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A Fully Abstract Game Semantics for Countable Nondeterminism

Authors W. John Gowers , James D. Laird

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  • 18 pages

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Author Details

W. John Gowers
  • Computer Science Department, University of Bath, Claverton Down Road, Bath. BA2 7QY, United Kingdom
James D. Laird
  • Department of Computer Science, University of Bath, Claverton Down Road, Bath. BA2 7QY, United Kingdom

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W. John Gowers and James D. Laird. A Fully Abstract Game Semantics for Countable Nondeterminism. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 24:1-24:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


The concept of fairness for a concurrent program means that the program must be able to exhibit an unbounded amount of nondeterminism without diverging. Game semantics models of nondeterminism show that this is hard to implement; for example, Harmer and McCusker's model only admits infinite nondeterminism if there is also the possibility of divergence. We solve a long standing problem by giving a fully abstract game semantics for a simple stateful language with a countably infinite nondeterminism primitive. We see that doing so requires us to keep track of infinitary information about strategies, as well as their finite behaviours. The unbounded nondeterminism gives rise to further problems, which can be formalized as a lack of continuity in the language. In order to prove adequacy for our model (which usually requires continuity), we develop a new technique in which we simulate the nondeterminism using a deterministic stateful construction, and then use combinatorial techniques to transfer the result to the nondeterministic language. Lastly, we prove full abstraction for the model; because of the lack of continuity, we cannot deduce this from definability of compact elements in the usual way, and we have to use a stronger universality result instead. We discuss how our techniques yield proofs of adequacy for models of nondeterministic PCF, such as those given by Tsukada and Ong.

Subject Classification

ACM Subject Classification
  • Theory of computation → Denotational semantics
  • semantics
  • nondeterminism
  • games and logic


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