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presented at the 4 th Workshop on Principles and Practice of Semantic Web Reasoning (PPSWR 2006)

Automated Reasoning Support for First-Order Ontologies. presented at the 4 th Workshop on Principles and Practice of Semantic Web Reasoning (PPSWR 2006). Peter Baumgartner (National ICT Australia, Canberra/Australia) Fabian M. Suchanek (Max-Planck-Institute for CS, Saarbruecken/Germany).

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presented at the 4 th Workshop on Principles and Practice of Semantic Web Reasoning (PPSWR 2006)

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  1. Automated Reasoning Support for First-Order Ontologies presented at the 4th Workshop on Principles and Practice of Semantic Web Reasoning (PPSWR 2006) Peter Baumgartner (National ICT Australia, Canberra/Australia) Fabian M. Suchanek (Max-Planck-Institute for CS, Saarbruecken/Germany) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  2. Automated Reasoning Support for FOLOs • رMotivation • ر Our Transformation from FOL to DLPs • ر Existentially Quantified Formulae • ر Equality • ر Conclusion PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  3. Model Computation Model (set of derivable facts) Ontology (set of FOL formulae) sings(elvis). singer(elvis). • x singer(x) => sings(x) singer(elvis) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  4. Use of Model Computation Model computation can be used for: رFinding contradictions in the ontology (there is a model only iff the ontology is consistent) ر Debugging the ontology ر Proving/disproving conjectures PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  5. Undecidability of Model Computation Model Computation is only semi-decidable for FOL: رwe can (in principle) always detect unsatisfiability رwe cannot always detect satisfiability (for principal reasons) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  6. Existing Approaches There exist model generation systems (e.g. MACE4 and Paradox). Problems: رThey try to map all constants to the same domain element رThey have difficulties if there are many distinct constants PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  7. Existing Approaches There exist model generation systems (e.g. MACE4 and Paradox). Problems: رThey try to map all constants to the same domain element رThey have difficulties if there are many distinct constants Fails for n>8 p(c1,...cn). p(x1,...,xi-1, x ,xi+1...,xj-1, x ,xj+1,...xn). for all 1<i<j<n PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  8. Our Approach Transform DLP KRHyper/ smodels / dlv PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  9. Disjunctive Logic Programs (DLPs) Rule: r(a) v p(x,y) :- q(x,y), r(z), not(s(a,x)). A Disjunctive Logic Program (DLP) is a set of rules. PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  10. Our Transformation from FOL to DLPs x inCharge(x) => onLeave(x) vy refersTo(x,y) Prenex Negation Normal Form x[…]y[…] [Qz…]inCharge(x) v onLeave(x) v refersTo(x,y) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  11. Our Transformation from FOL to DLPs x inCharge(x) => onLeave(x) vy refersTo(x,y) x[…] y[…] [Qz…]inCharge(x) v onLeave(x) v refersTo(x,y) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  12. Our Transformation from FOL to DLPs x inCharge(x) => onLeave(x) vy refersTo(x,y) x[…] y[…] [Qz…]inCharge(x) v onLeave(x) v refersTo(x,y) Disjuncts without existential or following variables Disjuncts with existential or following variables PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  13. Our Transformation from FOL to DLPs x inCharge(x) => onLeave(x) vy refersTo(x,y) x[…] y[…] [Qz…]inCharge(x) v onLeave(x) v refersTo(x,y) Disjuncts without existential or following variables Disjuncts with existential or following variables PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  14. Our Transformation from FOL to DLPs x inCharge(x) => onLeave(x) vy refersTo(x,y) negative literals x[…] y[…] [Qz…]inCharge(x) v onLeave(x) v refersTo(x,y) Disjuncts without existential or following variables Disjuncts with existential or following variables PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  15. Our Transformation from FOL to DLPs x y inCharge(x) v onLeave(x) v refersTo(x,y) Option 1: Usual Skolemization prev(x) v is_sat(x) v onLeave(x) v refersTo(x,sk(x)) :- inCharge(x). Problem: onLeave(Smith) refersTo(Smith,Miller) refersTo(Smith,sk(Smith)). