xx is over is over VV = { = {aa, , bb}, begins with }, begins with a,a,

followed by any number of followed by any number of bb, possible none} , possible none}

(note: (note: LL4 = 4 = LL3 = 3 = LL2)2)




Ways to define languages (3/3) ---Ways to define languages (3/3) ---

Description byDescription by integer parameters integer parameters --- ---

LL5 = {5 = {ababnn | | n n 0} 0}

(note: (note: LL5 = 5 = LL4 = 4 = LL3 = 3 = LL2)2)

*** *** bbnn = power of a symbol = power of a symbol



1.5.3a Operations on Languages 1.5.3a Operations on Languages (supplemental)(supplemental)(1/2)(1/2)

Languages are sets, and operations of sets may be Languages are sets, and operations of sets may be

applied to them:applied to them:

(1) union --- (1) union --- AA∪∪BB = { = {aa | | aa AA or or aa BB}}

(2) intersection --- (2) intersection --- AA∩∩BB = { = {aa | | aa AA and and aa BB}}

(3) difference --- (3) difference --- AA B B = {= {aa | | aa AA and and aa BB}}

A



1.5.3a Operations on Languages 1.5.3a Operations on Languages (supplemental)(supplemental)(2/2)(2/2)

Languages are sets, and operations of sets may be Languages are sets, and operations of sets may be

applied to them:applied to them:

(4) product --- (4) product --- AA BB = {( = {(aa, , bb) | ) | aa AA and and bb BB}}

(5) complement --- (5) complement --- ĀĀ = {= {aa | | aa UU and and aa AA}}

(6)(6) power set --- power set --- 22AA = { = {BB | | BB AA}}

Note:Note: UU is the is the universal setuniversal set, like , like which is the which is the closureclosure of an of an

alphabetalphabet

A



1.5.3b More Operations on Languages 1.5.3b More Operations on Languages (supplement(supplement

al)(1/2)al)(1/2)

Concatenation of two languages Concatenation of two languages LL11 and and LL22 --- ---

LL11LL22 = { = {xx11xx2 2 | | xx11 LL11 and and xx22 LL22}}

Power of a language Power of a language L ---L --- Defined directly --- Defined directly ---

LLkk = { = {xx11xx22……xxkk | x | x11, , xx22, …, , …, xxkk LL}}

Defined by recursion --- Defined by recursion --- LL00 = { = {}}

LLii == LL LLii-1-1

A



1.5.3b More Operations on Languages 1.5.3b More Operations on Languages (supplemental)(2/2)(supplemental)(2/2)

ClosureClosure of language of language LL --- ---

LL** = = = = LL∪∪ LL∪∪ LL∪∪ ……

Positive closurePositive closure of a language of a language L ---L ---

LL++ = = LL∪∪ LL∪∪ ……

It can be shown thatIt can be shown that L L LLLL – – {{} } when when LL

A

0

i

i

L



1.5.4 Problems1.5.4 ProblemsA problem in automata theory ---A problem in automata theory ---

a question of deciding whether a given string is a question of deciding whether a given string is

a member of some particular language.a member of some particular language.

That is, if That is, if is an alphabet, and is an alphabet, and LL is a is a language over language over , the , the problem problem LL is: is:

given a string given a string ww in in **, decide if, decide ifw w LL or or notnot

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1 Chapter 1 Automata: the Methods & the Madness Angkor Wat, Cambodia