The Structure of First-Order Causality

TL;DR

This paper develops a novel game semantics framework for first-order logic that captures dependencies induced by quantifiers, using categorical algebra to characterize definable strategies and their algebraic structure.

Contribution

It introduces a diagrammatic, algebraic approach to characterize definable strategies in game semantics for first-order logic, bridging algebra and denotational semantics.

Findings

Finite axiomatization of strategy equality

Finite generators for definable strategies

Categorical algebraic structure of strategies

Abstract

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.