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  16. Our Transformation from FOL to DLPs x y inCharge(x) v onLeave(x) v refersTo(x,y) Option 2: Recyling prev(x) v is_sat(x) v onLeave(x) v refersTo(x,sk(x)) :- inCharge(x). false :- is_sat(x), not(refersTo(x,y)). Problem: onLeave(Smith) refersTo(Smith,Miller) refersTo(Smith,sk(Smith)). (simplified. See paper) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  17. refersTo(sk(sk(sk(sk(Smith)))),sk(sk(sk(sk(sk(Smith)))))). refersTo(sk(sk(sk(sk(sk(Smith))))),sk(sk(sk(sk(sk(sk(Smith))))))). Our Transformation from FOL to DLPs refersTo(sk(sk(sk(sk(sk(sk(Smith)))))),sk(sk(sk(sk(sk(sk(sk(Smith)))))))). ersTo(sk(sk(sk(sk(sk(sk(sk(Smith))))))),sk(sk(sk(sk(sk(sk(sk(sk(Smith))))))))). x y inCharge(x) v onLeave(x) v refersTo(x,y) k(sk(sk(sk(sk(sk(sk(sk(Smith)))))))),sk(sk(sk(sk(sk(sk(sk(sk(sk(Smith)))))))))). Option 2: Recyling (sk(sk(sk(sk(sk(sk(Smith)))))))),sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(Smith))))))))))). prev(x) v is_sat(x) v onLeave(x) v refersTo(x,sk(x)) :- inCharge(x). sk(sk(sk(sk(sk(Smith)))))))),sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(Smith)))))))))))). sk(sk(sk(sk(Smith)))))))),sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(Smith))))))))))))). false :- is_sat(x), not(refersTo(x,y)). sk(sk(sk(Smith)))))))),sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(Smith)))))))))))))). Problem: x onLeave(x) k(sk(Smith)))))))),sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(Smith))))))))))))))). sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk( refersTo(Smith,sk(Smith)). refersTo(sk(Smith),sk(sk(Smith))). sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(sk refersTo(sk(sk(Smith)),sk(sk(sk(Smith)))). sk(sk(sk(sk(sk(sk(sk(sk(sk(sk(s refersTo(sk(sk(sk(Smith))),sk(sk(sk(sk(Smith))))). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  18. Our Transformation from FOL to DLPs x y inCharge(x) v onLeave(x) v refersTo(x,y) Option 2: Recyling prev(x) v is_sat(x) v onLeave(x) v refersTo(x,sk(x)) :- inCharge(x). false :- is_sat(x), not(refersTo(x,y)). Problem: x onLeave(x) refersTo(Smith,sk(Smith)). (simplified. See paper) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  19. Our Transformation from FOL to DLPs x y inCharge(x) v onLeave(x) v refersTo(x,y) Option 3: Loop check prev(x) v is_sat(x) v onLeave(x) v refersTo(x,sk(x)) :- inCharge(x). false :- is_sat(x), not(refersTo(x,y)). Problem: x onLeave(x) refersTo(x,sk(z)) :- prev(x), refersTo(z,sk(z)). refersTo(Smith,sk(Smith)). refersTo(sk(Smith),sk(Smith)). (simplified. See paper) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  20. Our Transformation from FOL to DLPs DLP prev , is_sat Transform The model is preserved (modulo the fresh predicate symbols). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  21. Treating Equality equal(dog(smith), bonzo). inCharge(dog(smith)). inCharge(bonzo). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  22. Treating Equality equal(dog(smith), bonzo). inCharge(dog(smith)). inCharge(Y):-inCharge(X),equal(X,Y). equal(dog(X),dog(Y)) :- equal(X,Y). equal(X,X). equal(X,Y) :- equal(Y,X). equal(X,Z) :- equal(X,Y), equal(Y,Z). Substitutionaxioms Equivalenceaxioms PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  23. Treating Equality equal(dog(dog(dog(dog(dog(dog(dog(dog(dog(dog(dog(elvi equal(dog(dog(dog(dog(dog(dog(dog(dog(dog(dog(smith))))) equal(dog(smith), bonzo). inCharge(dog(smith)). equal(dog(dog(dog(dog(dog(dog(dog(dog(dog(smith))))))))),w equal(dog(dog(dog(dog(dog(dog(dog(dog(smith)))))))),dog(wi inCharge(Y):-inCharge(X),equal(X,Y). equal(dog(X),dog(Y)) :- equal(X,Y). equal(X,X). equal(X,Y) :- equal(Y,X). equal(X,Z) :- equal(X,Y), equal(Y,Z). Substitutionaxioms equal(dog(dog(dog(dog(dog(dog(dog(smith))))))),dog(dog(wif equal(dog(dog(dog(dog(dog(dog(smith)))))),dog(dog(dog(wif Equivalenceaxioms equal(dog(dog(dog(dog(dog(smith))))),dog(dog(dog(dog(pris equal(dog(dog(dog(dog(smith)))),dog(dog(dog(bonzo)))). equal(dog(dog(dog(smith))),dog(dog(bonzo))). equal(dog(dog(smith)),dog(bonzo)). