A RESOLUTION DECISION PROCEDURE FOR SHOIQ

Transcription

1 A RESOLUTION DECISION PROCEDURE FOR SHOIQ Yevgeny Kazakov and Boris Motik The University of Manchester August 20, 2006

2 SHOIQ IS A DESCRIPTION LOGIC! Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 2/18

3 DESCRIPTION LOGICS: a family language for knowledge representation: HappyFather Human ( 2 haschild) haschild.(famous Rich) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 2/18

4 DESCRIPTION LOGICS: a family language for knowledge representation: HappyFather Human ( 2 haschild) haschild.(famous Rich) Distinguished by: Formal semantics (set-theoretic) Decidability for key reasoning problems (satisfiability, subsumption, instance) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 2/18

5 DESCRIPTION LOGICS: a family language for knowledge representation: HappyFather Human ( 2 haschild) haschild.(famous Rich) Distinguished by: Formal semantics (set-theoretic) Decidability for key reasoning problems (satisfiability, subsumption, instance) Related to: (Multi-) Modal Logics, Dynamic Logics Fragments of First-Order Logic (guarded, two-variable) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 2/18

6 APPLICATION OF DESCRIPTION LOGICS Databases (Schema Integration) Ontologies (Knowledge Bases): Rigorous description of terms in specific domains (Anatomy, Food, Cars) Access information by performing queries:?- Car hastransmittion.automatic haspart.(engene ( 6 haspart.cylider)) Semantic Web: Ontology Web Language OWL (W3C standard) Annotation of enteries using semantic mark-up Provide the meaning of entries <owl:class rdf:id="http: // Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 3/18

7 WHAT IS IT ABOUT SHOIQ? Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 4/18

8 WHAT IS IT ABOUT SHOIQ? DL SHOIQ is a logical counterpart of OWL DL Development of OWL DL-ontologies requires reasoning: computation of class trees (Heart Organ) evaluation of queries (?- Car... ) reasoning (OWL) = theorem proving (SHOIQ) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 4/18

9 WHAT IS IT ABOUT SHOIQ? DL SHOIQ is a logical counterpart of OWL DL Development of OWL DL-ontologies requires reasoning: computation of class trees (Heart Organ) evaluation of queries (?- Car... ) reasoning (OWL) = theorem proving (SHOIQ) Reasoning in SHOIQ can be reduced to C 2 (the two variable fragment with counting) C 2 is decidable [Grädel et al., 1997] C 2 is NExpTime-compete [Pacholski et al., 2000], [Pratt-Hartmann, 2005] Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 4/18

10 WHAT IS IT ABOUT SHOIQ? DL SHOIQ is a logical counterpart of OWL DL Development of OWL DL-ontologies requires reasoning: computation of class trees (Heart Organ) evaluation of queries (?- Car... ) reasoning (OWL) = theorem proving (SHOIQ) Reasoning in SHOIQ can be reduced to C 2 (the two variable fragment with counting) C 2 is decidable [Grädel et al., 1997] C 2 is NExpTime-compete [Pacholski et al., 2000], [Pratt-Hartmann, 2005] but these procedures are not practical ( guess-and-check ) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 4/18

11 WHAT IS IT ABOUT SHOIQ? DL SHOIQ is a logical counterpart of OWL DL Development of OWL DL-ontologies requires reasoning: computation of class trees (Heart Organ) evaluation of queries (?- Car... ) reasoning (OWL) = theorem proving (SHOIQ) Reasoning in SHOIQ can be reduced to C 2 (the two variable fragment with counting) C 2 is decidable [Grädel et al., 1997] C 2 is NExpTime-compete [Pacholski et al., 2000], [Pratt-Hartmann, 2005] but these procedures are not practical ( guess-and-check ) [Horrocks & Sattler, 2005] the first (and the only up until now) goal-directed procedure for SHOIQ Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 4/18

12 WHAT IS IT ABOUT SHOIQ? DL SHOIQ is a logical counterpart of OWL DL Development of OWL DL-ontologies requires reasoning: computation of class trees (Heart Organ) evaluation of queries (?- Car... ) reasoning (OWL) = theorem proving (SHOIQ) Reasoning in SHOIQ can be reduced to C 2 (the two variable fragment with counting) C 2 is decidable [Grädel et al., 1997] C 2 is NExpTime-compete [Pacholski et al., 2000], [Pratt-Hartmann, 2005] but these procedures are not practical ( guess-and-check ) [Horrocks & Sattler, 2005] the first (and the only up until now) goal-directed procedure for SHOIQ Now we can decide SHOIQ also by resolution! Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 4/18

