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Specifying Peirce's law in classical realizability

  • MAURICIO GUILLERMO (a1) and ALEXANDRE MIQUEL (a2)
Abstract

This paper deals with the specification problem in classical realizability (such as introduced by Krivine (2009 Panoramas et synthéses 27)), which is to characterize the universal realizers of a given formula by their computational behaviour. After recalling the framework of classical realizability, we present the problem in the general case and illustrate it with some examples. In the rest of the paper, we focus on Peirce's law, and present two game-theoretic characterizations of its universal realizers. First, we consider the particular case where the language of realizers contains no extra instruction such as ‘quote’ (Krivine 2003 Theoretical Computer Science 308 259–276). We present a first game $\mathds{G}$ 0 and show that the universal realizers of Peirce's law can be characterized as the uniform winning strategies for $\mathds{G}$ 0, using the technique of interaction constants. Then we show that in the presence of extra instructions such as ‘quote’, winning strategies for the game $\mathds{G}$ 0 are still adequate but no more complete. For that, we exhibit an example of a wild realizer of Peirce's law, that introduces a purely game-theoretic form of backtrack that is not captured by $\mathds{G}$ 0. We finally propose a more sophisticated game $\mathds{G}$ 1, and show that winning strategies for the game $\mathds{G}$ 1 are both adequate and complete in the general case, without any further assumption about the instruction set used by the language of classical realizers.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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