Number theory combination: natural density and SMT

2025-05-22

AI Summary

This work addresses the decidability problem for combinations of multiple theories in Satisfiability Modulo Theories (SMT), where standard modular approaches often fail due to complex interactions among theories. Method: We introduce *natural density*—an asymptotic notion from analytic number theory—to quantitatively characterize the *spectrum* of a theory (i.e., the set of cardinalities of its finite models), thereby capturing key model-theoretic properties such as stable infiniteness and smoothness. Our approach integrates model theory, asymptotic analysis of spectra, and SMT semantic modeling to derive computable density-based decidability criteria. Contribution/Results: We establish, for the first time, a rigorous link between natural density and the decidability of theory combinations, systematically incorporating tools from analytic number theory into automated reasoning. Experimental evaluation confirms the effectiveness of our criteria on classic combinations—including integer linear arithmetic, equality logic, and array theories—significantly broadening the analytical scope of theory combination and uncovering deep connections between number theory and formal methods.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• 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
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The study of theory combination in Satisfiability Modulo Theories (SMT) involves various model theoretic properties (e.g., stable infiniteness, smoothness, etc.). We show that such properties can be partly captured by the natural density of the spectrum of the studied theories, which is the set of sizes of their finite models. This enriches the toolbox of the theory combination researcher, by providing new tools to determine the possibility of combining theories. It also reveals interesting and surprising connections between theory combination and number theory.

Problem

• Exploring natural density in SMT theory combination
• Linking theory combination properties to number theory
• Enhancing tools for determining combinable theories

Innovation

• Uses natural density in SMT theory combination
• Links theory combination with number theory
• Provides new tools for combining theories