Integrating DLs with Logic Programming Boris Motik, University of Manchester Joint work with Riccardo Rosati, University of Rome

2/21 Contents Description Logics and OWL What is Missing in DLs? Hybrid MKNF Knowledge Bases Reasoning Algorithm Conclusion

3/21 Description Logics and OWL OWL (Web Ontology Langage)  language for ontology modeling in the Semantic Web  standard of the W3C ( 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? UK cities are in UK regions. UKCity v 9 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) 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:

4/21 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 OWL LP RIF Working Group

5/21 Contents Description Logics and OWL What is Missing in DLs? Hybrid MKNF Knowledge Bases Reasoning Algorithm Conclusion

6/21 Relational expressivity  OWL can express only tree-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” 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

7/21 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 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 v 9 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)

8/21 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

9/21 Contents Description Logics and OWL What is Missing in DLs? Hybrid MKNF Knowledge Bases Reasoning Algorithm Conclusion

10/21 Main Idea (Researcher t Programmer)(Boris) Researcher v Employed Programmer v Employed ² Employed(Boris) ² Researcher(Boris) ² Programmer(Boris) 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 OWA vs. CWA  CWA requires introspection – reasoning about own beliefs Modal logics allow reasoning about consequences  KB ² Aiff KB ² K A  KB ² A iffKB ² :K A ( looks like CWA

11/21 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  (I,M,N) ² AiifA 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!

12/21 Hybrid MKNF Knowledge Bases 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 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: 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) Æ Æ r 2 P 8 x 1,…,x n : H 1 Ç … Ç H n ½ B 1 Æ … Æ B m

13/21 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

14/21 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

15/21 Compatibility Our formalism is fully compatible with 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 Our formalism is fully compatible with LP (;,P) ² A iff P ² Afor A a ground atom  already shown by Lifschitz The combination seems quite intuitive  …as long as we do not mix modal and nonmodal atoms

16/21 Contents Description Logics and OWL What is Missing in DLs? Hybrid MKNF Knowledge Bases Reasoning Algorithm Conclusion

17/21 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 (  ): ob K,P = O [ { A | K A 2 P }  our main task is to find a partition (P,N) that defines a model

18/21 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.

19/21 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

20/21 Contents Description Logics and OWL What is Missing in DLs? Hybrid MKNF Knowledge Bases Reasoning Algorithm Conclusion

21/21 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