The rewriting calculus as a semantics to strategy application. Application to ELAN

Objects like rewrite rules and strategies are of very general use and interest in computer science. They are especially useful when modeling inference based systems like logical frameworks, theorem provers, constraint solvers, expert systems, transition systems. In order to formalize the concept of rule application, we have designed a calculus of explicit rule application called the rewritingor ρ-calculus. It gives a precise meaning to the definition of strategies and to the application of a strategy to an object. We present the ρ-calculus and show why it embeds in a uniform settings standard term rewriting as well as lambda-calculus. This allows us to describe conveniently strategies as well as strategy application even in the case of non-deterministic strategies. The ELAN [2] language has two first class objects: rules and strategies. We show how the semantics of the language can be described in a subset of the ρ-calculus. Rewrite rules allow to describe the computational part as well as the deductive part of logical framework. They are consequently very attractive when dealing with deduction modulo [5] or the Poincaré principle [1]. ELAN allows also to realize in a quite natural way the deduction modulo paradigm by implementing computations with a builtin leftmost-innermost strategy and deduction with user defined strategies. The language concepts and expressiveness have been successfully applied to theorem proving [4, 6], constraint programming as well as contraints solving [3]. This will be demonstrated on several examples. The ELAN environment together with its documentation and a lots of examples is freely available at http://www.loria.fr/ELAN.