The Boolean formula value problem is in ALOGTIME

Abstract

The Boolean formula value problem is to determine the truth value of a variable-free Boolean formula, or equivalently, to recognize the true Boolean sentences. N. Lynch [ll] gave log space algorithms for the Boolean formula value problem and for the more general problem of recognizing a parenthesis context-free grammar. This paper shows that these problems have alternating log time algorithms. This answers the question of Cook [5] of whether the Boolean formula value problem is log space complete it is not, unless log space and alternating log time are identical. Our results are optimal since, for an appropriately defined notion of fog time reductions, the Boolean formula value problem is complete for alternating log time under deterministic log time reductions; consequently, it is al30 complete for alternating log time under AC0 reductions. It follows that the Boolean formula value problem is not in the log time hierarchy. There are two reasons why the Boolean formula value problem is interesting. First, a Boolean (or propositional) formula is a very fundamental concept of logic. The computational complexity of evaluating a Boolean formula is therefore of interest. Indeed, the results below will give a precise characterisation of the computational complexity of determining the truth value of a Boolean formula. Second, the existence of an alternating log time algorithm for the Boolean formula problem implies the existence of log depth, polynomial size circuits for this problem and hence there are (at least theoretically) good parallel algorithms for determining the value of a Boolean sentence. As mentioned above, N. Lynch [ll] first studied the complexity of the Boolean formula problem. It follows from Lynch’s work that the Boolean formula value problem is in NC’, since Borodin [l] showed that LOCSPACE C NCZ. Another early significant result on this problem was due to Spira [17] who showed that for every formula of size n, there is an equivalent formula of size O(n2) and depth O(log n). An improved construction, which also applied to the evaluation of rational expressions, was obtained by Brent [z]. Spira’s result WAS significant in part because because it implied that there might be a family of polynomial size, log depth circuits for recognizing true Boolean formulas. In other words, that the Boolean formula value problem might be in (non-uniform) NC’. However, it was not known if the transformations of formulas defined by Brent and Spira could