Limiting Negations in Bounded-Depth Circuits: An Extension of Markov's Theorem

From a theorem of Markov, the minimum number of negation gates in a circuit sufficient to compute any collection of Boolean functions on n variable is l = ⌈log(n + 1) ⌉. Santha and Wilson [SIAM Journal of Computing 22(2):294–302 (1993)] showed that in some classes of bounded-depth circuits l negation gates are no longer sufficient for some explicitly defined Boolean function. In this paper, we consider a general class of bounded-depth circuits in which each gate computes an arbitrary monotone Boolean function or its negation. Our purpose is to extend the theorem of Markov for such a general class of circuits. We first show that a lower bound shown by Santha and Wilson becomes an extension of Markov’s lower bound by a small refinement. Then, we present tight upper bounds on the number of negations for computing an arbitrary collection of Boolean functions.