Bringing closure to theory combination properties

2026-05-28

AI Summary

This study investigates the closure properties of classical theory combination characteristics—stably infiniteness, smoothness, and shininess—under intersection and combinability. Through formal logical analysis, theory intersection operations, and fixed-point iteration, the work systematically enumerates all possible combinations of these three properties under intersection and iteratively computes the maximal sets of combinable theories supported by each resulting intersection until convergence. For the first time, it fully characterizes the closure structure of these properties with respect to intersection and combinability, precisely determines the number of attainable property combinations, and establishes the ultimate limits of theory combination, thereby providing a completeness guarantee for the combination of decidable theories.

Abstract

We consider the closure of three classical combination properties, namely, stable infiniteness, gentleness and shininess (or, equivalently for decidable theories, strong politeness), under intersection and combinability. We compute every possible intersection, and then compute the maximal set of theories that can be combined with each resulting intersection. We iterate this process until no new sets are identified. How many properties will we end up with?

Problem

• theory combination
• stable infiniteness
• gentleness
• shininess
• closure

Innovation

• theory combination
• stable infiniteness
• gentleness
• shininess
• closure