Games and Full Completeness for Multiplicative Linear Logic

We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass et al. Contents 1 Introduction 3 1.1 Overview of Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 MLL+MIX 4 2.1 An aside: Units : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Proof nets for MLL+MIX : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3 The Game Semantics 6..