1 research outputs found
Addition is exponentially harder than counting for shallow monotone circuits
Let denote the Boolean function which takes as input strings of
bits each, representing numbers in
, and outputs 1 if and only if Let THR denote a monotone unweighted threshold gate,
i.e., the Boolean function which takes as input a single string and outputs if and only if . We refer
to circuits that are composed of THR gates as monotone majority circuits.
The main result of this paper is an exponential lower bound on the size of
bounded-depth monotone majority circuits that compute . More
precisely, we show that for any constant , any depth- monotone
majority circuit computing must have size
. Since can be computed by a single
monotone weighted threshold gate (that uses exponentially large weights), our
lower bound implies that constant-depth monotone majority circuits require
exponential size to simulate monotone weighted threshold gates. This answers a
question posed by Goldmann and Karpinski (STOC'93) and recently restated by
Hastad (2010, 2014). We also show that our lower bound is essentially best
possible, by constructing a depth-, size- monotone majority
circuit for .
As a corollary of our lower bound, we significantly strengthen a classical
theorem in circuit complexity due to Ajtai and Gurevich (JACM'87). They
exhibited a monotone function that is in AC but requires super-polynomial
size for any constant-depth monotone circuit composed of unbounded fan-in AND
and OR gates. We describe a monotone function that is in depth- AC but
requires exponential size monotone circuits of any constant depth, even if the
circuits are composed of THR gates