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Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods Laura Giordano 1 , Valentina Gliozzi 2 , Nicola Olivetti 3 , and Gian Luca Pozzato 2 1 Dipartimento di Informatica - Universit` a del Piemonte Orientale A.

  1. Logic of typicality T ( Student ) ⊑ ¬ TaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ ∃ HasChild . ⊤ ) ⊑ ¬ TaxPayer ABox: 1. Student ( john ) 2. Student ( john ) , Worker ( john ) 3. Student ( john ) , Worker ( john ) , ∃ HasChild . ⊤ ( john ) expected conclusions: 1. ¬ TaxPayer ( john ) 2. TaxPayer ( john ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

  2. Logic of typicality T ( Student ) ⊑ ¬ TaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ ∃ HasChild . ⊤ ) ⊑ ¬ TaxPayer ABox: 1. Student ( john ) 2. Student ( john ) , Worker ( john ) 3. Student ( john ) , Worker ( john ) , ∃ HasChild . ⊤ ( john ) expected conclusions: 1. ¬ TaxPayer ( john ) 2. TaxPayer ( john ) 3. ¬ TaxPayer ( john ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

  3. Logic of typicality We have defined a nonmonotonic inference based on a minimal model semantics For DL + T = ALC + T nonmonotonic inference has a high complexity, namely CO -N EXP NP , comparable however with that one of other NMR DL (circumscription) We are interested in applying our approach to low-complexity DLs EL ⊥ and DL-Lite core . Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 8

  4. The logic EL + ⊥ T Extension of our approach to Low Complexity DL EL ⊥ Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

  5. The logic EL + ⊥ T Extension of our approach to Low Complexity DL EL ⊥ Logic EL ⊥ of the EL family Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

  6. The logic EL + ⊥ T Extension of our approach to Low Complexity DL EL ⊥ Logic EL ⊥ of the EL family allows for conjunction ( ⊓ ) and existential restriction ( ∃ R.C ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

  7. The logic EL + ⊥ T Extension of our approach to Low Complexity DL EL ⊥ Logic EL ⊥ of the EL family allows for conjunction ( ⊓ ) and existential restriction ( ∃ R.C ) allows for ⊥ Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

  8. The logic EL + ⊥ T Extension of our approach to Low Complexity DL EL ⊥ Logic EL ⊥ of the EL family allows for conjunction ( ⊓ ) and existential restriction ( ∃ R.C ) allows for ⊥ relevant for several applications, in particular in the bio-medical domain (GALEN Medical Knowledge Base, Systemized Nomenclature of Medicine, Gene Ontology) formalized in small extensions of EL Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

  9. The logic EL + ⊥ T Extension of our approach to Low Complexity DL EL ⊥ Logic EL ⊥ of the EL family allows for conjunction ( ⊓ ) and existential restriction ( ∃ R.C ) allows for ⊥ relevant for several applications, in particular in the bio-medical domain (GALEN Medical Knowledge Base, Systemized Nomenclature of Medicine, Gene Ontology) formalized in small extensions of EL reasoning in EL is polynomial-time decidable Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

  10. Language of EL ⊥ T min Alphabet of concept names C role names R individuals O Given A ∈ C and r ∈ R , we define: := A | ⊤ | ⊥ | C ⊓ C C C R := C | C R ⊓ C R | ∃ r.C C L := C R | T ( C ) TBox contains a finite set of concept inclusions C L ⊑ C R Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 10

  11. Example The reformulation of the previous example in EL + ⊥ T gives the following KB: TaxPayer ⊓ NotTaxPayer ⊑ ⊥ Parent ⊑ ∃ HasChild . ⊤ ∃ HasChild . ⊤ ⊑ Parent T ( Student ) ⊑ NotTaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ Parent ) ⊑ NotTaxPayer Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 11

