In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and quantum lambda-calculi.

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.

Abstract. We consider different classes of combinatory structures related to Krivine realizability. We show, in the precise sense that they give rise to the same class of triposes, that they are equivalent for the purpose of modeling higher-order logic. We center our attentions in the role of a special kind of Ordered Combinatory Algebras-that we call the Krivine ordered combinatory algebras ( K OCAs)-that we propose as the foundational pillars for the categorical perspective of Krivine's classical realizability as presented by Streicher in [23].Our procedure is the following: we show that each of the considered combinatory structures gives rise to an indexed preorder, and describe a way to transform the different structures into each other that preserves the associated indexed preorders up to equivalence. Since all structures give rise to the same indexed preorders, we only prove that they are triposes once: for the class of K OCA s. We finish showing that in K OCA s, one can define realizability in every higher-order language and in particular in higher-order arithmetic.

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In this work, we continue our consideration of the constructions presented in the paperKrivine's Classical Realizability from a Categorical Perspectiveby Thomas Streicher. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented,mutatis mutandis, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.