Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Lecture 17 of 42 Wednesday, 08 October

Computing & Information SciencesKansas State University

Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence

Lecture 17 of 42

Wednesday, 08 October 2008

William H. Hsu

Department of Computing and Information Sciences, KSU

KSOL course page: http://snipurl.com/v9v3

Course web site: http://www.kddresearch.org/Courses/Fall-2008/CIS730

Instructor home page: http://www.cis.ksu.edu/~bhsu

Reading for Next Class:

Chapter 8, Russell & Norvig 2nd edition

Description Logics, OntologiesDiscussion: Knowledge Representation



Lecture Outline

Last Wednesday’s Reading: Sections 10.4 – 10.6, R&N 2e

Last Friday’s Reading: Sections 10.7 – 10.9, R&N 2e

Today: Knowledge Rep, Ontologies, Situational Calculus

This Week Temporal logic

Semantic networks

Description Logics

Next Week Defeasible reasoning: nonmonotonic logic

Intro to Planning

Midterm Exam: Mon 20 Oct 2008 Remote students: have exam agreement faxed to DCE

Exam will be faxed to proctors Wednesday or Friday


Basics of Reasoning in Description Logics

Basics of Reasoning in Description Logics

Jie Bao

Iowa State University

Feb 7, 2006



An ontology of this talkAn ontology of this talk

People

Student

Jie Bao

present

Description Logic

DL reasoning

Knowledge Representation

Topic

© 2006 J. Bao


RoadmapRoadmap

What is Description Logics (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm


Description LogicsDescription Logics

A formal logic-based knowledge representation language “Description" about the world in terms of concepts (classes), roles

(properties, relationships) and individuals (instances)

Decidable fragments of FOL Widely used in database (e.g., DL CLASSIC) and semantic web

(e.g., OWL language)


A “Family” Knowledge BaseA “Family” Knowledge Base

Person include Man(Male) and Woman(Female),

A Man is not a Woman

A Father is a Man who has Child

A Mother is a Woman who has Child

Both Father and Mother are Parent

Grandmother is a Mother of a Parent

A Wife is a Woman and has a Husband( which as Man)

A Mother Without Daughter is a Mother whose all Child(ren) are not Women



DL for Family KBDL for Family KB



DL BasicsDL Basics

Concepts (unary predicates/formulae with one free variable) E.g., Person, Father, Mother

Roles (binary predicates/formulae with two free variables) E.g., hasChild, hasHudband

Individual names (constants) E.g., Alice, Bob, Cindy

Subsumption (relations between concepts) E.g. Female Person

Operators (for forming concepts and roles) And(Π) , Or(U), Not (¬) Universal qualifier ( Existent qualifier() Number restiction : Inverse role (-), transitive role (+), Role hierarchy



More for “Family” Ontology More for “Family” Ontology

(Inverse Role) hasParent = hasChild-

hasParent(Bob,Alice) -> hasChild(Alice, Bob)

(Transitive Role)hasBrother hasBrother(Bob,David), hasBrother(David, Mack) -> hasBrother(Bob,Mack)

(Role Hierarchy) hasMother hasParent hasMother(Bob,Alice) -> hasParent(Bob, Alice)

HappyFather Father Π hasChild.Woman Π hasChild.Man


DL ArchitectureDL Architecture

Knowledge Base

Tbox (schema)

Abox (data)

HappyFather Person Π hasChild.Woman Π hasChild.Man

Happy-Father(Bob)

Infe

ren

ce S

yste

m

Inte

rface

(Example from Ian Horrocks, U Manchester, UK)


DL RepresentivesDL Representives

ALC: the smallest DL that is propositionally closed Constructors include booleans (and, or, not),

Restrictions on role successors

SHOIQ = OWL DL

S=ALCR+: ALC with transitive role

H = role hierarchy

O = nomial .e.g WeekEnd = {Saturday, Sunday}

I = Inverse role

Q = qulified number restriction e.g. >=1 hasChild.Man

N = number restriction e.g. >=1 hasChild


RoadmapRoadmap

What is Description Logic (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm


InterpretationsInterpretations

DL Ontology: is a set of terms and their relations Interpretation of a DL Ontology: A possible world ("model") that materalizes the

ontology

People

Student

Jie Bao

present

Description Logic

DL reasoning

Knowledge Representation

Topic

Ontology:

Student PeopleStudent Present.TopicKR TopicDL KR

Interpretation



DL SemanticsDL Semantics

DL semantics defined by interpretations: I = (I, .I), where I is the domain (a non-empty set)

.I is an interpretation function that maps: Concept (class) name A -> subset AI of I

Role (property) name R -> binary relation RI over I

Individual name i -> iI element of I

Interpretation function .I tells us how to interpret atomic concepts, properties and individuals. The semantics of concept forming operators is given by extending the interpretation

function in an obvious way.



DL Semantics: exampleDL Semantics: example

I = (I, .I) I = {Jie_Bao, DL_Reasoning} PeopleI=StudentI={Jie_Bao} TopicI=KRI=DLI={DL_Reasoning} PresentI={(Jie_Bao, DL_Reasoning)}

An interpretation that satisifies all axioms in an DL ontology is also called a model of the ontology.



Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,



Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,


RoadmapRoadmap



What is Reasoning?What is Reasoning?

