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Deduktionstreffen 2005. Model Generation Theorem Proving for First-Order Logic Ontologies. Peter Baumgartner Fabian M. Suchanek. Max-Planck Institute for Computer Science Saarbrücken/Germany. Overview. Model Generation for Ontologies Our Contribution Treating Equality - PowerPoint PPT Presentation
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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