The Structure of First-Order Causality (extended version)

Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages. Denotational semantics were introduced to provide useful abstract invariants of proofs and programs modulo cut-elimination or reduction. In particular, game semantics, introduced in the nineties, have been very successful in capturing precisely the interactive behavior of programs. In these semantics, every type is interpreted as a game (that is as a set of moves that can be played during the game) together with the rules of the game (formalized by a partial order on the moves of the game indicating the dependencies between them). Every move is to be played by one of the two players, called Proponent and Opponent, who should be thought respectively as the program and its environment. The interactions between these two players are sequences of moves respecting the partial order of the game, called plays. In this setting, a program is characterized by the set of plays that it can exchange with its environment during an execution and thus defines a strategy reflecting the interactive behavior of the program inside the game specified by the type of the program. The notion of pointer game, introduced by Hyland and Ong [HO00], gave one of the first fully abstract models of PCF (a simply-typed λ-calculus extended with recursion, conditional branching and arithmetical constants). It has revealed that PCF programs generate strategies with partial memory, called This work was has been supported by the CHOCO (“Curry Howard pour la Concurrence”, ANR-07-BLAN-0324) French ANR project.