Integrating DLs with Logic Programming

1. Integrating DLs withLogic Programming Boris Motik, University of Manchester Joint work with Riccardo Rosati, University of Rome

2. Contents • Description Logics and OWL • What is Missing in DLs? • Hybrid MKNF Knowledge Bases • Reasoning Algorithm • Conclusion

3. UK cities are in UK regions. UKCity v9 isIn.UKRgn 8 x : UKCity(x) !9 y : isIn(x,y) Æ UKRgn(x) UK regions are EU regions. UKRgn v EURgn 8 x : UKRgn(x) ! EURgn(x) 8 x : [9 y : isIn(x,y) Æ EURgn(y)] ! EUPart(x) Things in EU are parts of EU. 9 isIn.EURgn v EUPart We can conclude: 8 x : UKCity(x) ! EUPart(x) UK cities are parts of EU. UKCity v EUPart 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/) • 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 OWL?

4. OWL LP RIF Working Group DLs vs. Logic Programming • OWL… • …has been very successful • …but many features are still needed in practice • Logic programming seems to address many needs • Political dimension: The battle for The Semantic Web Language • This work provides an integration framework for DLs and LP • fully compatible with both systems

5. Contents • Description Logics and OWL • What is Missing in DLs? • Hybrid MKNF Knowledge Bases • Reasoning Algorithm • Conclusion

6. x S x1 R R x2 x3 Missing Features (I) • Relational expressivity • OWL can express onlytree-like axioms • Polyadic predicates • e.g., Flight(From, To, Airline) • Exceptions • the heart is on the left, but in some cases it is on the right • Human v HeartOnLeft, Dextrocardiac v Human,Dextrocardiac v:HeartOnLeft • the class Dextrocardiac is unsatisfiable • we want to say “with no contrary evidence, the heart is on the left” 9S.(9 R.C u9 R.D) v Q , 8x:{[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)} , 8x,x1,x2,x3:{ S(x,x1) Æ R(x1,x2) Æ C(x2) Æ R(x1,x3) Æ D(x3) ! Q(x) }

7. Missing Features (II) – Closed Worlds Question: is there a flight from MAN to MUC? flight(MAN,STR) flight(MAN,LHR) flight(MAN,FRA) flight(FRA,ZAG) Open worlds (=OWL): Don’t know! We did not specify thatwe know information aboutall possible flights. Closed worlds (=LP): No. If we cannot prove something, it must be false. • Partial solution: close off the predicate 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 v9 father.Person Person(Peter) >v { Peter,Paul } • Peter and Paul are now the only objects (the domain is closed) • we do not have CWA (e.g, we cannot derive :father(Peter,Paul)

8. 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 v9 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

9. Contents • Description Logics and OWL • What is Missing in DLs? • Hybrid MKNF Knowledge Bases • Reasoning Algorithm • Conclusion

10. Main Idea • OWA vs. CWA • CWA requires introspection – reasoning about own beliefs • Modal logics allow reasoning about consequences • KB ² A iff KB ² K A • KB ² A iff KB ²:K A ( looks likeCWA (Researcher t Programmer)(Boris) Researcher v Employed Programmer v Employed ² Employed(Boris) ² Researcher(Boris) ² Programmer(Boris) ² K Employed(Boris) ² :KResearcher(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

11. Minimal Knowledge and Negation as Failure • [Lifschitz; IJCAI ’91, Artificial Intelligence ’95] • Syntax: FOL with modal operators K and not • Semantics: • an FO interpretation I and two sets of FO interpretations M and N • M is a model of  if: • (I,M,M) ² and • for each M’ ¾ M, there is some I’ 2 M’ such that (I’,M’,M) ² Gelfond-Lifschitz reduct!

12. Hybrid MKNF Knowledge Bases • MKNF Rule: • DL-safety: • each variable in each rule must occur in a body non-DL-K-atom • makes rules applicable only to named objects • necessary for decidability H1Ç … Ç Hnà B1, …, Bm • Hi are first-order or K-atoms • Bi are first-order, K-, or not-atoms P(t1, …, tn) - first-order atom K P(t1, …, tn) - K-atom not P(t1, …, tn) - not-atom • Hybrid 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) ÆÆr 2 P8 x1,…,xn : H1Ç … Ç Hn½ B1Æ … Æ Bm

13. Example (I) default rule • 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

14. 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

15. Compatibility • Our formalism is fully compatible with DLs (O,;) ² iff O² for any FOL formula  • to achieve this, we modified MKNF slightly • we must treat equality in a special way • Our formalism is fully compatible with LP (;,P) ² A iff P² A for A a ground atom • already shown by Lifschitz • The combination seems quite intuitive • …as long as we do not mix modal and nonmodal atoms

16. Contents • Description Logics and OWL • What is Missing in DLs? • Hybrid MKNF Knowledge Bases • Reasoning Algorithm • Conclusion

17. How to Represent Models • An MKNF model M is a set of interpretations • = typically infinite! • we need a finite representation • We represent M by a FOL formula such that M = { I | I ² } • We can consider only K-atoms from P • (P,N) – a partition of all K-atoms into positive and negative • objective knowledge (): obK,P = O[ { A | K A 2 P } • our main task is to find a partition (P,N) that defines a model

18. The General Case 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.

19. Data Complexity • 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 answering ground atomic queries: • schema is fixed • data is variable • The notion of stratification is rather complex • it must take into account recursion through the DL KB • difficult to check (= undecidable) and relatively weak

20. Contents • Description Logics and OWL • What is Missing in DLs? • Hybrid MKNF Knowledge Bases • Reasoning Algorithm • Conclusion

21. Conclusion • Hybrid MKNF rules… • …generalize most known combinations of DLs and rules • they generalize Rosati’s DL-log • they do not generalize Eiter’s approach • …are fully compatible with both DLs and LP • …are intuitive • …have nice complexity • Future work: • well-founded semantics • not trivial, because MKNF is a two-valued logic • implementation