  12. Language of DL-Lite c T min Alphabet of concept names C role names R individuals O Given A ∈ C and r ∈ R , we define: := A | ∃ R. ⊤ | T ( A ) C L := r | r − R := A | ¬ A | ∃ R. ⊤ | ¬∃ R. ⊤ C R TBox contains a finite set of concept inclusions C L ⊑ C R Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 12

  13. Monotonic Semantics A model M is a structure � ∆ , <, I � , where ∆ is the domain and for each extended concept C , C I ⊆ ∆ , and for each role R R I ⊆ ∆ × ∆ < is an irreflexive and transitive relation over ∆ satisfying the Smoothness Condition (well-foundness) < is multilinear (or weakly connected): if u < z and v < z , then either u = v or u < v or v < u Semantics of the T operator: ( T ( C )) I = Min < ( C I ) . For the other operators C I is defined in the usual way ( in particular, ( r − ) I = { ( a, b ) | ( b, a ) ∈ r I } ) A model satisfying a Knowledge Base (TBox,ABox) is defined as usual Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 13

  14. Modal interpretation We introduce a new modality � Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

  15. Modal interpretation We introduce a new modality � we interpret the relation < as an accessibility relation Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

  16. Modal interpretation We introduce a new modality � we interpret the relation < as an accessibility relation by the Smoothness Condition (well-foundness), it turns out that � has the properties of Gödel-Löb modal logic G Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

  17. Modal interpretation We introduce a new modality � we interpret the relation < as an accessibility relation by the Smoothness Condition (well-foundness), it turns out that � has the properties of Gödel-Löb modal logic G ( � C ) I = { x ∈ ∆ | for every y ∈ ∆ , if y < x then y ∈ C I } Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

  18. Modal interpretation We introduce a new modality � we interpret the relation < as an accessibility relation by the Smoothness Condition (well-foundness), it turns out that � has the properties of Gödel-Löb modal logic G ( � C ) I = { x ∈ ∆ | for every y ∈ ∆ , if y < x then y ∈ C I } Thus T ( C ) I = ( C ⊓ � ¬ C ) I Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

  19. Weakness of monotonic semantics EL + ⊥ T allows one to reason about typicality Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 15

  20. Weakness of monotonic semantics EL + ⊥ T allows one to reason about typicality e.g. we can consistently express that student, working student and working student with children have a different status as taxpayers Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 15

  21. Weakness of monotonic semantics EL + ⊥ T allows one to reason about typicality e.g. we can consistently express that student, working student and working student with children have a different status as taxpayers but we cannot derive anything about the prototypical properties of a given individual, unless the KB contains explicit tipicality assumptions concerning this individual Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 15

  22. Weakness of monotonic semantics TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T ( Student ) ⊑ NotTaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ Parent ) ⊑ NotTaxPayer Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

  23. Weakness of monotonic semantics TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T ( Student ) ⊑ NotTaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ Parent ) ⊑ NotTaxPayer What can we conclude about john ? Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

  24. Weakness of monotonic semantics TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T ( Student ) ⊑ NotTaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ Parent ) ⊑ NotTaxPayer What can we conclude about john ? If T ( Student ⊓ Worker ⊓ Parent )( john ) ∈ ABox, then in EL + ⊥ T we can conclude NotTaxPayer ( john ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

  25. Weakness of monotonic semantics TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T ( Student ) ⊑ NotTaxPayer T ( Student ⊓ Worker ) ⊑ TaxPayer T ( Student ⊓ Worker ⊓ Parent ) ⊑ NotTaxPayer What can we conclude about john ? If T ( Student ⊓ Worker ⊓ Parent )( john ) ∈ ABox, then in EL + ⊥ T we can conclude NotTaxPayer ( john ) If ( Student ⊓ Worker ⊓ Parent )( john ) ∈ ABox, we cannot derive NotTaxPayer ( john ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

