Parametricity and Proving Free Theorems for Functional-Logic Languages

Abstract : The goal of this paper is to provide the required foundations forestablishing free theorems – statements about program equivalence,guaranteed by polymorphic types – for the functional-logic pro-gramming language Curry. For the sake of presentation we restrictourselves to a language fragment that we call CuMin, and that hasthe characteristic features of Curry (both functional and logic). Wepresent a new denotational semantics based on partially ordered setswithout limits. We then introduce an intermediate language calledSaLT that is essentially a lambda-calculus extended with an abstractset type, and again give a denotational semantics. We show that thestandard (logical relations) techniques can be applied to obtain ageneral parametricity theorem for SaLT and derive free theoremsfrom it. Via a translation from CuMin to SaLT that fits the respectivesemantics, we then derive free theorems for CuMin.