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 logicalsystem 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. Key words rewrite system · abstract rewriting · axiomatization · abstract deduction procedure · abstract completion procedure M. Aiguier (B) Université d'Évry, IBISC CNRS FRE 2873, 523 pl. des Terrasses,
Here, we propose to apply rewriting techniques to compute over a class of algebraic structures larger than congruences and preorders such as congruences closed under monotonicity and modus ponens (see Section 9 of the long version of this paper in REF ).
Structures for Abstract Rewriting
Journal of Automated Reasoning
Journal of Automated Reasoning
Mathematics Computer Science