Relational Parametricity for Linear System F◦

This paper presents a novel syntactic logical relation for System F◦, a simple variant of the linear polymorphic λcalculus. We define a logical relation for open values under both open linear and unrestricted contexts, then extend it for open terms with evaluation and open relation substitutions. Relations that instantiate type quantifiers are for open terms and types. We demonstrate the applicability of this logical relation through its soundness with respect to contextual equivalence, along with free theorems for linearity that are difficult to achieve by closed logical relations. When interpreting types on only closed terms, the model defaults to a closed logical relation that is both sound and complete with respect to contextual equivalence, and is sufficient to reason about isomorphisms of type encodings. The idea of using open logical relations also extends easily to more traditional formulations of polymorphic linear type systems that use the ! type constructor.