Logic Column 17: A Rendezvous of Logic, Complexity, and Algebra

consisting of a conjunction of clauses, where a clause is a disjunction of literals; a literal is either a variable v (a positive literal) or the negation of a variable ¬v (a negative literal). We are to decide if there is an assignment to the variables satisfying the formula, that is, an assignment under which every clause contains at least one true literal. The example formula is satisfied by the assignment f where f(s) = f(u) = false and f(t) = true. The SAT problem is famously regarded as the first natural problem to be identified as NP-complete. Two special cases of the SAT problem that are well-known to be decidable in polynomial time are the 2-SAT problem, in which every clause is a 2-clause, a clause having exactly two literals, as in the formula