15 research outputs found
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
A Casual Tour Around a Circuit Complexity Bound
I will discuss the recent proof that the complexity class NEXP
(nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial
size. The proof will be described from the perspective of someone trying to
discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News,
September 201
A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent
We show that the permanent cannot be computed by DLOGTIME-uniform threshold
or arithmetic circuits of depth o(log log n) and polynomial size.Comment: 11 page
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists
an explicit -variate and degree polynomial such
that if any depth four circuit of bounded formal degree which computes
a polynomial of bounded individual degree , that is functionally
equivalent to , then must have size .
The motivation for their work comes from Boolean Circuit Complexity. Based on
a characterization for circuits by Yao [FOCS, 1985] and Beigel and
Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that
functions in can also be computed by algebraic
circuits (i.e., circuits of the form -- sums
of powers of polynomials) of size. Thus they argued that a
"functional" lower bound for an explicit
polynomial against circuits would imply a
lower bound for the "corresponding Boolean function" of against non-uniform
. In their work, they ask if their lower bound be extended to
circuits.
In this paper, for large integers and such that , we show that any circuit of
bounded individual degree at most that
functionally computes Iterated Matrix Multiplication polynomial
() over must have size . Since Iterated
Matrix Multiplication over is functionally in
, improvement of the afore mentioned lower bound to hold for
quasipolynomially large values of individual degree would imply a fine-grained
separation of from
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
A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent
11 pagesWe show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size