Rajendra AkerkarVestlandsforsking, Norway   R. Akerkar: Reasoning in DL   10:58:57   1
   What is Description Logics ( )                  p       g    (DL)   Semantics of DL   Basic Tableau Algorithm   Adv...
              g                g    A formal logic-based knowledge    representation language    ◦ “Description" about th...
Person include Man(Male) and Woman(Female), Woman(Female)A Man is not a WomanA Father is a Man who has ChildA Mother i...
10:58:57   R. Akerkar: Reasoning in DL   5
   Concepts (unary predicates/formulae with one free variable)    ◦ E.g., Person, Father, Mother      E     P       F th ...
   (Inverse Role) hasParent = hasChild-    ◦ hasParent(Bob,Alice) -> hasChild(Alice, Bob)   (Transitive Role)hasBrother ...
Knowledge Base Tbox (schema)    HappyFather  Person Π                                                               ys...
   ALC: the smallest DL that is propositionally    closed    ◦ Constructors include booleans (and, or, not),      Restric...
   What is Description Logic ( )                  p       g (DL)   Semantics of DL   Basic Tableau Algorithm   Advance...
DL Ontology: is a set of terms and their              gy    relations    Interpretation of a DL Ontology: A possible    wo...
   DL semantics defined by interpretations: I = (I, .I),    where    ◦ I is the domain (a non-empty set)    ◦ .I is an ...
   I = (I, .I)   I = {Raj, DL_Reasoning}   PeopleI=StudentI={Raj}   TopicI=KRI=DLI={DL_Reasoning}   PresentI={(Raj,...
Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002                                       R. Akerkar: ...
Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002                                       R. Akerkar: ...
   What is Description Logic ( )                  p       g (DL)   Semantics of DL   Basic Tableau Algorithm   Advance...
   "Machine Understanding"  g   Find facts that are implicit in the ontology    given explicitly stated facts    ◦ Find ...
   Knowledge is correct (captures intuitions)    ◦ C subsumes D w.r.t. K iff for every model I of K, CI µ DI             ...
Many inference tasks can be reduced to subsumption     reasoning     Subsumption can be reduced to satisfiability         ...
              g    Tableau Algorithm is the de facto standard    reasoning algorithm used in DL   Basic intuitions    ◦ ...
   given: Wife Woman, Woman Person    question: if Wife Person   Reasoning process    ◦ T t if th      Test there is ...
   Transform C into negation normal form(NNF),    i.e. negation occurs only in front of concept    i        ti           ...
R. Akerkar: Reasoning in DL   10:58:57   23
Clash        R. Akerkar: Reasoning in DL   10:58:57   24
   An ABox is called complete if none of the    expansion rules applies to it.   An ABox is called consistent if no logi...
   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...
Search         R. Akerkar: Reasoning in DL   10:58:57   27
   Another explanation of tableaux algorithm    is that it works on a finite completion tree    whose    ◦ i di id l i th...
   Similar tableaux expansions can be    designed for more expressive DL    d i     df                i    languages.   ...
   What is Description Logic ( )                  p       g (DL)   Semantics of DL   Basic Tableau Algorithm   Advance...
   Rich literatures in the past decade.   Advanced techniques    ◦ Blocking (Subset Blocking, Pair Locking, Dynamic     ...
SHIQ Expansion Rules             R. Akerkar: Reasoning in DL   10:58:57   32
   F. Baader, W. Nutt. Basic Description Logics. In the Description    Logic Handbook, edited by F Baader, D. Calvanese, ...
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Reasoning in Description Logics

  1. 1. Rajendra AkerkarVestlandsforsking, Norway R. Akerkar: Reasoning in DL 10:58:57 1
  2. 2.  What is Description Logics ( ) p g (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm R. Akerkar: Reasoning in DL 10:58:57 2
  3. 3.  g g A formal logic-based knowledge representation language ◦ “Description" about the world in terms of concepts (classes), roles ( ( l ) l (properties, relationships) and ti l ti hi ) d individuals (instances) Decidable fragments of FOL g Widely used in database (e.g., DL CLASSIC) and semantic web (e.g., OWL language) R. Akerkar: Reasoning in DL 10:58:57 3
  4. 4. Person include Man(Male) and Woman(Female), 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 g whose all Child(ren) are not Women R. Akerkar: Reasoning in DL 10:58:57 4
  5. 5. 10:58:57 R. Akerkar: Reasoning in DL 5
  6. 6.  Concepts (unary predicates/formulae with one free variable) ◦ E.g., Person, Father, Mother E P F th M th 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 R. Akerkar: Reasoning in DL 10:58:57 6
  7. 7.  (Inverse Role) hasParent = hasChild- ◦ hasParent(Bob,Alice) -> hasChild(Alice, Bob) (Transitive Role)hasBrother ◦ h B h (B b D id) h B h (D id M k) hasBrother(Bob,David), hasBrother(David, Mack) -> hasBrother(Bob,Mack) (Role Hierarchy) hasMother  hasParent ◦ hasMother(Bob,Alice) -> hasParent(Bob, Alice) HappyFather  Father Π hasChild.Woman ppy Π hasChild.Man R. Akerkar: Reasoning in DL 10:58:57 7
  8. 8. Knowledge Base Tbox (schema) HappyFather  Person Π  ystem hasChild.Woman Π hasChild.Man face Inference Sy Interf Abox (data) Happy-Father(Bob) (Example taken from Ian Horrocks, U Manchester, UK) R. Akerkar: Reasoning in DL 10:58:57 8
  9. 9.  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 hasChild Man  N = number restriction e.g. >=1 hasChild R. Akerkar: Reasoning in DL 10:58:57 9
