On polymorphic sessions and functions: a tale of two (fully abstract) encodings

This work exploits the logical foundation of session types to determine what kind of type discipline for the Λ-calculus can exactly capture, and is captured by, Λ-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session π-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the Λ-calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation

Linear Abadi and Plotkin Logic

We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dualintuitionistic/linear type theory with fixed points, and show, followingPlotkin's suggestions, that it can be used to define a wide collection oftypes, including existential types, inductive types, coinductive types andgeneral recursive types. We show that the recursive types satisfy a universalproperty called dinaturality, and we develop reasoning principles for theconstructed types. In the case of recursive types, the reasoning principle is amixed induction/coinduction principle, with the curious property thatcoinduction holds for general relations, but induction only for a limitedcollection of ``admissible'' relations. A similar property was observed inPitts' 1995 analysis of recursive types in domain theory. In a future paper wewill develop a category theoretic notion of models of the logic presented here,and show how the results developed in the logic can be transferred to themodels

Linear Abadi and Plotkin Logic

We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic/linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a wide collection of types, including existential types, inductive types, coinductive types and general recursive types. We show that the recursive types satisfy a universal property called dinaturality, and we develop reasoning principles for the constructed types. In the case of recursive types, the reasoning principle is a mixed induction/coinduction principle, with the curious property that coinduction holds for general relations, but induction only for a limited collection of ``admissible'' relations. A similar property was observed in Pitts' 1995 analysis of recursive types in domain theory. In a future paper we will develop a category theoretic notion of models of the logic presented here, and show how the results developed in the logic can be transferred to the models

Operational semantics and models of linear Abadi-Plotkin logic. Manuscript

Abstract. We present a model of Linear Abadi and Plotkin Logic for parametricity [8] based on the operational semantics of LILY, a polymorphic linear lambda calculus endowed with an operational semantics [3]. We use it to formally prove definability of general recursive types in LILY and to derive reasoning principles for the recursive types.