  26. NonMonotonic semantics We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

  27. NonMonotonic semantics We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

  28. NonMonotonic semantics We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

  29. NonMonotonic semantics We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models we introduce a semantic entailment determined by minimal models Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

  30. NonMonotonic semantics We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models we introduce a semantic entailment determined by minimal models Informally, we prefer a model M to a model N if M contains more typical instances of concepts than N Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

  31. NonMonotonic semantics We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models we introduce a semantic entailment determined by minimal models Informally, we prefer a model M to a model N if M contains more typical instances of concepts than N Given a KB, we consider a finite set L T of concepts occurring in the KB, the typicality of whose instances we want to maximize L T contains at least all concepts C such that T ( C ) occurs in the KB or in the query Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

  32. NonMonotonic semantics M � − L T = { ( a, ¬ � ¬ C ) | a ∈ ( ¬ � ¬ C ) I , with a ∈ ∆ , C ∈ L T } Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

  33. NonMonotonic semantics M � − L T = { ( a, ¬ � ¬ C ) | a ∈ ( ¬ � ¬ C ) I , with a ∈ ∆ , C ∈ L T } Given two models M = � ∆ M , < M , I M � and N = � ∆ N , < N , I N � of KB, we say that M is preferred to N w.r.t. L T ( M < L T N ), if: Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

  34. NonMonotonic semantics M � − L T = { ( a, ¬ � ¬ C ) | a ∈ ( ¬ � ¬ C ) I , with a ∈ ∆ , C ∈ L T } Given two models M = � ∆ M , < M , I M � and N = � ∆ N , < N , I N � of KB, we say that M is preferred to N w.r.t. L T ( M < L T N ), if: ∆ M = ∆ N Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

  35. NonMonotonic semantics M � − L T = { ( a, ¬ � ¬ C ) | a ∈ ( ¬ � ¬ C ) I , with a ∈ ∆ , C ∈ L T } Given two models M = � ∆ M , < M , I M � and N = � ∆ N , < N , I N � of KB, we say that M is preferred to N w.r.t. L T ( M < L T N ), if: ∆ M = ∆ N M � − L T ⊂ N � − L T Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

  36. NonMonotonic semantics M � − L T = { ( a, ¬ � ¬ C ) | a ∈ ( ¬ � ¬ C ) I , with a ∈ ∆ , C ∈ L T } Given two models M = � ∆ M , < M , I M � and N = � ∆ N , < N , I N � of KB, we say that M is preferred to N w.r.t. L T ( M < L T N ), if: ∆ M = ∆ N M � − L T ⊂ N � − L T a I = a I ′ for all a ∈ O Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

  37. NonMonotonic semantics M � − L T = { ( a, ¬ � ¬ C ) | a ∈ ( ¬ � ¬ C ) I , with a ∈ ∆ , C ∈ L T } Given two models M = � ∆ M , < M , I M � and N = � ∆ N , < N , I N � of KB, we say that M is preferred to N w.r.t. L T ( M < L T N ), if: ∆ M = ∆ N M � − L T ⊂ N � − L T a I = a I ′ for all a ∈ O A model M is a minimal model for KB (with respect to L T ) if it is a model of KB and there is no a model M ′ of KB such that M ′ < L T M Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

  38. Nonmonotonic Semantics Query F : either a formula C ( a ) or a subsumption C ⊑ D Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 19

  39. Nonmonotonic Semantics Query F : either a formula C ( a ) or a subsumption C ⊑ D Minimal Entailment in EL ⊥ T min Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 19

  40. Nonmonotonic Semantics Query F : either a formula C ( a ) or a subsumption C ⊑ D Minimal Entailment in EL ⊥ T min A query F is minimally entailed from KB w.r.t. L T : KB | = EL ⊥ T min F if F holds in all models of KB minimal w.r.t. L T Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 19

