Structures for Abstract Rewriting |
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Authors: | Marc Aiguier Diane Bahrami |
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Affiliation: | (1) Université d’évry, IBISC CNRS FRE 2873, 523 pl. des Terrasses, 91000 évry, France;(2) CEA/LIST Saclay, 91191 Gif sur Yvette Cedex, France |
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Abstract: | When rewriting is used to generate convergent and complete rewrite systems in order to answer the validity problem for some
theories, all the rewriting theories rely on a same set of notions, properties, and methods. Rewriting techniques have been
used mainly to answer the validity problem of equational theories, that is, to compute congruences. Recently, however, they
have been extended in order to be applied to other algebraic structures such as preorders and orders. In this paper, we investigate
an abstract form of rewriting, by following the paradigm of logical-system independency. To achieve this purpose, we provide a few simple conditions (or axioms) under which rewriting (and then the set of classical
properties and methods) can be modeled, understood, studied, proven, and generalized. This enables us to extend rewriting
techniques to other algebraic structures than congruences and preorders such as congruences closed under monotonicity and
modus ponens. We introduce convergent rewrite systems that enable one to describe deduction procedures for their corresponding
theory, and we propose a Knuth-Bendix–style completion procedure in this abstract framework. |
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Keywords: | rewrite system abstract rewriting axiomatization abstract deduction procedure abstract completion procedure |
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