Rewriting modulo a rewrite system

We introduce rewriting with two sets of rules, the rst interpreted equa-tionally and the second not. A semantic view considers equational rules as deening an equational theory and reduction rules as deening a rewrite relation modulo this theory. An operational view considers both sets of rules as similar. We introduce suucient properties for these two views to be equivalent (up to diierent notions of equivalence). The paper ends with a collection of example showing the eeectiveness of this approach. Rewriting can be viewed simultaneously as the most basic symbol-manipulating method, and as a very expressive speciication framework, given the expressive power of rewriting modulo equations. It is a primary candidate to the role of a general logical framework Mes92, MOM93]. Historically, rewriting has been given an equational semantics, saying that a rewrite rule u ?! v is interpreted as u is equal to v. This is the case for instance when deening functions or solving the word problem in a equational theory. But rewriting is also useful for specifying non equational relations, such as transitions between states or deduction steps, in which case it is solely interpreted as reduction. However, a naive combination of these two kinds of semantics does not work, as shown by the non-deterministic choice example: The set of rules on the left specify addition in Peano arithmetics, and one would like to give it an equational interpretation (1 + 2 is equal to 3). However, extending the equa-tional interpretation to the two rules on the right, which specify a non-deterministic choice operator, would allow to deduce 0 = 1, and make the whole algebra collapse. The solution suggested by the work on rewriting logic is to keep all rules with an equa-tional interpretation as a set E of non-oriented equations, and consider the remaining rules as deening rewrite steps over classes modulo E (E can be thought of as deening a structure and R as computation over that structure). This is satisfactory on a semantic point of view, but disastrous on an operational one, since rewriting was precisely designed originally as a mean to handle such arbitrary big equational theories ! We propose here an extension to rewriting logic allowing for two kinds of rules, equational rules interpreted equationally and reduction rules not necessarily. The equational rules are given an equational semantics, but still have to be used only according to their orientation. The central notion …