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?

Abstract This is the first part of an analysis of the interplay between multiple properties that are related to combination methodologies for theories in the field of satisfiability modulo theories. We here focus on Nelson-Oppen and polite theory combinations, leading to a total of five model-theoretic properties to be considered: stable infiniteness, smoothness, finite witnessability, strong finite witnessability, and convexity. Our first result is an improvement on polite theory combination, showing that it is possible when only assuming stable infiniteness and strong finite witnessability, and thus implying smoothness is not a prerequisite for this method. Second, we provide examples of Boolean combinations of the aforementioned 5 properties whenever they are possible (e.g., a theory that admits all the properties, a theory that admits none, etc.), sharp in the sense that no theories within simpler signatures may exhibit the exact same properties, and prove which combinations cannot occur. Among these examples, the most surprising one is that of a polite yet not strongly polite theory in one sort, a combination whose previous example in the literature was two-sorted.

We study properties that allow first-order theories to be disjointly combined, including stable infiniteness, shininess, strong politeness, and gentleness. Specifically, we describe a Galois connection between sets of decidable theories, which picks out the largest set of decidable theories that can be combined with a given set of decidable theories. Using this, we exactly characterize the sets of decidable theories that can be combined with those satisfying well-known theory combination properties. This strengthens previous results and answers in the negative several long-standing open questions about the possibility of improving existing theory combination methods to apply to larger sets of theories. Additionally, the Galois connection gives rise to a complete lattice of theory combination properties, which allows one to generate new theory combination methods by taking meets and joins of elements of this lattice. We provide examples of this process, introducing new combination theorems. We situate both new and old combination methods within this lattice.

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.

Abstract Stable infiniteness, strong finite witnessability, and smoothness are model-theoretic properties relevant to theory combination in satisfiability modulo theories. Theories that are strongly finitely witnessable and smooth are called strongly polite and can be effectively combined with other theories. Toledo, Zohar, and Barrett conjectured that stably infinite and strongly finitely witnessable theories are smooth and therefore strongly polite. They called counterexamples to this conjecture unicorn theories, as their existence seemed unlikely. We prove that, indeed, unicorns do not exist. We also prove versions of the Löwenheim–Skolem theorem and the Łoś–Vaught test for many-sorted logic.

This is a part of an ongoing research project, with the aim of finding the connections between properties related to theory combination in Satisfiability Modulo Theories. In pre- vious work, 7 properties were analyzed: convexity, stable infiniteness, smoothness, finite witnessability, strong finite witnessability, the finite model property, and stable finiteness. The first two properties are related to Nelson-Oppen combination, the third and fourth to polite combination, the fifth to strong politeness, and the last two to shininess. However, the remaining key property of shiny theories, namely, the ability to compute the cardinal- ities of minimal models, was not yet analyzed. In this paper we study this property and its connection to the others.

Abstract We make two contributions to the study of theory combination in satisfiability modulo theories. The first is a table of examples for the combinations of the most common model-theoretic properties in theory combination, namely stable infiniteness, smoothness, convexity, finite witnessability, and strong finite witnessability (and therefore politeness and strong politeness as well). All of our examples are sharp, in the sense that we also offer proofs that no theories are available within simpler signatures. This table significantly progresses the current understanding of the various properties and their interactions. The most remarkable example in this table is of a theory over a single sort that is polite but not strongly polite (the existence of such a theory was only known until now for two-sorted signatures). The second contribution is a new combination theorem showing that in order to apply polite theory combination, it is sufficient for one theory to be stably infinite and strongly finitely witnessable, thus showing that smoothness is not a critical property in this combination method. This result has the potential to greatly simplify the process of showing which theories can be used in polite combination, as showing stable infiniteness is considerably simpler than showing smoothness.