The Inverse Satisfiability Problem

We study the complexity of telling whether a set of bit-vectors represents the set of all satisfying truth assignments of a Boolean expression of a certain type. We show that the problem is coNP-complete when the expression is required to be in conjunctive normal form with three literals per clause (3CNF). We also prove a dichotomy theorem analogous to the classical one by Schaefer, stating that, unless P=NP, the problem can be solved in polynomial time if and only if the clauses allowed are all Horn, or all anti-Horn, or all 2CNF, or all equivalent to equations modulo two.