On the Complexity of the CQF Hierarchy

where allRi and S are formulas in conjunctive normal form (CNF), and where in general all CNF formulas occurring in Fi contain the variable set Qi Each formula Fi is a Constrained Quantified Formula (CQF). For a given i ≥ 1, the problem of evaluating formula Fi is called CQF(i). Note that CQF(i) is at the ith complexity level of the polynomial hierarchy. In particular, CQF(i) is ΣPi complete if i is odd and ΠPi -complete if i is even. More recent work—evidently not aware of Truemper [7]—defines similar hierarchies. See, for example, Sabharwal, Ansotegui, Gomes, Hart, and Selman [4], and Benedetti, Lallouet, and Vautard [2]. In this paper, we determine the complexity of two subclasses of CQF. A Horn formula is a CNF formula where each clause contains at most one nonnegated literal, and a 2CNF formula is a CNF formula where each clause contains at most two literals. Then Horn CQF(i) is the problem to evaluate an instance of CQF(i) where all CNF formulas are Horn formulas. Analogously, 2CNF CQF(i) is the problem to evaluate an instance of CQF(i) where all CNF formulas are 2CNF formulas. In Section 2, we show that the complexity of Horn CQF(i) is reduced by one level in the polynomial hierarchy. In Section 3, we show that the complexity of 2CNF CQF(i) is reduced by two levels.