Specifying Peirce's law in classical realizability

This paper deals with the specification problem in classical realizability (such as introduced by Krivine [17]), which is to characterize the universal realizers of a given formula by their computational behavior. 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’ [16]. We present a first game G0 and show that the universal realizers of Peirce’s law can be characterized as the uniform winning strategies for G0, using the technique of interaction constants. Then we show that in presence of extra instructions such as ‘quote’, winning strategies for the game G0 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 G0. We finally propose a more sophisticated game G1, and show that winning strategies for the game G1 are both adequate and complete in the general case, without any further assumption about the instruction set used by the language of classical realizers.