Term collections in λ and ρ - calculi

The ρ-calculus, also called the rewriting calculus, originally emerged from di erent motivations and from a di erent community than the λ-calculus. It was introduced to make explicit all the ingredients of rewriting such as rule application and result [CK01]. In ne the ρ-calculus provides an extension of the λ-calculus with additional concepts originating from rewriting and functional programing, namely, pattern-matching, and a structure construction which provides collections of terms. There are several aspects of the ρ-calculus that have been studied so far. The dynamics of the computations has been studied [FMS05] by de ning interaction nets for the ρ-calculus. We can mention also the study of type systems [BCKL03,Wac04] and its application in a proof theory that handles rich proof-terms in the generalized deduction modulo [Wac05]. On a more practical side, the ρ-calculus has been used to give a semantics both to rewrite based languages [CK01] such as ELAN [BKK98] and to the atelier FOCAL [Mod,Pre03], an environment dedicated to the development of certied computer algebra libraries (ongoing works). Also, the ρ-calculus has been used to implement e cient decision procedures [SDK03]. The management of collections of terms is crucial in calculi like the ρ-calculus, in logic programming or in web query languages. Typically, matching constraints that are involved in the calculus may have more than one solution this is also the case for example in programming language like TOM [Tom], Maude [Mau], ASF+SDF [ASF] or ELAN [Ela] and thus generates a collection of results. As previously mentioned, the ρ-calculus extends the syntax and the operational semantics of the λ-calculus by providing matching constraints and collections of terms. For example, let + be a commutative symbol, x, y be variables and a, b constants. In the ρ-calculus, the pattern-matching constraint x[x + y a + b], that is the application of the matching constraint x+y a+ b to x, reduces to a collection of terms consisting of the two terms a and b and denoted a o b. In fact, the two solutions of the pattern matching problem x + y a + b, respectively {x← a, y ← b} and {x← b, y ← a}, are both applied to the body of the pattern matching constraint x and then we get the two results a and b. The corresponding evaluation rule is given by: