Section: Scientific Foundations

Rewriting

Rewriting has reached some maturity and the rewriting paradigm is now widely used for specifying, modelizing, programming and proving. It allows for easily expressing deduction systems in a declarative way, for expressing complex relations on infinite sets of states in a finite way, provided they are countable. Programming languages and environments have been developed, which have a rewriting based semantics. Let us cite ASF+SDF   [39] , Maude   [42] , and Tom   [69] .

For basic rewriting, many techniques have been developed to prove properties of rewrite systems like confluence, completeness, consistency or various notions of termination. In a weaker proportion, proof methods have also been proposed for extensions of rewriting like equational extensions, consisting of rewriting modulo a set of axioms, conditional extensions where rules are applied under certain conditions only, typed extensions, where rules are applied only if there is a type correspondence between the rule and the term to be rewritten, and constrained extensions, where rules are enriched by formulas to be satisfied  [34] , [45] , [74] .

An interesting aspect of the rewriting paradigm is that it allows automatable or semi-automatable correctness proofs for systems or programs. Indeed, properties of rewriting systems as those cited above are translatable to the deduction systems or programs they formalize and the proof techniques may directly apply to them.

Another interesting aspect is that it allows characteristics or properties of the modelized systems to be expressed as equational theorems, often automatically provable using the rewriting mechanism itself or induction techniques based on completion  [44] . Note that the rewriting and the completion mechanisms also enable transformation and simplification of formal systems or programs. Applications of rewriting-based proofs to computer security are various. Let us mention recent work using rule-based specifications for detection of computer viruses  [76] , [77] .