1. Semantics and Reasoning Algorithms for a Faithful Integration of Description Logics and Rules Boris Motik, University of Oxford

2. 2/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

3. 3/32 Description Logics and OWL OWL (Web Ontology Langage)  language for ontology modeling in the Semantic Web  standard of the W3C (http://www.w3.org/2004/OWL/)http://www.w3.org/2004/OWL/ OWL is based on Description Logics (DLs)  inspired by semantic networks  DLs have a precise semantics based on first-order logics  well-understood computational properties What can we say in DLs? UK cities are in UK regions. UKCity v 9 isIn.UKRgn 8 x : UKCity(x) ! 9 y : isIn(x,y) Æ UKRgn(y) UK regions are EU regions. UKRgn v EURgn 8 x : UKRgn(x) ! EURgn(x) Things in EU are parts of EU. 9 isIn.EURgn v EUPart 8 x : [ 9 y : isIn(x,y) Æ EURgn(y)] ! EUPart(x) UK cities are parts of EU. UKCity v EUPart 8 x : UKCity(x) ! EUPart(x) We can conclude:

4. 4/32 Relational expressivity  OWL can express only tree-like axioms Polyadic predicates  e.g., Flight(From, To, Airline) Can be addressed by rules (LP or ASP) Missing Features (I) 9 S.( 9 R.C u 9 R.D) v Q, 8 x:{[ 9 y: S(x,y) Æ ( 9 x: R(y,x) Æ C(x)) Æ ( 9 x: R(y,x) Æ D(x))] ! Q(x)}, 8 x,x 1,x 2,x 3 :{ S(x,x 1 ) Æ R(x 1,x 2 ) Æ C(x 2 ) Æ R(x 1,x 3 ) Æ D(x 3 ) ! Q(x) } x x1x1 S x2x2 x3x3 RR

5. 5/32 Missing Features (II) – Closed Worlds flight(MAN,STR) flight(MAN,LHR) flight(MAN,FRA) flight(FRA,ZAG) Question: is there a flight from MAN to MUC? Open worlds (=OWL): Don’t know! We did not specify that we know information about all possible flights. Closed worlds (=LP): No. If we cannot prove something, it must be false. Partial solution: close off flight 8 x,y: flight(x,y) $ (x ¼ MAN Æ y ¼ STR) Ç (x ¼ MAN Æ y ¼ LHR) Ç …  cannot express many things (e.g., transitive closure) Closed-world is orthogonal to closed-domain reasoning Person v 9 father.Person Person(Peter) > v { Peter,Paul } CWA is available in various LP formalisms (e.g., ASP)

6. 6/32 Missing Features (III) – Constraints “Each person must have an SSN”  naïve attempt: Person u : ( 9 hasSSN.SSN) v ?  in FOL, this is equivalent to: Person v 9 hasSSN.SSN  assume that only Person(Peter) is given  we expect the constraint to be violated (no SSN)  but KB is satisfiable: Peter has some unknown SSN FOL formulae…  …speak about the general properties of worlds  …cannot reason about their own knowledge Constraints can be expressed in LP

7. 7/32 “The heart is usually on the left, but in some cases it is on the right” Naïve approach:  Human v HeartOnLeft Dextrocardiac v Human Dextrocardiac v : HeartOnLeft  the class Dextrocardiac is unsatisfiable  “with no contrary evidence, the heart is on the left” Exceptions…  …cannot be expressed in FOL  …can be expressed in ASP Missing Features (IV)

8. 8/32 The Magic Formula DLs (= taxonomical reasoning) + LP Rules (= relational expressivity + nonmonotonic inferences) = The Winning Combination!