  41. Example Let L T = { Student , Student ⊓ Worker , Student ⊓ Worker ⊓ Parent } KB ∪ { Student ( john ) } | = EL ⊥ T min NotTaxPayer ( john ) KB ∪ { Student ( john ) , Worker ( john ) } | = EL ⊥ T min TaxPayer ( john ) KB ∪ { Student ( john ) , Worker ( john ) , Parent ( john ) } | = EL ⊥ T min NotTaxPayer ( john ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 20

  42. Complexity results for EL ⊥ T min Entailment for EL + ⊥ T is CoNP , but Theorem 3.1 in [GGOP]. Entailment in EL ⊥ T min is E XP T IME -hard . We need further restrctions One possibility: Left Local EL ⊥ T min (considered for circumscriptive extension [BLW06]) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 21

  43. Language of Left Local EL ⊥ T min Alphabet of concept names C role names R individuals O Given A ∈ C and r ∈ R , we define: := A | ⊤ | ⊥ | C ⊓ C C C R := C | C R ⊓ C R | ∃ r.C C LL := C | C LL ⊓ C LL | ∃ r. ⊤ | T ( C ) L L L TBox contains a finite set of concept inclusions C LL ⊑ C R L Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 22

  44. Complexity results for Left Local EL ⊥ T min Small model theorem (Theorem 3.11 in [GGOP]). KB | = EL ⊥ T min F if and only if F holds in all models of KB whose size is polynomial in the size of KB. Theorem 3.12 in [GGOP]. If KB is Left Local, the problem of = EL ⊥ T min F is in Π p deciding whether KB | 2 . A small model theorem and a similar complexity result can be proved for DL-Lite c T min [GGOP] Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 23

  45. Complexity results for Left Local EL ⊥ T min Small model theorem (Theorem 3.11 in [GGOP]). KB | = EL ⊥ T min F if and only if F holds in all models of KB whose size is polynomial in the size of KB. Theorem 3.12 in [GGOP]. If KB is Left Local, the problem of = EL ⊥ T min F is in Π p deciding whether KB | 2 . A small model theorem and a similar complexity result can be proved for DL-Lite c T min [GGOP] R R C ∃ R.C C ∃ R.C R R C ∃ R.C ∃ R.C R R Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 23 C

  46. The Tableau calculus TAB EL ⊥ T min Tableau calculus TAB EL ⊥ T for deciding whether a query F is min minimally entailed from a KB (TBox,ABox) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

  47. The Tableau calculus TAB EL ⊥ T min Tableau calculus TAB EL ⊥ T for deciding whether a query F is min minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

  48. The Tableau calculus TAB EL ⊥ T min Tableau calculus TAB EL ⊥ T for deciding whether a query F is min minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TAB EL ⊥ T tries to build an open branch representing a minimal min model satisfying KB ∪ {¬ F } Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

  49. The Tableau calculus TAB EL ⊥ T min Tableau calculus TAB EL ⊥ T for deciding whether a query F is min minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TAB EL ⊥ T tries to build an open branch representing a minimal min model satisfying KB ∪ {¬ F } two-phase computation: Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

  50. The Tableau calculus TAB EL ⊥ T min Tableau calculus TAB EL ⊥ T for deciding whether a query F is min minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TAB EL ⊥ T tries to build an open branch representing a minimal min model satisfying KB ∪ {¬ F } two-phase computation: 1. Phase 1: TAB EL ⊥ T verifies whether KB ∪{¬ F } is satisfiable P H 1 in an EL + ⊥ T model, building candidate models Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

  51. The Tableau calculus TAB EL ⊥ T min Tableau calculus TAB EL ⊥ T for deciding whether a query F is min minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TAB EL ⊥ T tries to build an open branch representing a minimal min model satisfying KB ∪ {¬ F } two-phase computation: 1. Phase 1: TAB EL ⊥ T verifies whether KB ∪{¬ F } is satisfiable P H 1 in an EL + ⊥ T model, building candidate models 2. Phase 2: TAB EL ⊥ T checks whether the candidate models P H 2 found in Phase 1 are minimal models of KB Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

