1,623 research outputs found
Challenges in computational lower bounds
We draw two incomplete, biased maps of challenges in computational complexity
lower bounds
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. Such sign representations are well-studied in computer
science and have applications to computational complexity and computational
learning theory. In this work, we present a systematic study of tradeoffs
between degree and sparsity of sign representations through the lens of the
parity function. We attempt to prove bounds that hold for any choice of set
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . We show
that a tradeoff exists between sparsity and degree, by exhibiting a sign
representation that has higher degree but lower sparsity. We show a lower bound
of on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . The main tool used is Descartes'
Rule of Signs, a classical result in algebra, relating the sparsity of a
polynomial to its number of real roots. As an application, we use bounds on
sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with
a Threshold Gate at the top. We use this to give a simple proof that such
circuits need size to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
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
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