A categorical account of composition methods in logic (extended version)

TL;DR

This paper develops a categorical framework for composition methods in finite model theory, extending game comonad semantics to unify and generalize Feferman--Vaught--Mostowski theorems across various logics and model classes.

Contribution

It introduces a parametric, categorical approach to FVM theorems using game comonads, enabling uniform results across different logics and structures.

Findings

Recovered classical theorems including 3-variable counting logic and cospectrality.

Provided conditions for FVM theorems in a uniform, logic-parametric manner.

Extended the framework to product structures, beyond specific game arguments.

Abstract

We present a categorical theory of the composition methods in finite model theory -- a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the composition methods are Feferman--Vaught--Mostowski (FVM) type theorems, which characterize how logical equivalence behaves under composition and transformation of models. Our results are developed by extending the recently introduced game comonad semantics for model comparison games. This level of abstraction allow us to give conditions yielding FVM type results in a uniform way. Our theorems are parametric in the classes of models, logics and operations involved. Furthermore, they naturally account for the existential and positive existential fragments, and extensions with counting quantifiers of these logics. We also reveal surprising connections between FVM type theorems, and classical concepts in the theory of monads. We illustrate our methods by recovering many classical theorems of practical interest, including a refinement of a previous result by Dawar, Severini, and Zapata concerning the 3-variable counting logic and cospectrality. To highlight the importance of our techniques being parametric in the logic of interest, we prove a family of FVM theorems for products of structures, uniformly in the logic in question, which cannot be done using specific game arguments. This is an extended version of the LiCS 2023 conference paper of the same name.