13 WHY A RESOLUTION-BASED DECISION PROCEDURE FOR SHOIQ? Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 5/18

14 WHY A RESOLUTION-BASED DECISION PROCEDURE FOR SHOIQ? different from the tableau-based approach search for proofs vs. search for models Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 5/18

15 WHY A RESOLUTION-BASED DECISION PROCEDURE FOR SHOIQ? different from the tableau-based approach search for proofs vs. search for models likely to behave differently for different types of problems: Tableau is good for reasoning with large schema (terminologies) Resolution is useful for reasoning with large data (assertions) [Hustadt, Motik & Sattler, 2004] Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 5/18

16 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

17 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Terminology Assertions Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

18 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Terminology Assertions Basic building blocks of DLs: Concept names Role names Individuals Operators Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

19 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Terminology Assertions Basic building blocks of DLs: Concept names sets: Researcher, Human, Paper Role names Individuals Operators Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

20 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Terminology Assertions Basic building blocks of DLs: Concept names sets: Role names binary relations: Researcher, Human, Paper produces Individuals Operators Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

21 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Terminology Assertions Basic building blocks of DLs: Concept names sets: Role names binary relations: Individuals constants: Researcher, Human, Paper produces Rob Operators Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

22 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SYNTAX AXIOMS Researcher Human produce.paper Researcher (Rob) Terminology Assertions Basic building blocks of DLs: Concept names sets: Role names binary relations: Individuals constants: Researcher, Human, Paper produces Rob Operators logical constructors: C 1 C 2, r.c, A C Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 6/18

23 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SEMANTICS AXIOMS Researcher Human produces.paper Researcher (Rob) Basic building blocks of DLs: Concept names Role names Individuals Researcher, Human, Paper produces Rob Operators C 1 C 2, r.c, A C Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 7/18

24 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SEMANTICS AXIOMS Researcher Human produces.paper Researcher (Rob) Basic building blocks of DLs: Concept names unary atoms: Role names Individuals Researcher, Human, Paper Researcher(x), Human(x), Paper(x) produces Rob Operators C 1 C 2, r.c, A C Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 7/18

25 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SEMANTICS AXIOMS Researcher Human produces.paper Researcher (Rob) Basic building blocks of DLs: Concept names unary atoms: Role names binary atoms: Individuals Researcher(x), Human(x), Paper(x) produces produces(x,y) Rob Operators C 1 C 2, r.c, A C Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 7/18

26 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SEMANTICS AXIOMS Researcher Human produces.paper Researcher (Rob) Basic building blocks of DLs: Concept names unary atoms: Researcher(x), Human(x), Paper(x) Role names binary atoms: produces(x,y) Individuals Rob constants: Rob Operators C 1 C 2, r.c, A C Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 7/18

27 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SEMANTICS AXIOMS Researcher Human produces.paper Researcher (Rob) Basic building blocks of DLs: Concept names unary atoms: Researcher(x), Human(x), Paper(x) Role names binary atoms: produces(x,y) Individuals constants: Rob Operators C 1 C 2, r.c, A C constructors: C 1 (x) C 2 (x), y.[r(x, y) C(y)], x.[a(x) C(x)] Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 7/18

28 Description Logics and Ontologies DLs: Basics DESCRIPTION LOGICS: SEMANTICS AXIOMS Researcher(x) Human(x) y.[produces(x, y) Paper(y)] Researcher (Rob) Basic building blocks of DLs: Concept names unary atoms: Researcher(x), Human(x), Paper(x) Role names binary atoms: produces(x,y) Individuals constants: Rob Operators constructors: C 1 (x) C 2 (x), y.[r(x, y) C(y)], x.[a(x) C(x)] Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 7/18

29 Description Logics and Ontologies DLs: Basics HIERARCHY OF DLS Basic Description Logic ALC:,,, r.c, r.c, Transitive Roles: Transitive(r) S Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 8/18

30 Description Logics and Ontologies DLs: Basics HIERARCHY OF DLS Basic Description Logic ALC:,,, r.c, r.c, Transitive Roles: Transitive(r) S Role Hierarchies: r 1 r 2 H Inverse Roles: r 2 I Qualified Number Restrictions: ( n r.c), ( n r.c) Q = SHIQ Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 8/18