9. 9/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

10. 10/32 Open vs. Closed Worlds In DLs we derive Person(a) The formula is equivalent to 8 x : [Father(x) ! Person(x)]  eliminates all models in which x is a father and not a person In LP, : is interpreted as default negation  read as “is not provable” The example is unsatisfiable Negation defined using minimal knowledge “It is illegal to state that someone is a father without stating that he is a person” 8 x : [Father(x) Æ : Person(x) ! ? ] Father(a)

11. 11/32 Idea of Minimal Knowledge DLsLP Father(a) M 1 Father(a) M 2 Father(a), Person(a) M Father(a) 8 x : [Father(x) Æ : Person(x) ! ? ]  kills all models in which the formula does not hold All models are of equal “quality”. This is the only minimal model. (There is no model M’ ½ M.) We are left with models that contain Person(a) We are left with no model

12. 12/32 Minimal Knowledge and Negation DLsRules Father(a) 8 x : [Father(x) Æ : Person(x) ! Cat(x)]  esures Cat(x) in each model where x is a father and not a person Does not entail Cat(a)Does entail Cat(a), Cat(a) M 1 Father(a) M 2 Father(a), Person(a) M Father(a), Cat(a) Nonmonotonic semantics typically prefer certain models.

13. 13/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

14. 14/32 First-Order Rule Formalisms First-order combinations of DLs and rules:  SWRL, CARIN, AL-log, DL-safe rules  A 1 Æ … Æ A n à B 1 Æ … Æ B m  concepts (classes) = unary predicates  roles (properties) = binary predicates  interpreted as first-order clauses Semantics is standard first-order  Woman(x) ! Person(x) and : Person(Lassie) imply : Woman(Lassie) Easily undecidable  decidability achieved by syntactic restrictions; e.g., DL-safety Issues addressed: relational expressivity and polyadic predicates  nonmonotonic features

15. 15/32 Loose Integration dl-programs  [Eiter, Ianni, Lukasiewicz, Schindlauer, Tompits, AIJ 2008] A Ã B 1 Æ … Æ B m Æ not B m+1 Æ … Æ not B n  A and B i are first-order atoms over non-DL-predicates  B i can additionally be a query atom of the form DL[ S 1 [ p 1, S 2 [ p 2, S 3 Å p 3 ; Q ]  S i – DL predicates  p i – non-DL-predicates  Q – a DL query  understand as conditional queries over a DL ontology Rules are layered over a DL KB  they do not contribute to DL consequences + - -

16. 16/32 Strong Integration A 1 Ç … Ç A k à B 1 Æ … Æ B m Æ not B m+1 Æ … Æ not B n DL +log [Rosati, KR2006]  DL-atoms cannot occur under negation as failure  semantics:  DL-predicates interpreted under OWA  non-DL-predicates interpreted under CWA  no nonmonotonic reasoning over DL-predicates dl-programs [Lukasiewicz, ESWC2007]  no classical negation  cannot capture ASP  faithful extension of LP and DLs only w.r.t. entailment of positive ground atoms  unclear how to extend the semantics to make if faithful w.r.t. arbitrary consequences

17. 17/32 Autoepistemic Logics LP can be encoded into first-order AEL  AEL by [Konolige, Fund. Inf. 1991]  Use AEL as a framework for integrating FOL and LP  [de Bruijn, Eiter, Polleres, Tompits, IJCAI 2007] Various encodings proposed with different levels of faithfulness  considers disjunctive datalog and not ASP No proof theory yet

18. 18/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

19. 19/32 Knowledge Operator K (Researcher t Programmer)(Boris) Researcher v Employed Programmer v Employed ² Employed(Boris) ² Researcher(Boris) ² Programmer(Boris) K allows us to reason about FO consequences  KB ² K AiffKB ² A  KB ² : K AiffKB ² A ² K Employed(Boris) ² : K Researcher(Boris) ² : K Programmer(Boris) K is nonmonotonic  if we assert Researcher(Boris), then…  K Researcher(Boris) holds  : K Researcher(Boris) does not hold any more Used in an algebra-like query language EQL-Lite  [Calvanese, De Giacomo, Lembo, Lenzerini, Rosati, IJCAI 2007]

20. 20/32 Default Negation Operator not Bird(Tweety) K Bird(Tweety) Æ not : Flies(Tweety) ! K Flies(Tweety) Interpreted as not consequence Read as:  if  “Tweety is a bird” is a consequence  and  “Tweety cannot fly” is not a consequence  then  “Tweety can fly” should be a consequence

