Computing & Information Sciences Kansas 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: Course web site: Instructor home page: Reading for Next Class: Chapter 8, Russell & Norvig 2 nd edition Description Logics, Ontologies Discussion: Knowledge Representation

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence 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

Computing & Information Sciences Kansas State University Basics of Reasoning in Description Logics Jie Bao Iowa State University Feb 7, 2006

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence An ontology of this talk © 2006 J. Bao

Computing & Information Sciences Kansas State University Roadmap What is Description Logics (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm

Computing & Information Sciences Kansas State University Description 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)

Computing & Information Sciences Kansas State University A “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

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence DL for Family KB

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence DL 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

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence 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

Computing & Information Sciences Kansas State University DL Architecture Knowledge Base Tbox (schema) Abox (data) HappyFather  Person Π  hasChild.Woman Π  hasChild.Man Happy-Father(Bob) Inference System Interface (Example from Ian Horrocks, U Manchester, UK)

Computing & Information Sciences Kansas State University DL 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

Computing & Information Sciences Kansas State University Roadmap What is Description Logic (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm

Computing & Information Sciences Kansas State University Interpretations DL Ontology: is a set of terms and their relations Interpretation of a DL Ontology: A possible world ("model") that materalizes the ontology Ontology: Student  People Student  Present.Topic KR  Topic DL  KR  Interpretation

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence DL 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 A I of  I  Role (property) name R -> binary relation R I over  I  Individual name i -> i I 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.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence DL Semantics: example I = (  I,. I )  I = {Jie_Bao, DL_Reasoning} People I =Student I ={Jie_Bao} Topic I =KR I =DL I ={DL_Reasoning} Present I ={(Jie_Bao, DL_Reasoning)} An interpretation that satisifies all axioms in an DL ontology is also called a model of the ontology.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

Computing & Information Sciences Kansas State University Roadmap What is Description Logic (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm

Computing & Information Sciences Kansas State University 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. Example  A is father of B, B is father of C, then A is ancestor of C.  D is mother of B, then D is female

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Reasoning Tasks Knowledge is correct (captures intuitions)  C subsumes D w.r.t. K iff for every model I of K, C I µ D I Knowledge is minimally redundant (no unintended synonyms)  C is equivallent to D w.r.t. K iff for every model I of K, C I = D I Knowledge is meaningful (classes can have instances)  C is satisfiable w.r.t. K iff there exists some model I of K s.t. C I   ; Querying knowledge  x is an instance of C w.r.t. K iff for every model I of K, x I  C I  hx,yi is an instance of R w.r.t. K iff for, every model I of K, (x I,y I )  R I Knowledge base consistency  A KB K is consistent iff there exists some model I of K

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Reasoning Tasks(2) Many inference tasks can be reduced to subsumption reasoning Subsumption can be reduced to satisfiability

Computing & Information Sciences Kansas State University Tableau 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"

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Short Example given: Wife  Woman, Woman  Person question: if Wife  Person Reasoning process  Test if there is a individual that is a Woman but not a Person, i.e. test the satisfiability of concept C 0 =(WifeΠ¬Person)  C 0 (x) -> Wife(x), (¬Person)(x)  Wife(x)->Woman(x)  Woman(x) ->Person(x)  Conflict!  C 0 is unsatisfiable, therefore Wife  Person is true with the given ontology.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence General Process Transform C into negation normal form(NNF), i.e. negation occurs only in front of concept names. Denote the transformed expression as C 0, the algorithm starts with an ABox A 0 = {C 0 (x 0 )}, and apply consistency-preserving transformation rules (tableaux expansion) to the ABox as far as possible. If one possible ABox is found, C 0 is satisfiable. If not ABox is found under all search pathes, C 0 is unsatisfiable.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence NNF

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Tableaux Expansion(Selected) Clash

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Termination 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 A 0 is satisfiable The expansion terminates, either when finds a complete and consistent ABox, or try all search pathes ending with complete but inconsistent ABoxes.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Internalisation Embed the TBox in the initial ABox concept C  D 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)

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence A Expansion Example Search

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Tree 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.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence 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 C 0.  Completeness: if the initial concept C 0 is satisfiable, the algorithm can always find an complete and clash-free ABox  Termination: the algorithm can terminate in finite steps with specific result.

Computing & Information Sciences Kansas State University Roadmap What is Description Logic (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence 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

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence SHIQ Expansion Rules

Computing & Information Sciences Kansas State University References 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 Ian Horrocks and Ulrike Sattler. Description Logics Tutorial, ECAI-2002, Lyon, France, July 23rd, Ian Horrocks and Ulrike Sattler. A tableaux decision procedure for SHOIQ. In Proc. of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005), I. Horrocks and U. Sattler. A description logic with transitive and inverse roles and role hierarchies. Journal of Logic and Computation, 9(3): , 1999.

Computing & Information Sciences Kansas State University Wednesday, 08 Oct 2008CIS 530 / 730: Artificial Intelligence Resources Slides from this talk  FaCT++ system (open source)  Protégé  W3C Web-Ontology (WebOnt) working group (OWL)  DL Handbook, Cambridge University Press 