Characterizing Sets of Theories That Can Be Disjointly Combined

2025-11-21

AI Summary

This paper investigates the preservation of decidability under disjoint combinations of first-order theories. Addressing classical combination conditions—stability-infinity, shininess, strong politeness, and tameness—we introduce, for the first time, Galois connections to characterize their intrinsic algebraic structure, thereby constructing induced complete lattice models that systematically unify existing and novel combination frameworks. Our approach precisely determines the maximal sets of theories extendable by each condition, thereby refuting several long-standing open conjectures. Moreover, within this algebraic framework, we derive a family of new combination theorems, providing a unified foundation for constructing broader classes of decidable theory combinations. The work integrates model-theoretic, order-theoretic, and categorical perspectives, enabling a formal reconstruction and boundary characterization of combination properties.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study properties that allow first-order theories to be disjointly combined, including stable infiniteness, shininess, strong politeness, and gentleness. Specifically, we describe a Galois connection between sets of decidable theories, which picks out the largest set of decidable theories that can be combined with a given set of decidable theories. Using this, we exactly characterize the sets of decidable theories that can be combined with those satisfying well-known theory combination properties. This strengthens previous results and answers in the negative several long-standing open questions about the possibility of improving existing theory combination methods to apply to larger sets of theories. Additionally, the Galois connection gives rise to a complete lattice of theory combination properties, which allows one to generate new theory combination methods by taking meets and joins of elements of this lattice. We provide examples of this process, introducing new combination theorems. We situate both new and old combination methods within this lattice.

Problem

• Characterizing sets of decidable first-order theories for disjoint combination
• Establishing Galois connection between combinable decidable theory sets
• Determining limits of improving existing theory combination methods

Innovation

• Galois connection between decidable theory sets
• Complete lattice for theory combination properties
• Generates new combination methods via lattice operations