31 Description Logics and Ontologies DLs: Basics HIERARCHY OF DLS Basic Description Logic ALC:,,, r.c, r.c, Transitive Roles: Transitive(r) S Role Hierarchies: r 1 r 2 H Inverse Roles: r 2 I Qualified Number Restrictions: ( n r.c), ( n r.c) Q = SHIQ Nominals: {i} O = SHOIQ Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 8/18

32 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n C {i 1 } {i 2 } {i n } C n Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

33 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n C {i 1 } {i 2 } {i n } C {i 1 } {i 2 } {i n } {i p } {i q }, p < q C n C n Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

34 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 0 {i} C 0 ( 2 r.c 1 ) i C 0 C 0 1 C 1 2 C 1 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

35 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 0 {i} C 0 ( 2 r.c 1 ) C 1 ( 2 r.c 2 ) C n 1 ( 2 r.c n ) i C 0 C 0 1 C 1 2 C 1 C 2 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

36 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 0 {i} C 0 ( 2 r.c 1 ) C 1 ( 2 r.c 2 ) C n 1 ( 2 r.c n ) ( 1 r. ) i C 0 C 0 1 C 1 2 C 1 C 2 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

37 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 2 n C 0 {i} C 0 ( 2 r.c 1 ) C 1 ( 2 r.c 2 ) C n 1 ( 2 r.c n ) ( 1 r. ) i C 0 C 0 1 C 1 2 C 1 C n 2 n C 2 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

38 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 2 n, C 2 n C 0 {i} C 0 ( 2 r.c 1 ) C 1 ( 2 r.c 2 ) C n 1 ( 2 r.c n ) ( 1 r. ) C 0 {i} C 1 ( 1 r.c 0 ) C 2 ( 1 r.c 1 ) C n ( 1 r.c n 1 ) ( 2 r. ) i C 0 C 0 1 C 1 2 C 1 C n 2 n C 2 i C 0 C 0 1 C 1 2 C 1 C n 2 n C 2 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

39 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 2 n, C 2 n C 0 {i} C 0 ( 2 r.c 1 ) C 1 ( 2 r.c 2 ) C n 1 ( 2 r.c n ) ( 1 r. ) C 0 {i} C 1 ( 1 r.c 0 ) C 2 ( 1 r.c 1 ) C n ( 1 r.c n 1 ) ( 2 r. ) i C 0 C 0 1 C 1 2 C 1 C n 2 n C 2 i C 0 C 0 1 C 1 2 C 1 C n 2 n C 2 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

40 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 2 n, C 2 n Huge cardinality restrictions: C 2 2n, C 2 2n Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

41 Description Logics and Ontologies SHOIQ EXPRESSIVE POWER OF SHOIQ Cardinality restrictions: C n, C n Large cardinality restrictions: C 2 n, C 2 n Huge cardinality restrictions: C 2 2n, C 2 2n B n B 0 {i} ( 1 r. ) i B n B 1 B 0 = 0 B n B 1 B 0 = 1 ( 2 r. ) B n B 1 B 0 = 2... B 0 r. B 0 = 2 B 0 r. B 0 B i+1 B i r. B i+1 bits count over r B i+1 B i r. B i+1 B i+1 B i r.[ ( B i+1 B i ) ( B i+1 B i ) ] B i+1 B i r.[ ( B i+1 B i ) ( B i+1 B i ) ] Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 9/18

42 Resolution Decision Procedures RESOLUTION-BASED PROCEDURES: THE BASIC PRINCIPLES Invented by Joyner Jr. (1976) Allows one to use existing automated theorem provers (SPASS, VAMPIRE) as decision procedures The general idea is as follows: 1 Define a clause class for the target fragment 2 Show that this class is closed under inferences 3 Show the class is finite for a fixed signature Many decision procedures are based on this principle: clause classes E, S +, E +, etc. [Fermüller et al., 1993] modal logics [Schmidt, 1997], [Hustadt, 1999], fragments of first-order logic [Bachmair et al., 1993], [Ganzinger & de Nivelle, 1999]. Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 10/18

43 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

44 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: EXAMPLE A(c) A r.a A(c) A(x) A(f(x)) A(f(c)) A(f(f(c))) Problem: the depth grows Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

45 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: EXAMPLE A(c) A r.a A(c) A(x) A(f(x)) A(f(c)) A(f(f(c))) Problem: the depth grows The reason: the selected literal is not the deepest one Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

46 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: EXAMPLE A(c) A r.a A(c) A(x) A(f(x)) A(f(c)) A(f(f(c))) Problem: the depth grows The reason: the selected literal is not the deepest one Solution: resolve on the depest literal Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