  52. The Tableau calculus TAB EL ⊥ T min Given a knowledge base (TBox,ABox), tableaux nodes of TAB EL ⊥ T are called constraint systems and have the form min � S | U | W � , where : Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

  53. The Tableau calculus TAB EL ⊥ T min Given a knowledge base (TBox,ABox), tableaux nodes of TAB EL ⊥ T are called constraint systems and have the form min � S | U | W � , where : R S = { a : C | C ( a ) ∈ ABox } ∪ { a − → b | aRb ∈ ABox } Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

  54. The Tableau calculus TAB EL ⊥ T min Given a knowledge base (TBox,ABox), tableaux nodes of TAB EL ⊥ T are called constraint systems and have the form min � S | U | W � , where : R S = { a : C | C ( a ) ∈ ABox } ∪ { a − → b | aRb ∈ ABox } U = { C ⊑ D ∅ | C ⊑ D ∈ TBox } Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

  55. The Tableau calculus TAB EL ⊥ T min Given a knowledge base (TBox,ABox), tableaux nodes of TAB EL ⊥ T are called constraint systems and have the form min � S | U | W � , where : R S = { a : C | C ( a ) ∈ ABox } ∪ { a − → b | aRb ∈ ABox } U = { C ⊑ D ∅ | C ⊑ D ∈ TBox } W is a set of labels x C used by existential rules Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

  56. Special Existential Rules The rule ( ∃ + ) is split in the following two rules: � S, u : ∃ R.C | U | W � ( ∃ + ) 1 R R R . . . � S, u − → x C , x C : C | U | W ∪ { x C } � � S, u − → y m , y m : C | U | W � � S, u − → y 1 , y 1 : C | U | W � if x C �∈ W and y 1 , . . . , y m are all the labels occurring in S � S, u : ∃ R.C | U | W � ( ∃ + ) 2 R R R . . . � S, u − → x C | U | W � � S, u − → y 1 , y 1 : C | U | W � � S, u − → y m , y m : C | U | W � if x C ∈ W and y 1 , . . . , y m are all the labels occurring in S Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 26

  57. Special Rule for ( � − ) S = S, u : ¬ � ¬ C 1 , . . . , u : ¬ � ¬ C n . S M u → y = { y : ¬ D, y : � ¬ D | u : � ¬ D ∈ S } and, for k = 1 , 2 , . . . , n , � − k u → y = { y : ¬ � ¬ C j ⊔ C j | u : ¬ � ¬ C j ∈ S ∧ j � = k } . S � S, u : ¬ � ¬ C 1 , ¬ � ¬ C 2 , . . . , u : ¬ � ¬ C n | U | W � ( � − ) � − k � S, x : C k , x : � ¬ C k , S M u → x | U | W � u → x , S � − k � − k . . . � S, y m : C k , y m : � ¬ C k , S M � S, y 1 : C k , y 1 : � ¬ C k , S M u → y m | U | W � u → y 1 , S u → y 1 | U | W � u → y m , S for all k = 1 , 2 , . . . , n , where y 1 , . . . , y m are all the labels occurring in S and x is new. Rule ( � − ) contains: n branches, one for each u : ¬ � ¬ C k in S ; other n × m branches, where m is the number of labels occurring in S , one for each label y i and for each u : ¬ � ¬ C k in S Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 27

