Presentation of a Game Semantics for First-Order Propositional Logic

TL;DR

This paper develops a novel game semantics framework for a fragment of first-order propositional logic, characterizing definable strategies through algebraic and categorical tools, and providing a finite, diagrammatic presentation.

Contribution

It introduces an original methodology combining game semantics, rewriting theory, and categorical algebra to characterize definable strategies with a finite axiomatization.

Findings

Finite set of atomic strategies generates all definable strategies.

Strategies satisfy laws similar to bialgebras, linking algebra and semantics.

Finite axiomatization of strategy equality achieved.

Abstract

Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies - that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of ``atomic'' strategies and that the equality between strategies generated in such a way admits a finite axiomatization. These generators satisfy laws which are a variation of bialgebras laws, thus bridging algebra and denotational semantics in a clean and unexpected way.