e krhyper a hyper tableau theorem prover with equality
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E-KRHyper A Hyper Tableau Theorem Prover with Equality. Björn Pelzer [email protected] Overview and Motivation. KRHyper theorem prover. E-KRHyper theorem prover. add equality. hyper tableau calculus. E-hyper tableau calculus. reasoning. KRHyper:

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overview and motivation
Overview and Motivation

KRHyper

theorem prover

E-KRHyper

theorem prover

add equality

hyper tableau calculus

E-hyper tableau calculus

reasoning

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

E-KRHyper

hyper tableau calculus overview
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.

E-KRHyper

hyper tableau calculus hyper extension
Hyper Tableau Calculus - Hyper Extension

p(b, f(a))

p(b, f(a))

p(x, y)  q(f(x), z)  p(z, f(a))  r(g(a))

p(z, f(a))

r(g(a))

 = {z b}

q(a, x)

r(g(x))

r(g(x))

 = {x a}

p(a, y)  q(f(a), b)  p(b, f(a))  r(g(a))

r(g(a))

q(f(a), b)

p(a, y)

p(b, f(a))

  • 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

E-KRHyper

e hyper tableau calculus overview
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

E-KRHyper

e hyper tableau calculus superposition
E-Hyper Tableau Calculus - Superposition

f(x)x 

q(f(x), b) p(f(a))

f(x)

f(x)x 

r(g(b)) p(b)

q(f(x), b) p(f(a))

f(a)

q(f(x), b) p(a)

q(f(x), b) p(a)

The superposition rules derive a new node by applying a positive equation unit to another clause from the same branch.

E-KRHyper

e hyper tableau calculus reflexivity and split
E-Hyper Tableau Calculus - Reflexivity and Split

p(f(x, y)) q(y)  g(b)g(x)

p(f(x, y)) q(y)  g(b)g(x)

p(f(b, y)) q(y) 

The split-rule uses a positive disjunction to split the branch.

p(f(b, a))

q(a) 

p(f(b, y)) q(y) 

 = {y a}

p(f(b, a))

q(a) 

The reflexivity-rule eliminates a trivial negative equation.

p(f(b, y)) q(y) 

E-KRHyper

e hyper tableau calculus handling redundancy
E-Hyper Tableau Calculus - Handling Redundancy

f(x)x 

q(f(a), b) 

q(f(a), b) 

f(x)x 

q(f(a), b) 

r(g(b)) p(b)

q(a, b) 

q(a, b) 

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

tt 

q(f(a), b) 

E-KRHyper

e krhyper overview
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

E-KRHyper

e krhyper specialities
E-KRHyper - Specialities

(1) f(a)a 

(2) g(a)a 

(3) f(g(x))g(f(x))

(4) p(f(x))  p(g(x))

  • 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)

E-KRHyper

e krhyper experiments and outlook
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!

E-KRHyper

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