Polynomial size constant depth circuits with a limited number of negations

Abstract

It follows from a theorem of Markov that the minimum number of negation gates in a circuit sufficient to compute any Boolean function on n variables is l=[log n]+1. It can be shown that, for functions computed by families of polynomial size, O(log n) depth and bounded fan-in circuits (N C ¹), the same result holds: on such circuits l negations are necessary and sufficient. In this paper we prove that this situation changes when polynomial size circuit families of constant depth are considered: l negations are no longer sufficient. For threshold circuits we prove that there are Boolean functions computable in constant depth (TC ⁰) such that no such threshold circuit containing o(n ^∈), for all ∈>0, negations can compute them. We have a matching upper bound: for any ε>0, everything computed by constant depth threshold circuits can be so computed using n ^ε negations asymptotically. We also have tight bounds for constant depth, unbounded fan-in circuits (AC ⁰): n/log^r n, for any r, negations are sufficient, and Ω(n/log^r n), for some r, are necessary.

Supported by NSF grant CCR-8810051