Politeness and Stable Infiniteness: Stronger Together

We make two contributions to the study of polite combination in satisfiability modulo theories. The first is a separation between politeness and strong politeness, by presenting a polite theory that is not strongly polite. This result shows that proving strong politeness (which is often harder than proving politeness) is sometimes needed in order to use polite combination. The second contribution is an optimization to the polite combination method, obtained by borrowing from the Nelson-Oppen method. The Nelson-Oppen method is based on guessing arrangements over shared variables. In contrast, polite combination requires an arrangement over all variables of the shared sorts. We show that when using polite combination, if the other theory is stably infinite with respect to a shared sort, only the shared variables of that sort need be considered in arrangements, as in the Nelson-Oppen method. The time required to reason about arrangements is exponential in the worst case, so reducing the number of variables considered has the potential to improve performance significantly. We show preliminary evidence for this by demonstrating a speed-up on a smart contract verification benchmark.