Second-Order Type Isomorphisms Through Game Semantics

TL;DR

This paper uses game semantics to characterize second-order type isomorphisms in the μ-calculus, providing a geometric perspective and extending to other polymorphic calculi.

Contribution

It introduces a game semantics model for second-order μ-calculus and characterizes type isomorphisms through arena equality, extending previous work on system F.

Findings

Type isomorphisms coincide with arena equality.

The model recovers known characterizations for system F.

The approach offers a geometric understanding of second-order type isomorphisms.

Abstract

The characterization of second-order type isomorphisms is a purely syntactical problem that we propose to study under the enlightenment of game semantics. We study this question in the case of second-order λ$\mu$-calculus, which can be seen as an extension of system F to classical logic, and for which we define a categorical framework: control hyperdoctrines. Our game model of λ$\mu$-calculus is based on polymorphic arenas (closely related to Hughes' hyperforests) which evolve during the play (following the ideas of Murawski-Ong). We show that type isomorphisms coincide with the "equality" on arenas associated with types. Finally we deduce the equational characterization of type isomorphisms from this equality. We also recover from the same model Roberto Di Cosmo's characterization of type isomorphisms for system F. This approach leads to a geometrical comprehension on the question of second order type isomorphisms, which can be easily extended to some other polymorphic calculi including additional programming features.