1. Model Generation Theorem Proving for First-Order Logic Ontologies
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. Model Generation Theorem Proving for FOL Ontologies
Overview
Model Generation for Ontologies
Our Contribution
Treating Equality
Achieving Termination
Evaluation
Model Generation Theorem Proving for FOL Ontologies

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

9. Treating Equality – Known Approaches
Approaches for treating equality
ر Naive approach: Add the equality axioms
Problem: Cumbersome function substitution axioms ( x = y ) f ^ a b :
ر Brand'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
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 p ( f g a ) Ã p ( f g a ) Ã p ( f x ) Ã = g a p ( x ) Ã = f y ^ g z a
Our transformation is complete and correct.
Model Generation Theorem Proving for FOL Ontologies

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

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

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

14. Model Generation Theorem Proving for FOL Ontologies
Blocking Technique c h a p t e r ( f x ) Ã b o k ^ d o m ( x ) Ã c h a p t e r ( x ) ^ b o k
book
chapter
chapter
book
book d o m f c h a p t e r ( b ) f b o k ( c h a p t e r ) b f c h a p t e r ( b ) f b o k ( )
: 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
Ontology
w/out Blocking
w/ Blocking
Galen
1.3 sec
4.0 sec
Wine
97.0 sec
timeout
Tambis (w/instances) 
66.0 sec
Model Generation Theorem Proving for FOL Ontologies

17. Evaluation – W3C Benchmark Proofs for OWL
System
Consistency
Incon'cy
Entailment
KRHyper
89%
90%
86%
FACT(DL)
42%
85% 7%
Hoolet(Vampire)
78%
94%
72%
FOWL(DL)
53% 4%
32%
Pellet(DL)
96%
98%
Euler 0%
100%
Cerebra(DL)
59%
61%
ConsVISor
77%
65% -
OWLP(DL)
50%
26%
Model Generation Theorem Proving for FOL Ontologies

18. Model Generation Theorem Proving for FOL Ontologies
Conclusion
Our approach for ontological reasoning
ر produces 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"
Model Generation Theorem Proving for FOL Ontologies