Curry-style type isomorphisms and game semantics

Abstract

Curry-style system F, that is, system F with no explicit types in terms, may be viewed as a core presentation of polymorphism from the point of view of programming languages.This paper gives a characterisation of type isomorphisms for this language using a game model, whose intuitions come from both the syntax and the game semantics universe. The model is composed of an untyped part to interpret terms, a notion of arena to interpret types and a typed part to express the fact that an untyped strategy σ plays on an arenaA.By analysing isomorphisms in the model, we prove that the equational system corresponding to type isomorphisms for Curry-style system F is the extension of the equational system for Church-style isomorphisms with a new, non-trivial equation: ∀X.AA[∀Y.Y/X] ifXappears only positively inA.