21. 21/32 Minimal Knowledge and Negation as Failure Satisfiability defined w.r.t. an MKNF structure (I,M,N)  I – a FOL interpretation  M and N – sets of FOL interpretations M is a model of  if:  (I,M,M) ²  with I 2 M and  for each M’ ¾ M, there is some I’ 2 M’ such that (I’,M’,M) ²   [Lifschitz, IJCAI 91; Artificial Intelligence 94] MKNF explains many nonmonotonic formalisms (I,M,N) ² A iifA is true in I (I,M,N) ² :  iif  is false in I (I,M,N) ²  1 Æ  2 iif both  1 and  2 are true in I (I,M,N) ² K  iif (J,M,N) ²  for each J 2 M (I,M,N) ² not  iif (J,M,N) ²  for some J 2 N Gelfond- Lifschitz reduct!

22. 22/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

23. 23/32 MKNF Knowledge Bases DL-safety:  the rules are applicable only to explicitly named objects H 1 Ç … Ç H n à B 1, …, B m H i are first-order or K-atoms B i are first-order, K-, or not-atoms P(t 1, …, t n )- first-order atom K P(t 1, …, t n )- K-atom not P(t 1, …, t n )- not-atom MKNF Rule: MKNF Knowledge Base K = ( O, P ) O – a FOL KB in some language DL P – a finite set of MKNF rules Semantics by translation into MKNF  K ) = K  ( O ) Æ Æ r 2 P 8 x 1,…,x n : H 1 Ç … Ç H n ½ B 1 Æ … Æ B m

24. 24/32 Example (I) We derive seasideCity(Barcelona)  assuming it does not lead to contradiction  deriving seasideCity(Hamburg) would cause a contraction We derive Suggest(Barcelona)  this involves standard DL reasoning  we do not know the name of the beach in Barcelona default rule

25. 25/32 Example (II) We treat ¼ in a special way  we minimize equality along with other predicates  this yields intuitive consequences The constraint is satisfied  HolyFamily is a church,  the architect of SagradaFamilia has been specified, and  HolyFamily and SagradaFamilia are synonyms constraint

26. 26/32 Faithfulness MKNF KBs are fully faithful w.r.t. DLs ( O, ; ) ²  iff O ²  for any FOL formula   to achieve this, we modified MKNF slightly  we must treat equality in a special way MKNF KBs are fully faithful w.r.t. ASP ( ;, P ) ² ( : )A iff P ² ( : )Afor A a ground atom  already shown by Lifschitz The combination seems quite intuitive

27. 27/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

28. 28/32 How to Represent Models A MKNF model is a set of interpretations  = typically infinite!  we need a finite representation Idea: represent models by FOL formulae  find a first-order formula  such that M = { I | I ²  } We represent  using K-atoms  (P,N) – a partition of all K-atoms into positive and negative  defines the consequences that must hold in an MKNF model  objective knowledge: ob K,P = O [ { A | K A 2 P }  our main task is to find a partition (P,N) that defines a model

29. 29/32 Characterization of MKNF Models Grounding Guess a partition that defines an MKNF model Check whether the rules are satisfied in this model. Check whether this model is consistent with the DL KB. Check whether this is the model of minimal knowledge. Check whether the query does not hold in the model. These are the extensions to the standard algorithm for disjunctive datalog.

30. 30/32 Data Complexity Reasoning is undecidable without DL-safety  different sources of undecidability than in the FO case If rules have special form, we can…  …find (P,N) in an easier way (e.g. deterministically) and/or  …check the minimality condition easier Data complexity of ground atom entailment:

31. 31/32 Contents Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases Reasoning and Complexity Conclusion

32. 32/32 Conclusion MKNF rules…  …generalize many known combinations of DLs and rules  …are fully compatible with both DLs and LP  …are intuitive  think of K as consequence  think of not as not consequence  …have nice complexity  defined by the DL and the LP fragment (in most cases) Future challenges  implementation  use in applications