  58. Phase 1: TAB EL ⊥ T PH 1 � S, x : C, x : ¬ C | U | W � � S, x : ¬ ⊤ | U | W � (Clash) (Clash) ¬ ⊤ � S, x : ⊥ | U | W � (Clash) ⊥ � S, x : C ⊔ D | U | W � � S, x : ¬ ( C ⊓ D ) | U | W � � S, x : C ⊓ D | U | W � ( ⊓ − ) ( ⊔ + ) ( ⊓ + ) � S, x : C, x : D | U | W � � S, x : ¬ C | U | W � � S, x : ¬ D | U | W � � S, x : C | U | W � � S, x : D | U | W � � S | U, C ⊑ D L | W � � S, x : T ( C ) | U | W � � S, x : ¬ T ( C ) | U | W � (Unfold) ( T + ) ( T − ) � S, x : ¬ C ⊔ D | U, C ⊑ D L,x | W � � S, x : C, x : � ¬ C | U | W � � S, x : ¬ C | U | W � � S, x : ¬ � ¬ C | U | W � if x occurs in S and x �∈ L � S, u : ∃ R.C | U | W � ( ∃ + ) 1 R R R . . . � S, u − → x C , x C : C | U | W ∪ { x C } � − → y 1 , y 1 : C | U | W � � S, u − → y m , y m : C | U | W � � S, u if x C �∈ W and y 1 , . . . , y m are all the labels occurring in S � S, u : ∃ R.C | U | W � ( ∃ + ) 2 R R R . . . � S, u − → x C | U | W � � S, u − → y 1 , y 1 : C | U | W � � S, u − → y m , y m : C | U | W � if x C ∈ W and y 1 , . . . , y m are all the labels occurring in S R � S, x : ¬ ∃ R.C, x − → y | U | W � � S | U | W � ( ∃ − ) ( cut ) R � S, x : ¬ � ¬ C | U | W � � S, x : � ¬ C | U | W � � S, x : ¬ ∃ R.C, x − → y, y : ¬ C | U | W � if y : ¬ C �∈ S if x : ¬ � ¬ C �∈ S and x : � ¬ C �∈ S x occurs in S C ∈ L T � S, u : ¬ � ¬ C 1 , ¬ � ¬ C 2 , . . . , u : ¬ � ¬ C n | U | W � ( � − ) � − k � S, x : C k , x : � ¬ C k , S M u → x | U | W � u → x , S � − k � − k . . . � S, y 1 : C k , y 1 : � ¬ C k , S M � S, y m : C k , y m : � ¬ C k , S M u → y m | U | W � u → y 1 , S u → y 1 | U | W � u → y m , S x new if y 1 , . . . , y m are all the labels occurring in S, y 1 � = u, . . . , y m � = u k = 1 , 2 , . . . , n Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 28

  59. Phase 2: TAB EL ⊥ T PH 2 for each open branch B built by TAB EL ⊥ T PH 1 , verifies if it is a minimal model of the KB Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

  60. Phase 2: TAB EL ⊥ T PH 2 for each open branch B built by TAB EL ⊥ T PH 1 , verifies if it is a minimal model of the KB Given an open branch B of a tableau built from TAB EL ⊥ T PH 1 , we define: Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

  61. Phase 2: TAB EL ⊥ T PH 2 for each open branch B built by TAB EL ⊥ T PH 1 , verifies if it is a minimal model of the KB Given an open branch B of a tableau built from TAB EL ⊥ T PH 1 , we define: D ( B ) as the set of labels occurring on B Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

  62. Phase 2: TAB EL ⊥ T PH 2 for each open branch B built by TAB EL ⊥ T PH 1 , verifies if it is a minimal model of the KB Given an open branch B of a tableau built from TAB EL ⊥ T PH 1 , we define: D ( B ) as the set of labels occurring on B B � − = { x : ¬ � ¬ C | x : ¬ � ¬ C occurs in B } Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

  63. Phase 2: TAB EL ⊥ T PH 2 A tableau of TAB EL ⊥ T is a tree whose nodes are triples of the PH 2 form � S | U | K � Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

  64. Phase 2: TAB EL ⊥ T PH 2 A tableau of TAB EL ⊥ T is a tree whose nodes are triples of the PH 2 form � S | U | K � � S | U � is a constraint system (as in TAB EL ⊥ T PH 1 ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