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  24. Treating Equality equal(dog(smith), bonzo). inCharge(dog(smith)). inCharge(Y):-inCharge(X),equal(X,Y). equal(dog(X),dog(Y)) :- equal(X,Y). equal(X,X). equal(X,Y) :- equal(Y,X). equal(X,Z) :- equal(X,Y), equal(Y,Z). Substitutionaxioms Equivalenceaxioms PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  25. Treating Equality equal(dog(smith), bonzo). inCharge(dog(smith)). equal(X,X). equal(X,Y) :- equal(Y,X). equal(X,Z) :- equal(X,Y), equal(Y,Z). Equivalenceaxioms PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  26. Treating Equality equal(dog(smith), bonzo). inCharge( X ) :- equal(X,dog(smith)). Flatten function terms (Brand 1975) equal(X,X). equal(X,Y) :- equal(Y,X). equal(X,Z) :- equal(X,Y), equal(Y,Z). Equivalenceaxioms PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  27. Treating Equality equal(dog(smith), bonzo). inCharge( X ) :- equal(X,dog(smith)). equal(X,X). equal(X,Y) :- equal(Y,X). equal(X,Z) :- equal(X,Y), equal(Y,Z). Equivalenceaxioms inCharge(bonzo). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  28. Treating Equality Transform DLP Iff the original DLP has a model that satisfies equality, then the transformed DLP has a model (which contains the original one) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  29. Preliminary Experiments with SUMO SUMO The Suggested Upper Merged Ontology (SUMO): ر is the largest public formal ontology available today ر uses first order logic with higher order features ر contains 1800 facts and rules if higher order features are stripped x human(x) => y mother(x,y) PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  30. Preliminary Experiments with SUMO Applying our approach to SUMO: رThe transformation to a DLP takes just a few seconds رThe model computation with KRHyper takes just a few seconds رThe model computation revealed numerous inconsistencies in SUMO PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  31. Preliminary Experiments with SUMO Results: رEquality transformation proved scalable and useful PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  32. Preliminary Experiments with SUMO Results: رEquality transformation proved scalable and useful We added the following conjectures to SUMO: orientation(germany,west, biggestTradingPartner(germany)). orientation(france, west, germany). F G BTP(G) equal(biggestTradingPartner(germany),france). = orientation(germany, east, france). orientation(germany, west, france). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  33. Preliminary Experiments with SUMO Results: رEquality transformation proved scalable and useful رRecycling of terms works as expected PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  34. Preliminary Experiments with SUMO Results: رEquality transformation proved scalable and useful رRecycling of terms works as expected We added the following conjecture to SUMO: instance(p,judicialProcess) SUMO axiom: x instance(x,judicialProcess) => y agent(x,y) agent(p,sk(p)). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  35. Preliminary Experiments with SUMO Results: رEquality transformation proved scalable and useful رRecycling of terms works as expected We added the following conjecture to SUMO: instance(p,judicialProcess) agent(p,smith) SUMO axiom: x instance(x,judicialProcess) => y agent(x,y) agent(p,smith). PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  36. Preliminary Experiments with SUMO Results: رEquality transformation proved scalable and useful رRecycling of terms works as expected رLoop check cannot always avoid infinite models PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

  37. Conclusion Our transformation for FOL ontologies ر allows to compute models for large ontologies ر supports equality رavoids unnecessary skolem terms ر(often) avoids infinite models PPSWR 2006 - Automated Reasoning Support for FOL Ontologies

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