Colors Make Theories Hard

The satisfiability problem for conjunctions of quantifier-free literals in first-order theories T of interest –“T -solving” for short– has been deeply investigated for more than three decades from both theoretical and practical perspectives, and it is currently a core issue of state-of-the-art SMT solving. Given some theory T of interest, a key theoretical problem is to establish the computational (in)tractability of T -solving, or to identify intractable fragments of T . In this paper we investigate this problem from a general perspective, and we present a simple and general criterion for establishing the NP-hardness of T solving, which is based on the novel concept of “colorer” for a theory T . As a proof of concept, we show the effectiveness and simplicity of this novel criterion by easily producing very simple proofs of the NP-hardness for many theories of interest for SMT, or of some of their fragments.