The Polymorphic Rewriting-calculus: [Type Checking vs. Type Inference]

The Rewriting-calculus (Rho-calculus), is a minimal framework embedding Lambdacalculus and Term Rewriting Systems, by allowing abstraction on variables and patterns. The Rho-calculus features higher-order functions (from Lambda-calculus) and pattern-matching (from Term Rewriting Systems). In this paper, we study extensively a second-order Rho-calculus à la Church (RhoF) that enjoys subject reduction, type uniqueness, and decidability of typing. Then we apply a classical type-erasing function to RhoF, to obtain an untyped Rho-calculus à la Curry (uRhoF). The related type inference system is isomorphic to RhoF and enjoys subject reduction. Both RhoF and uRhoF systems can be considered as minimal calculi for polymorphic rewriting-based programming languages. We discuss the possibility of a logic existing underneath the type systems via a Curry-Howard Isomorphism.