We present a possible world semantics for a call-by-value higherorder programming language with impredicative polymorphism, general references, and recursive types. The model is one of the first relationally parametric models of a programming language with all these features. To model impredicative polymorphism we define the semantics of types via parameterized (world-indexed) logical relations over a universal domain. It is well-known that it is non-trivial to show the existence of logical relations in the presence of recursive types. Here the problems are exacerbated because of general references. We explain what the problems are and present our solution, which makes use of a novel approach to modeling references. We prove that the resulting semantics is adequate with respect to a standard operational semantics and include simple examples of reasoning about contextual equivalence via parametricity.

Abstract. This paper presents a polymorphic type system for a language with delimited control operators, shift and reset. Based on the monomorphic type system by Danvy and Filinski, the proposed type system allows pure expressions to be polymorphic. Thanks to the explicit presence of answer types, our type system satisfies various important properties, including strong type soundness, existence of principal types and an inference algorithm, and strong normalization. Relationship to CPS translation as well as extensions to impredicative polymorphism are also discussed. These technical results establish the foundation of polymorphic delimited continuations.

Abstract. We study the equational theory of Parigot’s second-order λµ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λµ-terms. On the other hand, the unconstrained relational parametricity on the λµ-calculus turns out to be inconsistent. Following these facts, we propose to formulate the relational parametricity on the λµ-calculus in a constrained way, which might be called “focal parametricity”. Dedicated to Prof. Gordon Plotkin on the occasion of his sixtieth birthday 1.

We provide a syntactic analysis of contextual preorder and equivalence for a polymorphic programming language with effects. Our approach applies uniformly to arbitrary algebraic effects, and thus incorporates, as instances: errors, input/output, global state, nondeterminism, probabilistic choice, and combinations thereof. Our approach is to extend Plotkin and Power’s structural operational semantics for algebraic effects (FoSSaCS 2001) with a primitive “basic preorder ” on ground type computation trees. The basic preorder is used to derive notions of contextual preorder and equivalence on program terms. Under mild assumptions on this relation, we prove fundamental properties of contextual preorder (hence equivalence) including extensionality properties, a characterisation via applicative contexts, and machinery for reasoning about polymorphism using relational parametricity. 1.

Abstract. We present a logic for reasoning about parametric polymorphism in combination with arbitrary computational effects (nondeterminism, exceptions, continuations, side-effects etc.). As examples of reasoning in the logic, we show how to verify correctness of polymorphic type encodings in the presence of effects. 1

When one develops, implements, and studies type theories based on linear logic, for example, in the context of theorem proving, logic programming, and formal reasoning, one is immediately confronted with questions about their equational theory and how to deal with logic variables. In this paper, we propose linear contextual modal type theory that gives a mathematical account of the nature of logic variables. Our type theory is conservative over intuitionistic contextual modal type theory proposed by Nanevski, Pfenning, and Pientka. As a technical contribution we provide a proof of soundness, and, as a justification for its usefulness, we shed some light on the difficulties working with logic variables in linear logics that contain ⊤. 1