Positional Injectivity for Innocent Strategies

Authors Lison Blondeau-Patissier, Pierre Clairambault

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Author Details

Lison Blondeau-Patissier
  • Université Lyon, EnsL, UCBL, CNRS, LIP, F-69342, Lyon Cedex 07, France
Pierre Clairambault
  • Université Lyon, EnsL, UCBL, CNRS, LIP, F-69342, Lyon Cedex 07, France


We thank the reviewers, whose comments greatly helped improve the paper.

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Lison Blondeau-Patissier and Pierre Clairambault. Positional Injectivity for Innocent Strategies. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 17:1-17:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In asynchronous games, Melliès proved that innocent strategies are positional: their behaviour only depends on the position, not the temporal order used to reach it. This insightful result shaped our understanding of the link between dynamic (i.e. game) and static (i.e. relational) semantics. In this paper, we investigate the positionality of innocent strategies in the traditional setting of Hyland-Ong-Nickau-Coquand pointer games. We show that though innocent strategies are not positional, total finite innocent strategies still enjoy a key consequence of positionality, namely positional injectivity: they are entirely determined by their positions. Unfortunately, this does not hold in general: we show a counter-example if finiteness and totality are lifted. For finite partial strategies we leave the problem open; we show however the partial result that two strategies with the same positions must have the same P-views of maximal length.

Subject Classification

ACM Subject Classification
  • Theory of computation → Denotational semantics
  • Game Semantics
  • Innocence
  • Relational Semantics
  • Positionality


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