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E-KRHyper A Hyper Tableau Theorem Prover with Equality

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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

- 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

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

- 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

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

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

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.

tt

q(f(a), b)

E-KRHyper

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

(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

- 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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