"Machine Understanding" Find facts that are implicit in the ontology given explicitly stated facts

Find what you know, but you don't know you know it - yet.

ExampleA is father of B, B is father of C, then A is ancestor of C.D is mother of B, then D is female



Reasoning TasksReasoning Tasks

Knowledge is correct (captures intuitions) C subsumes D w.r.t. K iff for every model I of K, CI µ DI

Knowledge is minimally redundant (no unintended synonyms) C is equivallent to D w.r.t. K iff for every model I of K, CI = DI

Knowledge is meaningful (classes can have instances) C is satisfiable w.r.t. K iff there exists some model I of K s.t. CI ;

Querying knowledge x is an instance of C w.r.t. K iff for every model I of K, xI CI

hx,yi is an instance of R w.r.t. K iff for, every model I of K, (xI,yI) RI

Knowledge base consistency A KB K is consistent iff there exists some model I of K



Reasoning Tasks(2)Reasoning Tasks(2)

Many inference tasks can be reduced to subsumption reasoning

Subsumption can be reduced to satisfiability


Tableau AlgorithmTableau Algorithm

Tableau Algorithm is the de facto standard reasoning algorithm used in DL

Basic intuitions Reduces a reasoning problem to concept satisfiability problem Finds an interpretation that satisfies concepts in question. The interpretation is incrementally constructed as a "Tableau"



Short ExampleShort Example

given: WifeWoman, WomanPerson question: if WifePerson

Reasoning process Test if there is a individual that is a Woman but not a Person, i.e. test the

satisfiability of concept C0=(WifeΠ¬Person)

C0(x) -> Wife(x), (¬Person)(x)

Wife(x)->Woman(x) Woman(x) ->Person(x) Conflict! C0 is unsatisfiable, therefore WifePerson is true with the given ontology.



General ProcessGeneral Process

Transform C into negation normal form(NNF), i.e. negation occurs only in front of concept names.

Denote the transformed expression as C0, the algorithm starts with an ABox A0 = {C0(x0)}, and apply consistency-preserving transformation rules (tableaux expansion) to the ABox as far as possible.

If one possible ABox is found, C0 is satisfiable.

If not ABox is found under all search pathes, C0 is unsatisfiable.



NNFNNF



Tableaux Expansion(Selected)Tableaux Expansion(Selected)

Clash



Termination RulesTermination Rules

An ABox is called complete if none of the expansion rules applies to it. An ABox is called consistent if no logic clash is found. If any complete and consistent ABox is found, the initial ABox A0 is

satisfiable The expansion terminates, either when finds a complete and consistent

ABox, or try all search pathes ending with complete but inconsistent ABoxes.



InternalisationInternalisation

Embed the TBox in the initial ABox concept CD is equivalent T¬C U D (T is the "top" concept. It imeans ¬C U D

is the super concept for ANY concepts) E.g.

Given ontology: Mother Woman Π Parent, Woman Person Query: Mother Person The intitial ABox is : ¬Mother U(Woman Π Parent) Π (¬Woman U Person) Π

(Mother Π¬Person)



A Expansion Example

A Expansion Example

Search



Tree ModelTree Model

Another explanation of tableaux algorithm is that it works on a finite completion tree whose individuals in the tableau correspond to nodes and whose interpretation of roles is taken from the edge labels.



Requirments for Tab. Alg.Requirments for Tab. Alg.

Similar tableaux expansions can be designed for more expressive DL languages.

A tableau algorithm has to meet three requirements Soundness: if a complete and clash-free ABox is found by the algorithm, the

ABox must satisfies the initial concept C0.

Completeness: if the initial concept C0 is satisfiable, the algorithm can always find an complete and clash-free ABox

Termination: the algorithm can terminate in finite steps with specific result.


RoadmapRoadmap




Advanced Tableau Alg.Advanced Tableau Alg.

Rich literatures in the past decade. Advanced techniques

Blocking (Subset Blocking,Pair Locking, Dynamic Blocking) For more expressive languages: number restriction, transitive role, inverse

role, nomial, data type Detailed analysis of complexities.

Refer to references at the end of this presentation for details



SHIQ Expansion Rules


ReferencesReferences

F. Baader, W. Nutt. Basic Description Logics. In the Description Logic Handbook, edited by F. Baader, D. Calvanese, D.L. McGuinness, D. Nardi, P.F. Patel-Schneider, Cambridge University Press, 2002, pages 47-100.

Ian Horrocks and Ulrike Sattler. Description Logics Tutorial, ECAI-2002, Lyon, France, July 23rd, 2002.

Ian Horrocks and Ulrike Sattler. A tableaux decision procedure for SHOIQ. In Proc. of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005), 2005.

I. Horrocks and U. Sattler. A description logic with transitive and inverse roles and role hierarchies. Journal of Logic and Computation, 9(3):385-410, 1999.



ResourcesResources

Slides from this talk http://www.cs.man.ac.uk/~horrocks/Slides/HyLo06.ppt

FaCT++ system (open source) http://owl.man.ac.uk/factplusplus/

Protégé http://protege.stanford.edu/plugins/owl/

W3C Web-Ontology (WebOnt) working group (OWL) http://www.w3.org/2001/sw/WebOnt/

DL Handbook, Cambridge University Press http://books.cambridge.org/0521781760.htm

Documents

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Lecture 17 of 42 Wednesday, 08 October