A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas

Let F = Qlxr Qzxz l ** Qnx, C be a quantified Boolean formula with no free variables, where each Qi is either 3 or t, and C is in conjunctive normal form. That is, C is a conjunction of clauses, each clause is a disjunction of literals, and each literal is either a variable, xi, or the negation of a variable, Zi (1 < i f n). We shall use Ui to denote a literal equal to either Xi or Fi. The evaluation problem for quantified Boolean formulas is to determine whether such a formula F is true. The evaluation problem is complete in polynomial space [6], even if C is restricted to contain at most three literals per clause. The satisfiability problem, the special case in which all quantifiers are existential, is NP-complete [ 1,2,4] for formulas with three literals per clause. However, the satisfiability problem for formulas with only two literals per clause is solvable in polynomial time [ 1,2,4] ; Even, Itai, and Shamir [3] outline a linear-time algorithm. Schaefer [5] claims a polynomial time bound for the evaluation problem with two literals per clause, although he gives no proof. In this note we present a simple constructive algorithm for the evaluation of formulas having two literals per clause, which runs in linear time on a random access machine.