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1. Basics of Reasoning in Description Logics Jie Bao Iowa State University Feb 7, 2006

2. An ontology of this talk

3. Roadmap • What is Description Logics (DL) • Semantics of DL • Basic Tableau Algorithm • Advanced Tableau Algorithm

4. 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)

5. 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

6. DL for Family KB

7. 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

8. 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 1 hasChild.Woman Π 1 hasChild.Man

9. DL Architecture Knowledge Base Tbox (schema) HappyFather  Person Π 1 hasChild.Woman Π 1 hasChild.Man Interface Inference System Abox (data) Happy-Father(Bob) (Example from Ian Horrocks, U Manchester, UK)

10. 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 = qualified number restriction e.g. >=1 hasChild.Man • N = number restriction e.g. >=1 hasChild

11. Roadmap • What is Description Logic (DL) • Semantics of DL • Basic Tableau Algorithm • Advanced Tableau Algorithm

12. 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

13. DL Semantics • DL semantics defined by interpretations: I = (DI, .I), where • DI is the domain (a non-empty set) • .I is an interpretation function that maps: • Concept (class) name A -> subset AI of DI • Role (property) name R -> binary relation RI over DI • Individual name i -> iI element of DI • 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.

14. DL Semantics: example • I = (DI, .I) • DI = {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.

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

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

17. Roadmap • What is Description Logic (DL) • Semantics of DL • Basic Tableau Algorithm • Advanced Tableau Algorithm

18. 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

19. Reasoning Tasks • Knowledge is correct (captures intuitions) • C subsumes D w.r.t. K iff for every modelI of K, CI µ DI • Knowledge is minimally redundant (no unintended synonyms) • C is equivallent to D w.r.t. K iff for every modelI of K, CI = DI • Knowledge is meaningful (classes can have instances) • C is satisfiable w.r.t. K iff there exists some modelI of K s.t. CI; • Querying knowledge • x is an instance of C w.r.t. K iff for every modelI of K, xICI • hx,yi is an instance of R w.r.t. K iff for, every modelI of K, (xI,yI) RI • Knowledge base consistency • A KB K is consistent iff there exists some modelI of K

20. Reasoning Tasks(2) • Many inference tasks can be reduced to subsumption reasoning • Subsumption can be reduced to satisfiability

21. 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"

22. Short Example • given: Wife Woman, Woman Personquestion: 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 C0=(WifeΠ¬Person) • C0(x) -> Wife(x), (¬Person)(x) • Wife(x)->Woman(x) • Woman(x) ->Person(x) • Conflict! • C0 is unsatisfiable, therefore Wife Person is true with the given ontology.

23. General 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.

24. NNF

25. 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 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.

26. 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)

27. A Expansion Example Search

28. 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.

29. 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.

30. Roadmap • What is Description Logic (DL) • Semantics of DL • Basic Tableau Algorithm • Advanced Tableau Algorithm

31. 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

32. SHIQ Expansion Rules

33. 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 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.