E-KRHyper A Hyper Tableau Theorem Prover with Equality

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E-KRHyperA Hyper Tableau Theorem Prover with
Equality
Björn Pelzer bpelzer_at_uni-koblenz.de
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Overview and Motivation

• KRHyper
• theorem prover for first-order logic
• implements hyper tableau calculus
• designed for embedding in knowledge-representation
  applications
• is used in e-learning, document management,
  database schema processing, ontology
  reasoning,...
• limitation for use with modal and description
  logics no equality reasoning

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Hyper Tableau Calculus - Overview

• theorem proving and model generation method for
  FOL clauses
• based on clausal normal form tableaux

• Technique
• Given a set of clauses,
• constructs a literal tree,
• uses a single rule for attaching nodes hyper
  extension.

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Hyper Tableau Calculus - Hyper Extension

• Given some branch in a tableau
• select a clause whose negative literals unify
  with branch literals

p(x, y) ? q(f(x), b) ? p(b, f(a)) ? r(g(a))

• if positive literals from the clause share
  variables, apply some ground substitution

• attach the substituted literals as new nodes

• branches with negative leaves are closed and
  cannot be extended any further

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E-Hyper Tableau Calculus - Overview

• joint work with Peter Baumgartner and Ulrich
  Furbach
• combines hyper tableaux with superposition-based
  handling of equality
• sound and complete
• Differences to hyper tableaux
• clause tree instead of literal tree
• four extension rules instead of one
• adds term ordering
• adds redundancy handling

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E-Hyper Tableau Calculus - Superposition
The superposition rules derive a new node by
applying a positive equation unit to another
clause from the same branch.
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E-Hyper Tableau Calculus - Reflexivity and Split
The reflexivity-rule eliminates a trivial
negative equation.
p(f(b, y)) ? q(y) ?
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E-Hyper Tableau Calculus - Handling Redundancy

• If a clause...
• is subsumed, or
• follows from smaller clauses,
• then it can be removed.

t?t ?
q(f(a), b) ?
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E-KRHyper - Overview

• E-hyper tableau is built depth-first, one branch
  at a time
• splitting delayed as long as possible
• iterative deepening bounded by term weight
• enumerates models
• backward compatible to KRHyper

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E-KRHyper - Specialities

• satisfiable
• yet can cause termination problems for some
  provers
• p(g(f(x))) ? p(g(g(x))) ?
• p(g(g(f(x)))) ? p(g(g(g(x)))) ?
• ...

• E-KRHyper
• purification creates ground instances
• (1) and (2) allow detection of redundancy
• terminates with model p(a) ?

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E-KRHyper - Experiments and Outlook

• works best so far on problems that are
  range-restricted and satisfiable (solves 74 of
  the subset in TPTP)
• early experiments with blocking transformation
  for bottom-up model generation
• for the future performance optimization
• Thanks!