Revisiting the Equivalence of Shininess and Politeness

The Nelson-Oppen method [4] allows the combination of satisfiability procedures of stably infinite theories with disjoint signatures. Due to its importance, several attempts to extend the method to different and wider classes of theories were made. In 2005, it was shown that shiny [9] and polite [6] theories could be combined with an arbitrary theory (the relationship between these classes was analysed in [6]). Later, a stronger notion of polite theory was proposed, see [3], in order to overcome a subtle issue with a proof in [6]. In this paper, we analyse the relationship between shiny and strongly polite theories in the onesorted case. We show that a shiny theory with a decidable quantifier-free satisfiability problem is strongly polite and provide two different sufficient conditions for a strongly polite theory to be shiny. Based on these results, we derive a combination method for the union of a polite theory with an arbitrary theory.