Rho calculus

Abstract: The rewriting calculus, or Rho Calculus, is a simple calculus that uniformly integrates abstraction on patterns and non-determinism. Therefore, it fully integrates rewriting and lambda-calculus. The original presentation of the calculus was untyped. In this paper we present a uniform way to decorate the terms of the calculus with types. This gives raise to a new presentation a la Church, together with nine (8+1) type systems which can be placed in a RHO-cube that extends the Lambda-cube of Barendregt. Due to the matching capabilities of the calculus, the type systems use only one abstraction mechanism and therefore gives an original answer to the identification of the standard ``lambda'' and ``pi'' abstractors. As a consequence, this brings matching and rewriting as the first class concepts of the Rho-versions of the Logical Framework (LF) of Harper-Honsell-Plotkin, and of the Calculus of Constructions (CC) of Coquand-Huet.

Abstract: In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of the calculus, and we focus on its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. Furthermore, the calculus allows us to get object oriented constructions unreachable in other calculi. In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework.

Abstract: This paper presents the design, the implementation, and experiments of the integration of syntactic, conditional possibly associative-commutative term rewriting into proof assistants based on constructive type theory. Our approach is called external because it consists in performing term rewriting in a specific and efficient environment and checking the computations later in a proof assistant. Two typical systems are considered in this work: ELAN, based on the rewriting calculus, as the term rewriting-based environment, and Coq, based on the calculus of inductive constructions as the proof assistant. We first formalize the proof terms for deduction by rewriting and strategies in ELAN using the rewriting calculus with explicit substitutions. We then show how these proof terms can soundly be translated into Coq syntax where they can be directly type checked. For the method to be applicable for rewriting modulo associativity and commutativity, we provide an effective method to prove equalities modulo these axioms in Coq using ELAN. These results have been integrated into an ELAN-based rewriting tactic in Coq.

Abstract: The rewriting calculus, also called rho-calculus, is a framework embedding Lambda-calculus and rewriting capabilities, by allowing abstraction not only on variables but also on patterns. The higher-order mechanisms of the Lambda-calculus and the pattern matching facilities of the rewriting are then both available at the same level. Many type systems for the Lambda-calculus can be generalized to the rho-calculus: in this paper, we study extensively a first-order rho-calculus a la Church, called ρ stk . The type system of ρ → stk allows one to type (object oriented flavored) fixpoints, leading to an expressive and safe calculus. In particular, using pattern matching, one can encode and typecheck term rewriting systems in a natural and automatic way. Therefore, we can see our framework as a starting point for the theoretical basis of a powerful typed rewriting-based language.