On SAT Modulo Theories and Optimization Problems

Solvers for SAT Modulo Theories (SMT) can nowadays handle large industrial (e.g., formal hardware and software verification) problems over theories such as the integers, arrays, or equality. Here we show that SMT approaches can also efficiently solve problems that, at first sight, do not have a typical SMT flavor. In particular, here we deal with SAT and SMT problems where models M are sought such that a given cost function f(M) is minimized. For this purpose, we introduce a variant of SMT where the theory T becomes progressively stronger, and prove it correct using the Abstract DPLL Modulo Theories framework. We discuss two different examples of applications of this SMT variant: weighted Max-SAT and weighted Max-SMT. We show how, with relatively little effort, one can obtain a competitive system that, in the case of weighted Max-SMT in the theory of Difference Logic, can even handle well-known hard radio frequency assignment problems without any tailored heuristics. These results seem to indicate that Max-SAT/SMT techniques can already be used for realistic applications.