  65. Phase 2: TAB EL ⊥ T PH 2 A tableau of TAB EL ⊥ T is a tree whose nodes are triples of the PH 2 form � S | U | K � � S | U � is a constraint system (as in TAB EL ⊥ T PH 1 ) K contains formulas of the form x : ¬ � ¬ C , with C ∈ L T Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

  66. Phase 2: TAB EL ⊥ T PH 2 A tableau of TAB EL ⊥ T is a tree whose nodes are triples of the PH 2 form � S | U | K � � S | U � is a constraint system (as in TAB EL ⊥ T PH 1 ) K contains formulas of the form x : ¬ � ¬ C , with C ∈ L T Basic idea: given an open B built by TAB EL ⊥ T PH 1 , K is initialized with B � − in order to build smaller models Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

  67. Phase 2: TAB EL ⊥ T PH 2 � S, x : ¬⊤ | U | K � (Clash) ¬ ⊤ � S, x : C, x : ¬ C | U | K � � S, x : ⊥ | U | K � (Clash) ⊥ (Clash) � S | U , C ⊑ D L | K � � S | U | ∅� � S, x : ¬ � ¬ C | U | K � (Clash) ∅ (Clash) � − (Unfold) � S, x : ¬ C ⊔ D | U, C ⊑ D L,x | K � if x : ¬ � ¬ C �∈ K x ∈ D ( B ) and x �∈ L � S, x : T ( C ) | U | K � � S, x : ¬ ( C ⊓ D ) | U | K � � S, x : C ⊓ D | U | K � ( T + ) ( ⊓ + ) ( ⊓ − ) � S, x : C, x : D | U | K � � S, x : ¬ C | U | K � � S, x : C, x : � ¬ C | U | K � � S, x : ¬ D | U | K � � S, x : ¬ T ( C ) | U | K � � S | U | K � ( cut ) ( T − ) � S, x : � ¬ C | U | K � � S, x : ¬ � ¬ C | U | K � � S, x : ¬ C | U | K � � S, x : ¬ � ¬ C | U | K � if x : ¬ � ¬ C �∈ S and x : � ¬ C �∈ S x ∈ D ( B ) C ∈ L T � S, u : ∃ R.C | U | K � ( ∃ + ) R R . . . − → y 1 , y 1 : C | U | K � − → y m , y m : C | U | K � � S, u � S, u if D ( B ) = { y 1 , . . . , y m } � S, u : ¬ � ¬ C 1 , . . . , u : ¬ � ¬ C n | U | K, u : ¬ � ¬ C 1 , . . . , u : ¬ � ¬ C n � ( � − ) � − k � − k � S, y 1 : C k , y 1 : � ¬ C k , S M u → y 1 | U | K � . . . � S, y m : C k , y m : � ¬ C k , S M u → y 1 , S u → y m , S u → y m | U | K � if D ( B ) = { y 1 , . . . , y m } and y 1 � = u, . . . , y m � = u Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 31

  68. The Tableau calculus TAB EL ⊥ T min � S | U | ∅� is the corresponding constraint system of KB F = query S ′ = set of constraints obtained by adding to S the constraint corresponding to ¬ F The calculus TAB EL ⊥ T checks whether a query F is minimally min entailed from a KB by means of the following procedure: is applied to � S ′ | U | ∅� ; (phase 1) the calculus TAB EL ⊥ T PH 1 if, for each branch B built by TAB EL ⊥ T PH 1 , either (i) B is closed or (ii) (phase 2) the tableau built by the calculus TAB EL ⊥ T PH 2 for � S | U | B � − � is open, then KB | = EL ⊥ T min F , otherwise KB �| = EL ⊥ T min F . Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 32

  69. An example {∃ hc.S ( j ) , T ( S ) ⊑ NTP } | = EL ⊥ T min ∃ hc.NTP ( j ) Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 33

