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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

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Model Generation Theorem Proving for First-Order Logic Ontologies

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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


Overview

Overview

Model Generation for Ontologies

Our Contribution

Treating Equality

Achieving Termination

Evaluation

Model Generation Theorem Proving for FOL Ontologies


Ontologies

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


Ontologies1

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


Types of provers

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


Model generation provers

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


Model generation for ontologies

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


Equality

Equality

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Model Generation Theorem Proving for FOL Ontologies


Treating equality known approaches

Treating Equality – Known Approaches

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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


Treating equality our approach

Treating Equality – Our Approach

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Our transformation is complete and correct.

Model Generation Theorem Proving for FOL Ontologies


Treating equality comparison with brand

Treating Equality – Comparison with Brand

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n-fold branching

O(n2n)-fold branching

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still exponential)

Model Generation Theorem Proving for FOL Ontologies


Cycles in existential roles

Cycles in Existential Roles

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Model Generation Theorem Proving for FOL Ontologies


Cycles in existential roles1

Cycles in Existential Roles

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Model Generation Theorem Proving for FOL Ontologies


Blocking technique

Blocking Technique

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

Model Generation Theorem Proving for FOL Ontologies


Blocking technique results

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


Evaluation consistency checks

Evaluation – Consistency Checks

Model Generation Theorem Proving for FOL Ontologies


Evaluation w3c benchmark proofs for owl

Evaluation – W3C Benchmark Proofs for OWL

Model Generation Theorem Proving for FOL Ontologies


Conclusion

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


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