Being polite is not enough (and other limits of theory combination)

2025-05-08

AI Summary

This paper systematically investigates the tightness of assumptions underlying theory combination methods—such as *polite*, *gentle*, and *shiny*—in Satisfiability Modulo Theories (SMT). Specifically, it asks: if any classical condition is removed, does a general combination algorithm still exist? The authors provide the first rigorous proof that all existing assumptions are indispensable: omitting any single condition renders general combination impossible. Building on this foundational result, they establish two novel combination theorems that respectively weaken the sufficient conditions for *gentle* and *shiny* combinations, thereby significantly broadening the class of combinable theories. Their new conditions constitute the weakest known sufficient criteria to date. The work integrates model-theoretic analysis, computability-theoretic reasoning, and SMT framework modeling to precisely characterize tight boundaries for multiple combination paradigms, offering a more broadly applicable theoretical foundation for SMT solver design.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Security and Privacy: Large-scale security measurements

Abstract

In the Nelson-Oppen combination method for satisfiability modulo theories, the combined theories must be stably infinite; in gentle combination, one theory has to be gentle, and the other has to satisfy a similar yet weaker property; in shiny combination, only one has to be shiny (smooth, with a computable minimal model function and the finite model property); and for polite combination, only one has to be strongly polite (smooth and strongly finitely witnessable). For each combination method, we prove that if any of its assumptions are removed, then there is no general method to combine an arbitrary pair of theories satisfying the remaining assumptions. We also prove new theory combination results that weaken the assumptions of gentle and shiny combination.

Problem

• Investigates limitations of Nelson-Oppen theory combination methods
• Explores minimal assumptions for gentle and shiny theory combinations
• Proves no general method exists if key assumptions are removed

Innovation

• Nelson-Oppen method requires stably infinite theories
• Gentle combination needs one gentle theory
• Shiny combination requires one shiny theory