  70. An example {∃ hc.S ( j ) , T ( S ) ⊑ NTP } | = EL ⊥ T min ∃ hc.NTP ( j ) � j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP ∅ | ∅� (Unfold) � j : ¬ T ( S ) ⊔ NTP, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� ( ⊔ + ) � j : NTP, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j : ¬ T ( S ) , j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� ( cut ) ( T − ) � j : ¬ � ¬ S, j : NTP, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j : ¬ � ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� ( cut ) ( cut ) � j : � ¬ S, j : ¬ � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� (Clash) � j : ¬ � ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j : ¬ � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j : � ¬ S, j : NTP, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� ( ∃ + 1 ) hc → x S , x S : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | x S � hc → j, j : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | ∅� � j − � j − (Clash) ( ∃ − ) → x S , x S : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | x S � hc � x S : ¬ NTP, j − !""#$%&'&#()!*+%&'#+",'& . . . '$!$-+#)."&' hc → x S , x S : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | x S � � x S : ¬ � ¬ S, x S : ¬ NTP, j − ( � − ) → x S , x S : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | x S � → x S , x S : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | T ( S ) ⊑ NTP { j } | x S � hc hc � y : S, y : S, x S : ¬ NTP, j − � j : S, . . . , x S : ¬ NTP, j − . . . (Clash) '$!$-+#)."&' T ( S ) ⊑ NTP { j,x S ,y } | x S � hc � y : NTP, y : S, y : S, x S : ¬ NTP, j − → x S , x S : S, j : � ¬ S, j : ¬ S, j : ∃ hc.S, j : ¬ ∃ hc.NTP | ,/&*#()!*+% Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 33

  71. An example � j : ∃ hc.S | T ( S ) ⊑ NTP ∅ | x S : ¬ � ¬ S � D ( B ) = { j, y, x S } (Unfold) � j : ¬ T ( S ) ⊔ NTP, j : ∃ hc.S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � ( ⊔ + ) � j : ¬ T ( S ) , j : ∃ hc.S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � � j : NTP, j : ∃ hc.S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � . . . ( T − ) � j : ¬ S, j : ∃ hc.S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � � j : ¬ � ¬ S, j : ∃ hc.S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � (Clash) � − ( ∃ + ) → y, y : S, j : ¬ S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � → j, j : S, j : ¬ S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � hc hc − − � j � j (Clash) . . . → x S , x S : S, j : ¬ S | T ( S ) ⊑ NTP { j } | x S : ¬ � ¬ S � hc � j − (Unfold) → x S , x S : S, j : ¬ S | T ( S ) ⊑ NTP { j,x S } | x S : ¬ � ¬ S � hc � x S : ¬ T ( S ) ⊔ NTP, j − ( ⊔ + ) → x S , x S : S, j : ¬ S | T ( S ) ⊑ NTP { j,x S } | x S : ¬ � ¬ S � → x S , x S : S, j : ¬ S | T ( S ) ⊑ NTP { j,x S } | x S : ¬ � ¬ S � hc hc � x S : ¬ T ( S ) , j − � x S : NTP, j − . . . . . . ( cut ) and static rules . . . → x S , x S : S, j : ¬ S | T ( S ) ⊑ NTP { j,x S ,y } | x S : ¬ � ¬ S � hc � y : � ¬ S, j : � ¬ S, x S : � ¬ S, x S : NTP, j − !"#$%&'&("$&)(*+,)!- Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 34

  72. The Tableau calculus TAB EL ⊥ T min Theorem : TAB EL ⊥ T is a sound and complete decision min procedure for verifying if KB | = EL ⊥ T min F . Proposition : Given a KB and a query F , the problem of checking whether KB ∪{¬ F } is satisfiable is in NP. Theorem : The problem of deciding whether KB | = EL ⊥ T min F by means of TAB EL ⊥ T is in Π p 2 . (matching known complexity) min Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 35

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