Model Generation Theorem Proving for First-Order Logic Ontologies

1. Deduktionstreffen 2005 Model Generation Theorem Proving for First-Order Logic Ontologies Peter Baumgartner Fabian M. Suchanek Max-Planck Institute for Computer Science Saarbrücken/Germany Model Generation Theorem Proving for FOL Ontologies

2. Overview Model Generation for Ontologies Our Contribution Treating Equality Achieving Termination Evaluation Model Generation Theorem Proving for FOL Ontologies

3. Ontologies Ontologies OWL DL (Tambis, Wine, Galen) OWL FOL (SUMO/MILO, OpenCyc) FrameNet } DL-Provers Reasoning Tasks Satisfiability Subsumption Entailment Instance retrieval Model Generation Theorem Proving for FOL Ontologies

4. Ontologies Ontologies OWL DL (Tambis, Wine, Galen) OWL FOL (SUMO/MILO, OpenCyc) FrameNet FOL Refutational Provers Reasoning Tasks Satisfiability Subsumption Entailment Instance retrieval Model Generation Theorem Proving for FOL Ontologies

5. Types of Provers Shortcomings of Refutational Provers: رThey often cannot produce models (but models are useful as counterexamples or overviews) ر They may not terminate on satisfiable formula sets (but termination is highly desirable) • Proposal: • Use Model Generation Provers instead Model Generation Theorem Proving for FOL Ontologies

6. Model Generation Provers Model Generation Provers compute models for satisfiable formula sets (iff the set is satisfiable and the prover terminates). • Existing Model Generation Provers include: • s-models • KRHyper (HyperTableaux) • Darwin (Model-Evolution) Model Generation Theorem Proving for FOL Ontologies

7. Model Generation for Ontologies Ontologies OWL DL (Tambis, Wine, Galen) OWL FOL (SUMO/MILO, OpenCyc) FrameNet FOL Clause Form Reasoning Tasks Satisfiability Subsumption Entailment Instance retrieval Model Generation Prover Model Model Generation Theorem Proving for FOL Ontologies

8. Equality Equality comes in e.g. رfor nominals ("one of") ( ) ( ) ( ) ( ) ( ) h h h h 8 h d h d h h h W W C C L L C C F F G G S C C P i i i i i i i i i 4 1 5 v · v t t t t t ^ ^ ^ ^ t t ¢ ¢ ¢ a a o o e e n n x o o r r e e x a s a m s m a a a r e a g e r e g r x r o e o m m x r r a a p p e e x a y u a v s g n o a n r g e x x e n n n o ) ) ; : ; ; h S C P i i i 1 2 1 3 4 5 t _ _ _ _ _ ¢ ¢ ¢ x y x a u v x g n o n x y e x n n x y n o = = = = = = ر for cardinality restrictions Model Generation Theorem Proving for FOL Ontologies

9. Treating Equality – Known Approaches ( ( ) ( ) ) f f b ^ x y x y a = ) = = ( ) ( ) f f b a ) = ( ( ) ) ( ( ) ) f f f f b a ) = ( ( ( ) ) ) ( ( ( ) ) ) f f f f f f b a ) = : : : Approaches for treating equality رNaive approach: Add the equality axioms Problem: Cumbersome function substitution axioms رBrand's Transformation (1975, later improved) Works fine, but can be optimized in our case Model Generation Theorem Proving for FOL Ontologies

10. Treating Equality – Our Approach ( ( ( ( ) ( ( ( ( ( ) ) ) ) ) ) ) ) ( ) ( ) ( ) f f f f ^ ^ p p p p x x g g a a x x y g a y g z z a à à = à à = = = 1. Add equivalence axioms for = 2. Add predicate substitution axioms 3. Flatten the clauses A clause is flat iff all proper subterms are constants or variables Our transformation is complete and correct. Model Generation Theorem Proving for FOL Ontologies

11. Treating Equality – Comparison with Brand t t t t t _ _ _ _ _ s s s s s = = = = = 1 1 1 1 1 1 n n n n : : : : : : t t t _ _ _ s s s = = = 1 1 2 2 n n : : : t t t _ _ _ s s s = = = 1 1 2 2 n n : : : t t t _ _ _ s s s = = = 1 1 2 2 n n : : : : : : t t t t _ _ _ s s = = = 1 1 2 2 n n : : : Our transformation induces a smaller search space 2n n-fold branching O(n2n)-fold branching (with regularity constraint: still exponential) Model Generation Theorem Proving for FOL Ontologies

12. Cycles in Existential Roles ( ( ) ) ( ) b h k f 9 h f b k O v t t t c o o a p e r x p a r c a p o e o r x à b k o o : ( ( ) ) ( ) b k f 9 f h h h O v t t t p o a o r x a s x c a p e c r a p e r x à b k o o ; : ( ( ) ) ( ) h f b k t c a p e r x o o x à h t c a p e r ( ( ) ) ( ) h f b k a s x x o o x à h t c a p e r ; Model Generation Theorem Proving for FOL Ontologies

13. Cycles in Existential Roles ( ( ( ) ( ( ( ( ( ( ( ( ) ( ( ) ) ) ( ( ) ( ( ( ) ( ) ) ) ) ) ( ) ) ) ( ) ) ) ) ) ) b b b h h h k k k f b f f f f f f b f f b f f f f f b b f b t t t c c c o o o o o o a a a p p p e e e r r r b b k k h h h h h b b b k k k h h h b k h t t t t t t t t t o o o o c c c a a a c c p p p a a e e e r r r p p e e r r o o o o o o c c a a c p p a e e r r p e r o o c a p e r Model Generation Theorem Proving for FOL Ontologies

14. Blocking Technique ( ( ( ( ) ) ( ( ) ( ( ) ( ) ) ) ( ) ) ) ( ) ( ) h d f f d b f f h f b f f b b b b k b k t t ^ ^ c a o p m o e m c r x a p e r x x o o o x o x à à b b h h k k h h h t t t t t c c o o o o a a p p e e c r r c a a p c p e e a r r p e r ( ( ) ( ) ) ( ) b d k f h t ^ o o o m x x c a p e r x à b k o o book chapter chapter book book : rewrite relation This search is encoded in the DLP (see paper for details). Model Generation Theorem Proving for FOL Ontologies

15. Blocking Technique – Results Our blocking transformation ر ensures termination in many cases ر is complete and correct ر can be applied to arbitrary formula sets (not just DL) Model Generation Theorem Proving for FOL Ontologies

16. Evaluation – Consistency Checks Model Generation Theorem Proving for FOL Ontologies

17. Evaluation – W3C Benchmark Proofs for OWL Model Generation Theorem Proving for FOL Ontologies

18. Conclusion Our approach for ontological reasoning رproduces a model in case of satisfiability ر can be applied to arbitrary ontologies (not just DL) ر is competitive with existing systems For details, see our paper "Model Generation Theorem Proving for First-Order Logic Ontologies" http://www.mpi-sb.mpg.de/~baumgart/publications/model-generation-ontologies.pdf Model Generation Theorem Proving for FOL Ontologies