Limiting Negations in Constant Depth Circuits

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 $l = \lfloor {\log n} \rfloor + 1$. It can be shown that, for functions computed by families of polynomial size, $O(\log n)$ depth and bounded fan-in circuits $(NC^1 )$, the same result holds: on such circuits l negations are necessary and sufficient. In this paper it is proven that this situation changes when polynomial size circuit families of constant depth are considered: l negations are no longer sufficient. For threshold circuits it is proven that there are Boolean functions computable in constant depth $(TC^0 )$ such that no such threshold circuit containing $o(n^\epsilon )$, for all $\epsilon > 0$, negations can compute them. There is a matching upper bound: for any $\epsilon > 0$, everything computable by constant depth threshold circuits can be computed by constant depth threshold circuits using $n^\epsilon $ negations asymptotically. There are also tight bounds for constant depth, unbounded fan-in circuits $(AC^0 ):{n / {\log ^r }}n$, for any r, negations are sufficient, and $\Omega ({n / {\log ^r n}})$, for some r, are necessary.