47 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: EXAMPLE A(c) A R.A A(c) A(x) R(x, y) A(y) R(c, y 1 ) A(y 1 ) R(c, y 1 ) R(y 1, y 2 ) A(y 2 )... Problem: variables got duplicated Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

48 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: EXAMPLE A(c) A R.A A(c) A(x) R(x, y) A(y) R(c, y 1 ) A(y 1 ) R(c, y 1 ) R(y 1, y 2 ) A(y 2 )... Problem: variables got duplicated The reason: the unified expression does not contain all variables of the clause Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

49 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: EXAMPLE A(c) A R.A A(c) A(x) R(x, y) A(y) R(c, y 1 ) A(y 1 ) R(c, y 1 ) R(y 1, y 2 ) A(y 2 )... Problem: variables got duplicated The reason: the unified expression does not contain all variables of the clause Solution: resolve on the expression with all variables Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

50 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: depth or no. of variables grows Decidability is typically a consequence that all expressions in clauses are covering: Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

51 Resolution Decision Procedures HOW TO TURN RESOLUTION INTO A DECISION PROCEDURE? Tweak the parameters of a prover (ordering and selection function) so that the size of clauses does not grow Problematic situations: depth or no. of variables grows Decidability is typically a consequence that all expressions in clauses are covering: every functional term of an expression contains all variables EXAMPLE A(x) r(x, f(x, y)) A(x) x c term f(x, y) is covering term c is not covering Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 11/18

52 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) 3. O(x) O(f (x)) 1 r. 4. r(x, y) x g(y) i Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

53 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i not covering O r.o 2. O(x) r(x, f (x)) 3. O(x) O(f (x)) 1 r. 4. r(x, y) x g(y) not covering Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

54 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) 3. O(x) O(f (x)) 1 r. 4. r(x, y) x g(y) OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

55 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) 3. O(x) O(f (x)) 1 r. 4. r(x, y) x g(y) OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) of the same form Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

56 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) 3. O(x) O(f (x)) 1 r. 4. r(x, y) x g(y) OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) O(x) x g(g(i)) O(x) x g(g(g(i))) of the same form produces deeper clauses Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

57 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) 3. O(x) O(f (x)) 1 r. 4. r(x, y) x g(y) OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) add new: 8. O(x) i g(i) consequence of 1 and 7 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

58 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) follows from 1 and 8 8. O(x) i g(i) consequence of 1 and 7 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

59 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) follows from 1 and 8 larger than 1, 8. O(x) i g(i) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

60 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) follows from 1 and 8 larger than 1, but not larger than 8! 8. O(x) i g(i) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

61 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

62 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) wait a bit... OR[7; 3] : 8. O(x) f (x) g(i) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

63 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) wait a bit... OR[7; 3] : 8. O(x) f (x) g(i) add: 9. O(x) i g(i) consequence of 5 and 8 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

64 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) wait a bit... OR[7; 3] : 8. O(x) f (x) g(i) follows from 5 and 9 larger than 5, and larger than 9! 9. O(x) i g(i) consequence of 5 and 8 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

65 Deciding SHOIQ by Resolution DIFFICULTIES WITH SHOIQ IN RESOLUTION EXAMPLE O {i} 1. O(x) x i O r.o 2. O(x) r(x, f (x)) REDUNDANCY FOR CLAUSES 3. O(x) O(f (x)) 1 r. A4. r(x, clausey) is redundant x g(y) if it follows from smaller clauses OR[1; 3] : 5. O(x) f (x) i OR[2; 4] : 6. O(x) x g(f (x)) OP[5; 6] : 7. O(x) x g(i) wait a bit... OR[7; 3] : 8. O(x) f (x) g(i) remove! 9. O(x) i g(i) The saturation procedure terminates! consequence of 5 and 8 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 12/18

66 Deciding SHOIQ by Resolution NOMINAL GENERATION The idea is developed into a new simplification rule that introduces constants NOMINAL GENERATION α(x) n i=1 f (x) t i α(x) n i=1 f (x) c i α(x) n j=1 c i t j 1 i n where (i) c i are fresh constants for t i and α Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 13/18

67 Deciding SHOIQ by Resolution NOMINAL GENERATION The idea is developed into a new simplification rule that introduces constants the constants are reused when the rule has been applied to α(x) and f (x) before. NOMINAL GENERATION α(x) n i=1 f (x) t i α(x) k i=1 f (x) c i α(x) n j=1 c i t j 1 i k where (i) c i are fresh constants for t i and α, (ii) k=n for the first application of rule for α(x) and f (x), otherwise k and c i are reused Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 13/18

