AbstractThe Rewriting-calculus (Rho-calculus), is a minimal framework embedding Lambda-calculus and Term Rewriting Systems, by allowing abstraction on variables and patterns. The Rho-calculus fea- tures 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 program- ming languages. We discuss the possibility of a logic existing underneath the type systems via a Curry-Howard Isomorphism.