5 research outputs found
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates
We consider the class of constant depth AND/OR circuits augmented with
a layer of modular counting gates at the bottom layer, i.e circuits. We show that the following
holds for several types of gates : by adding a gate of type at
the output, it is possible to obtain an equivalent randomized depth 2
circuit of quasipolynomial size consisting of a gate of type at
the output and a layer of modular counting gates, i.e circuits. The types of gates we consider are modular
counting gates and threshold-style gates. For all of these, strong
lower bounds are known for (deterministic)
circuits
Circuits on Cylinders
We consider the computational power of constant width polynomial size cylindrical circuits and nondeterministic branching programs. We show that every function computed by a Pi_2 o MOD o AC^0 circuit can also be computed by a constant width polynomial size cylindrical nondeterministic branching program (or cylindrical circuit) and that every function computed by a constant width polynomial size cylindrical circuit belongs to ACC^0
Size and Energy of Threshold Circuits Computing Mod Functions
Abstract. Let C be a threshold logic circuit computing a Boolean function MODm : {0, 1} n → {0, 1}, where n ≥ 1 and m ≥ 2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} n to C is a multiple of m and, otherwise, C outputs "1." The function MOD2 is the so-called PARITY function, and MODn+1 is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output "1" for any input x ∈ {0, 1} n . In the paper, we prove that a very simple inequality n/(m − 1) ≤ s e holds for every circuit C computing MODm. The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MODm, and yields a lower bound e = Ω((log n − log m)/ log log n) on e if s = O(polylog(n)). We actually obtain a general result on the so-called generalized mod function, from which the result on the ordinary mod function MODm immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits
Smaller ACC0 Circuits for Symmetric Functions
What is the power of constant-depth circuits with gates, that can
count modulo ? Can they efficiently compute MAJORITY and other symmetric
functions? When is a constant prime power, the answer is well understood:
Razborov and Smolensky proved in the 1980s that MAJORITY and require
super-polynomial-size circuits, where is any prime power not
dividing . However, relatively little is known about the power of
circuits for non-prime-power . For example, it is still open whether every
problem in can be computed by depth- circuits of polynomial size and
only gates.
We shed some light on the difficulty of proving lower bounds for
circuits, by giving new upper bounds. We construct circuits computing
symmetric functions with non-prime power , with size-depth tradeoffs that
beat the longstanding lower bounds for circuits for prime power .
Our size-depth tradeoff circuits have essentially optimal dependence on and
in the exponent, under a natural circuit complexity hypothesis.
For example, we show for every that every symmetric
function can be computed with depth-3 circuits of
size, for a constant depending only on
. That is, depth- circuits can compute any symmetric
function in \emph{subexponential} size. This demonstrates a significant
difference in the power of depth- circuits, compared to other models:
for certain symmetric functions, depth- circuits require
size [H{\aa}stad 1986], and depth-
circuits (for fixed prime power ) require size
[Smolensky 1987]. Even for depth-two circuits,
lower bounds were known [Barrington Straubing Th\'erien 1990].Comment: 15 pages; abstract edited to fit arXiv requirement
Violating Constant Degree Hypothesis Requires Breaking Symmetry
The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to
study some extensions of -groups by nilpotent groups and the power of these
groups in a certain computational model. In its simplest formulation, it
establishes exponential lower bounds for circuits computing AND of unbounded arity (for
constant integers and a prime ). While it has been proved in some
special cases (including ), it remains wide open in its general form for
over 30 years.
In this paper we prove that the hypothesis holds when we restrict our
attention to symmetric circuits with being a prime. While we build upon
techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric
version of their Degree Decreasing Lemma and apply it in a highly non-trivial
way. Moreover, to establish the result we perform a careful analysis of
automorphism groups of subcircuits and
study the periodic behaviour of the computed functions.
Finally, our methods also yield lower bounds when is treated as a
function of