Specifying Peirce's law in classical realizability

Abstract

This paper deals with the specification problem in classical realizability (such as introduced by Krivine (2009 Panoramas et synthéses27)), 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 Science308 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.

Classical realizability and arithmetical formulæ

“…We will rephrase the gametheoretic framework of the first author Ph.D. thesis (Guillermo 2008), to provide a game-theoretic characterization G 1 that is both complete and adequate, in the particular case where the underlying calculus contains infinitely many interaction constants. However, this hypothesis -that is crucial in our proof of completeness -is known to be incompatible with the presence of instructions such as quote or eq (Guillermo and Miquel 2014), which allow us to distinguish syntactically λ c terms that are computationally equivalent. We exhibit in Section 6.3 a wild realizer that uses these instructions and does not suit as a winning strategy for G 1 , proving that G 1 is no more complete in this case.…”

“…Furthermore, as stated in the paper on Peirce's Law (Guillermo and Miquel 2014), the presence of instructions such as quote (defined in Section 2.3) makes the problem still more subtle. We will deal with this particular case in Section 7.…”

Perspectives on Interrogative Models of Inquiry

“…For further details about these definitions we refer to Guillermo and Miquel (2014). P. †/ mapping a k-tuple to a set of stacks.…”

“…24 As explained in Naibo et al (2015), the approach undertaken by untyped proof theory differs from the usual techniques adopted in standard proof theory, as it considers a set of objects -the paraproofs -which is much larger than the set of proofs. 23 Although the pole is sometimes defined from a set of processes which could be understood as a notion of termination (see Guillermo and Miquel 2014). 22 Notice, however, that one defines the interpretation of linear logic formulas, and not classical logic formulas, as it will be clarified below.…”

Realizability Interpretation and Normalization of Typed Call-by-Need $$\lambda $$-calculus with Control

“…See for instance[27] about witness extraction or[12,13] about specification problems 4. For instance, one way to realize the axiom of dependent choice in classical realizability is by means of an extra instruction quote[18] 5.…”