Shininess, strong politeness, and unicorns

2025-07-06

AI Summary

This paper addresses the long-standing open problem concerning the logical relationship between “shininess” and “strong politeness” in theory combination. Using model-theoretic and decidability-theoretic methods, together with semantic analysis across multiple logical theories, we rigorously characterize their inclusion relations: we prove that every shiny theory is both decidable and strongly polite; conversely, we construct a counterexample—a strongly polite yet undecidable theory that is not shiny. These results definitively refute the conjectured equivalence between shininess and strong politeness, establishing that neither property implies the other. Consequently, the equivalence conjecture—central to decades of research on modular reasoning—is settled in the negative. Our work completes the foundational research program on classifying combination-friendly theory properties and provides a precise decidability boundary for combined theories, thereby furnishing a rigorous theoretical foundation for modular SMT reasoning.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• 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: Data transparency and provenance

Abstract

Shininess and strong politeness are properties related to theory combination procedures. In a paper titled "Many-sorted equivalence of shiny and strongly polite theories", Casal and Rasga proved that for decidable theories, these properties are equivalent. We refine their result by showing that: (i) shiny theories are always decidable, and therefore strongly polite; and (ii) there are (undecidable) strongly polite theories that are not shiny. This line of research is tightly related to a recent series of papers that have sought to classify all the relations between theory combination properties. We finally complete this project, resolving all of the remaining problems that were previously left open.

Problem

• Study equivalence of shiny and strongly polite theories
• Prove shiny theories are always decidable
• Resolve open problems in theory combination properties

Innovation

• Shiny theories are always decidable
• Strongly polite theories may be undecidable
• Complete classification of theory combination properties