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

2024-04-30

arXiv.org

AI Summary

This paper addresses the foundational challenge of unifying characterizations of logical equivalence preservation under model composition—such as products, disjoint unions, and sums—in finite model theory. Method: It systematically incorporates Feferman–Vaught–Mostowski (FVM)-style theorems into a game-based comonadic semantics framework, extending coalgebraic semantics for model comparison games to enable parametric modeling of model classes, logical fragments, and composition operations. Contributions: (i) A unified reconstruction of several classical FVM theorems; (ii) the first parametric FVM theorem for arbitrary logics over structural products; (iii) an identification of deep connections between FVM phenomena and monad theory; and (iv) an improvement upon Dawar et al.’s results on C³ logic and cospectrality, thereby providing categorical foundations for modular logical reasoning. The approach yields a principled, compositional account of logical invariance across model constructions, advancing both finite model theory and categorical logic.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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.

Problem

• Develops categorical theory for composition methods in finite model theory
• Provides uniform conditions for Feferman-Vaught-Mostowski type theorems
• Extends game comonad semantics for model comparison games

Innovation

• Extends game comonad semantics for model comparison
• Provides uniform conditions for FVM type theorems
• Parametric in models, logics, and operations