On moderately exponential time for SAT

Can sat be solved in “moderately exponential” time, i.e., in time p(|F|) 2cn for some polynomial p and some constant c<1, where F is a CNF formula of size |F| over n variables? This challenging question is far from being resolved. In this paper, we relate the question of moderately exponential complexity of sat to the question of moderately exponential complexity of problems defined by existential second-order sentences. Namely, we extend the class SNP (Strict NP) that consists of Boolean queries defined by existential second-order sentences where the first-order part has a universal prefix. The extension is obtained by allowing a ∀...∀∃...∃ prefix in the first-order part. We prove that if sat can be solved in moderately exponential time then all problems in the extended class can also be solved in moderately exponential time.