Number theory combination: natural density and SMT

The study of theory combination in Satisfiability Modulo Theories (SMT) involves various model theoretic properties (e.g., stable infiniteness, smoothness, etc.). We show that such properties can be partly captured by the natural density of the spectrum of the studied theories, which is the set of sizes of their finite models. This enriches the toolbox of the theory combination researcher, by providing new tools to determine the possibility of combining theories. It also reveals interesting and surprising connections between theory combination and number theory.