Model Generation Theorem Proving for FOL Ontologies 1Fabian M. Suchanek

Model Generation Theorem Proving

for First-Order Logic Ontologies

Peter Baumgartner

Fabian M. Suchanek

Max-Planck Institute for Computer Science Saarbrücken/Germany

Deduktionstreffen 2005

Model Generation Theorem Proving for FOL Ontologies 2Fabian M. Suchanek

Overview

1. Model Generation for Ontologies

2. Our Contribution

1. Treating Equality

2. Achieving Termination

3. Evaluation

Model Generation Theorem Proving for FOL Ontologies 3Fabian M. Suchanek

Ontologies

Ontologies

OWL DL (Tambis, Wine, Galen)

OWL

FOL (SUMO/MILO, OpenCyc)

FrameNet

Reasoning Tasks

Satisfiability

Subsumption

Entailment

Instance retrieval

} DL-Provers

Model Generation Theorem Proving for FOL Ontologies 4Fabian M. Suchanek

Ontologies

Ontologies

OWL DL (Tambis, Wine, Galen)

OWL

FOL (SUMO/MILO, OpenCyc)

FrameNet

Reasoning Tasks

Satisfiability

Subsumption

Entailment

Instance retrieval

FOL Refutational Provers

Model Generation Theorem Proving for FOL Ontologies 5Fabian M. Suchanek

Types of Provers

Shortcomings of Refutational Provers:

Proposal:

Use Model Generation Provers instead ر

They often cannot produce models ر (but models are useful as counterexamples or overviews)

They may not terminate on satisfiable formula sets ر (but termination is highly desirable)

Model Generation Theorem Proving for FOL Ontologies 6Fabian M. Suchanek

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 ر

KRHyper (HyperTableaux) ر

Darwin (Model-Evolution) ر

Model Generation Theorem Proving for FOL Ontologies 7Fabian M. Suchanek

Model Generation for Ontologies

Ontologies

OWL DL (Tambis, Wine, Galen)

OWL

FOL (SUMO/MILO, OpenCyc)

FrameNet

Reasoning Tasks

Satisfiability

Subsumption

Entailment

Instance retrieval

FOL

Clause Form

Model Generation Prover

Model

Model Generation Theorem Proving for FOL Ontologies 8Fabian M. Suchanek

Equality

Equality comes in e.g.

for nominals ("one of") ر

for cardinality restrictions ر

WhiteLoire(x) ^madeF romGrape(x;y) )y = Sauvignon _ y= Chenin _ y= P inot

Cation v · 4 hasCharge

Cation(x) ^hasCharge(x;x1) ^¢¢¢̂ hasCharge(x;x5) )x1= x2_ x1= x3_ ¢¢¢_ x4= x5

WhiteLoirev 8madeF romGrape:Sauvignon t Chenin t P inot

Model Generation Theorem Proving for FOL Ontologies 9Fabian M. Suchanek

Treating Equality – Known Approaches

Approaches for treating equality

Naive approach: Add the equality axioms ر

Problem: Cumbersome function substitution axioms(x = y ) f (x) = f (y)) ^ a= b) f (a) = f (b)) f (f (a)) = f (f (b))) f (f (f (a))) = f (f (f (b))): : :

Brand's Transformation (1975, later improved) ر

Works fine, but can be optimized in our case

Model Generation Theorem Proving for FOL Ontologies 10Fabian M. Suchanek

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

Our transformation is complete and correct.

p(f (x)) Ã x = g(a)

p(f (g(a))) Ã

p(x) Ã x = f (y) ^ y= g(z) ^ z = a

p(f (g(a))) Ã

Model Generation Theorem Proving for FOL Ontologies 11Fabian M. Suchanek

Treating Equality – Comparison with Brand

Our transformation induces a smaller search space

s1 = t1 _ s1 = t1 _ :: : _ sn = tnt1 = s1 _ s2 = t2 _ :: : _ sn = tns1 = t1 _ t2 = s2 _ :: : _ sn = tnt1 = s1 _ t2 = s2 _ :: : _ sn = tn: : :t1 = s1 _ t2 = t2 _ :: : _ sn = tn

s1 = t1 _ :: : _ sn = tn

n-fold branching O(n2n)-fold branching (with regularity constraint:still exponential)

2n

Model Generation Theorem Proving for FOL Ontologies 12Fabian M. Suchanek

Cycles in Existential Roles

chapter v 9 partOf : bookbook v 9 has : chapter

book(f book(x)) Ã chapter(x)partOf(x;f book(x)) Ã chapter(x)

chapter(f chapter (x)) Ã book(x)has(x;f chapter (x)) Ã book(x)

Model Generation Theorem Proving for FOL Ontologies 13Fabian M. Suchanek

Cycles in Existential Roles

book(b)

chapter(f chapter (b))

book(f book(f chapter (b)))

book(f book(f chapter (f book(f chapter (b)))))

chapter(f chapter (f book(f chapter (f book(f chapter (b))))))

chapter(f chapter (f book(f chapter (b))))

Model Generation Theorem Proving for FOL Ontologies 14Fabian M. Suchanek

Blocking Technique

^dom(x)^dom(x)

b

dom

f book(b) f chapter (f chapter (b))

f chapter (b)

: rewrite relation

f book(f chapter (b))

à chapter(x) ^book(x)

This search is encoded in the DLP (see paper for details).

chapter(f chapter (x)) Ã book(x)book(f book(x)) Ã chapter(x)

book chapterchapter bookbook

Model Generation Theorem Proving for FOL Ontologies 15Fabian M. Suchanek

Blocking Technique – Results

Our blocking transformation

ensures termination in many cases ر

is complete and correct ر

can be applied to arbitrary formula sets (not just DL) ر

Model Generation Theorem Proving for FOL Ontologies 16Fabian M. Suchanek

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 17Fabian M. Suchanek

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% 86%

Euler 0% 98% 100%

Cerebra(DL) 90% 59% 61%

ConsVISor 77% 65% -

OWLP(DL) 50% 26% 53%

Model Generation Theorem Proving for FOL Ontologies 18Fabian M. Suchanek

Conclusion

Our approach for ontological reasoning

produces a model in case of satisfiability ر

can be applied to arbitrary ontologies (not just DL) ر

is competitive with existing systems ر

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