68 Deciding SHOIQ by Resolution NOMINAL GENERATION The idea is developed into a new simplification rule that introduces constants the constants are reused when the rule has been applied to α(x) and f (x) before. there is a second variant of this rule for a different type of clauses NOMINAL GENERATION 1 α(x) n i=1 f (x) t i α(x) k i=1 f (x) c i α(x) n j=1 c i t j 1 i k where (i) c i are fresh constants for t i and α, (ii) k=n for the first application of rule for α(x) and f (x), otherwise k and c i are reused NOMINAL GENERATION 2 α(x) n i=1 f (x) t i n i=1 x c i Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 13/18

69 Deciding SHOIQ by Resolution TERMINATION AND COMPLEXITY ANALYSIS Every application of the rule can increase the number of constants by at most a polynomial factor NOMINAL GENERATION α(x) n i=1 f (x) t i α(x) k i=1 f (x) c i α(x) n j=1 c i t j 1 i k where (i) c i are fresh constants for t i and α, (ii) k=n for the first application of rule for α(x) and f (x), otherwise k and c i are reused Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 14/18

70 Deciding SHOIQ by Resolution TERMINATION AND COMPLEXITY ANALYSIS Every application of the rule can increase the number of constants by at most a polynomial factor There are at most exponentially many applications possible (exponentially many pairs α(x) and f (x)) NOMINAL GENERATION α(x) n i=1 f (x) t i α(x) k i=1 f (x) c i α(x) n j=1 c i t j 1 i k where (i) c i are fresh constants for t i and α, (ii) k=n for the first application of rule for α(x) and f (x), otherwise k and c i are reused Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 14/18

71 Deciding SHOIQ by Resolution TERMINATION AND COMPLEXITY ANALYSIS Every application of the rule can increase the number of constants by at most a polynomial factor There are at most exponentially many applications possible (exponentially many pairs α(x) and f (x)) Hence the procedure terminates, with the upper bound: 3EXPTIME NOMINAL GENERATION α(x) n i=1 f (x) t i α(x) k i=1 f (x) c i α(x) n j=1 c i t j 1 i k where (i) c i are fresh constants for t i and α, (ii) k=n for the first application of rule for α(x) and f (x), otherwise k and c i are reused Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 14/18

72 Deciding SHOIQ by Resolution WHY IS IT SO HARD? In SHOIQ it is possible to express very large cardinality restrictions like C 2 2n, D 2 2m. Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 15/18

73 Deciding SHOIQ by Resolution WHY IS IT SO HARD? In SHOIQ it is possible to express very large cardinality restrictions like C 2 2n, D 2 2m. Hence, it is possible to encode combinatorial constraints involving very big numbers: EXAMPLE A B 2 2n, A C 2 2m +k, B C 2 2k, C 2 n Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 15/18

74 Deciding SHOIQ by Resolution WHY IS IT SO HARD? In SHOIQ it is possible to express very large cardinality restrictions like C 2 2n, D 2 2m. Hence, it is possible to encode combinatorial constraints involving very big numbers Such problems (in particular, the pigeon hole principle) are known to be hard for resolution since it is not really capable to deal with numbers Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 15/18

75 Conclusions CONCLUSIONS We have found a decision procedure for SHOIQ based on basic superposition calculus which runs in 3ExpTime High complexity is due to combination of: nominals + number restrictions + inverse roles The restriction of the procedure to simpler languages (SHOIQ, ALC) behaves like procedures known before hence it exhibits pay as you go behaviour The restricted version for SHIQ has proved itself in practice in system KAON2 1 No additional degree of non-determinism is introduced by NOMINAL GENERATION rules Future developments: Integration of algebraic reasoning into resolution? 1 Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 16/18

76 Conclusions Thank You! Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 17/18

77 COMPARISON WITH THE TABLEAU PROCEDURE Constants introduced by Nominal Generation correspond (in some way) to nominal nodes. The exact number of different constants is not guessed, but equality constraints are generated Blocking is native in resolution by subsumption deletion No yo-yo effect in resolution, since deletion of clauses is permanent (A picture from the presentation by Horrocks & Sattler on A Tableau Decision Procedure for SHOIQ [2005]) Yevgeny Kazakov and Boris Motik A Resolution Decision Procedure for SHOIQ 18/18