  10. 10.  What is Description Logic ( ) p g (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm R. Akerkar: Reasoning in DL 10:58:57 10
  11. 11. DL Ontology: is a set of terms and their gy relations Interpretation of a DL Ontology: A possible world ("model") that materializes the ontology Ontology: Student  People Student  Present Topic Present.Topic KR  Topic DL  KR10:58:57 R. Akerkar: Reasoning in DL 11
  12. 12.  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  I di id l name i -> iI element of I Individual l t f Interpretation function .I tells us how to interpret atomic concepts, properties and individuals. p ,p p ◦ The semantics of concept forming operators is given by extending the interpretation function in an obvious way. R. Akerkar: Reasoning in DL 10:58:57 12
  13. 13.  I = (I, .I) I = {Raj, DL_Reasoning} PeopleI=StudentI={Raj} TopicI=KRI=DLI={DL_Reasoning} PresentI={(Raj, DL_Reasoning)}An interpretation that satisifies all axioms in an DLontology is also called a model of the ontology. R. Akerkar: Reasoning in DL 10:58:57 13
  14. 14. Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002 R. Akerkar: Reasoning in DL 10:58:57 14
  15. 15. Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002 R. Akerkar: Reasoning in DL 10:58:57 15
  16. 16.  What is Description Logic ( ) p g (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm R. Akerkar: Reasoning in DL 10:58:57 16
  17. 17.  "Machine Understanding" g Find facts that are implicit in the ontology given explicitly stated facts ◦ Find what you know, but you dont 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 R. Akerkar: Reasoning in DL 10:58:57 17
  18. 18.  Knowledge is correct (captures intuitions) ◦ C subsumes D w.r.t. K iff for every model I of K, CI µ DI wrt K 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 i meaningful ( l K l d is i f l (classes can h have instances) i t ) ◦ 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 R. Akerkar: Reasoning in DL 10:58:57 18
  19. 19. Many inference tasks can be reduced to subsumption reasoning Subsumption can be reduced to satisfiability p y10:58:57 R. Akerkar: Reasoning in DL 19
  20. 20.  g 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 p p question. ◦ The interpretation is incrementally constructed as a "Tableau" Tableau R. Akerkar: Reasoning in DL 10:58:57 20
  21. 21.  given: Wife Woman, Woman Person question: if Wife Person Reasoning process ◦ T t if th Test there is a individual th t i a W i i di id l that is Woman b t not but t a Person, i.e. test the satisfiability of concept C0=(WifeΠ¬Person) ◦ C0(x) -> Wife(x), (¬Person)(x) ◦ Wife(x)->Woman(x) ◦W ( ) >P Woman(x) ->Person(x) ( ) ◦ Conflict! ◦ C0 is unsatisfiable, therefore Wife Person is true with the given ontology. R. Akerkar: Reasoning in DL 10:58:57 21
  22. 22.  Transform C into negation normal form(NNF), i.e. negation occurs only in front of concept i ti l i f t f t names. Denote the transformed expression as C0, the p 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 f f d d ll h h found under all search pathes, C0 is unsatisfiable. R. Akerkar: Reasoning in DL 10:58:57 22
  23. 23. R. Akerkar: Reasoning in DL 10:58:57 23
  24. 24. Clash R. Akerkar: Reasoning in DL 10:58:57 24
  25. 25.  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. l hi f d If any complete and consistent ABox is found, found the initial ABox A0 is satisfiable The expansion terminates, either when finds a complete and consistent ABox or ABox, try all search pathes ending with complete but inconsistent ABoxes. R. Akerkar: Reasoning in DL 10:58:57 25
  26. 26.  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 f ANY concepts) t for t ) E.g. ◦ Given ontology: Mother  Woman Π Parent Parent, Woman  Person ◦ Query: Mother  Person y ◦ The intitial ABox is : ¬Mother U(Woman Π Parent) Π (¬Woman U Person) Π (Mother Π ¬Person) R. Akerkar: Reasoning in DL 10:58:57 26
  27. 27. Search R. Akerkar: Reasoning in DL 10:58:57 27
  28. 28.  Another explanation of tableaux algorithm is that it works on a finite completion tree whose ◦ i di id l i th t bl individuals in the tableau correspond t nodes d to d ◦ and whose interpretation of roles is taken from the edge labels. g R. Akerkar: Reasoning in DL 10:58:57 28
  29. 29.  Similar tableaux expansions can be designed for more expressive DL d i df i 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 algorithm satisfies the initial concept C0. ◦ Completeness: if the initial concept C0 is satisfiable, the algorithm can always fi d an i fi bl h l ih l find complete and clash-free ABox ◦ Termination: the algorithm can terminate in finite steps with specific result. R. Akerkar: Reasoning in DL 10:58:57 29
  30. 30.  What is Description Logic ( ) p g (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm R. Akerkar: Reasoning in DL 10:58:57 30
  31. 31.  Rich literatures in the past decade. Advanced techniques ◦ Blocking (Subset Blocking, Pair Locking, Dynamic Blocking) ◦ For more expressive languages: number restriction, inverse role, transitive role, nomial, data type ◦ Detailed analysis of complexities. R. Akerkar: Reasoning in DL 10:58:57 31
  32. 32. SHIQ Expansion Rules R. Akerkar: Reasoning in DL 10:58:57 32
  33. 33.  F. Baader, W. Nutt. Basic Description Logics. In the Description Logic Handbook, edited by F Baader, D. Calvanese, D.L. Handbook 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. R. Akerkar: Reasoning in DL 10:58:57 33

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