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Decomposable theories

Published online by Cambridge University Press:  01 September 2007

KHALIL DJELLOUL
KHALIL DJELLOUL*
Affiliation:
Parc scientifique et technologique de Luminy, Laboratoire d'Informatique Fondamentale de Marseille, 163 avenue de Luminy - Case 901, 13288 Marseille, cedex 9, France (e-mail: [email protected])
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Abstract

We present in this paper a general algorithm for solving first-order formulas in particular theories called decomposable theories. First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory T. The algorithm is given in the form of five rewriting rules. It transforms a first-order formula ϕ, which can possibly contain free variables, into a conjunction φ of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if ϕ has no free variables then φ is either the formula true or ¬true. The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in with more than 160 nested alternated quantifiers.

Keywords

Logical first-order formulaComplete theoryRewriting rulesTheory of trees
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 7 , Issue 5 , September 2007 , pp. 583 - 632
Copyright
Copyright © Cambridge University Press 2007

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