A Logic of Sequentiality

Game semantics has been used to interpret both proofs and functional programs: an important further development on the programming side has been to model higher-order programs with state by allowing strategies with "history-sensitive" behaviour. In this paper, we develop a detailed analysis of the structure of these strategies from a logical perspective by showing that they correspond to proofs in a new kind of affine logic. We describe the semantics of our logic formally by giving a notion of categorical model and an instance based on a simple category of games. Using further categorical properties of this model, we prove a full completeness result: each total strategy is the semantics of a unique cut-free core proof in the system. We then use this result to derive an explicit cut-elimination procedure.

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