Game Semantics and Uniqueness of Type Inhabitance in the Simply-Typed λ-Calculus

Abstract

To cite this version:Pierre Bourreau, Sylvain Salvati. Game semantics and uniqueness of type inhabitance in the simply typed lambda-calculus. Typed Lambda-Calculi and Applications, Apr 2011, Novi Sad, Serbia. 6878, pp.61-75, 2011 Game semantics and uniqueness of type inhabitance in the simply-typed λ-calculus We also fully characterize a family of λ-terms for Aoto's theorem. The use of games also leads to a new characterization of principal typings for simply-typed λ-terms. These results show that game models can help proving strong structural properties in the simply-typed λ-calculus.

A short note on type-inhabitation: Formula-trees vs. game semantics

“…In this section, we describe a method for studying type-inhabitation, based on game-semantics, introduced in [5]. We begin recalling some notions from game-semantics, in the context of type inhabitation.…”

“…We begin recalling some notions from game-semantics, in the context of type inhabitation. The definitions and results in this section are the ones presented in [5], following the restriction in [21]. However, at times we adapt notation according to what we defined in Section 2.…”

“…However, at times we adapt notation according to what we defined in Section 2. Note also that, for the reasons given in Section 2, our presentation of this method is in terms of inhabitation of types, contrary to [5], where it was presented in terms of inhabitation of typings. This choice simplifies the exposition and also makes the establishment of the relationship between the two methods easier.…”

“…In [5] it was shown that it is possible to establish a bijection between the set of winning strategies in the arena of a type, and the set of its long normal inhabitants, through an interpretation of winning strategies as -terms. The preceding relation on sequences is defined by s 1 s 2 i↵ there exists i, j 2 N + and s 2 N ?…”

“…More recently, another method for studying type inhabitation was given, through the use of game semantics [5], that also explores the relation between the structure of types and terms. In that context, type inhabitants are seen as the interpretation of winning strategies in the arena given by a particular typing.…”