4,070 research outputs found
AC0(MOD2) lower bounds for the Boolean inner product
AC0 ◦MOD2 circuits are AC0 circuits augmented with a layer of parity gates just above the input layer. We study AC0 ◦ MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC0 ◦ MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an Ω( ˜ n 2 ) lower bound for the special case of depth-4 AC0 ◦ MOD2. Our proof of the depth-4 lower bound employs a new “moment-matching” inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match
Sums of products of polynomials in few variables : lower bounds and polynomial identity testing
We study the complexity of representing polynomials as a sum of products of
polynomials in few variables. More precisely, we study representations of the
form such that each is
an arbitrary polynomial that depends on at most variables. We prove the
following results.
1. Over fields of characteristic zero, for every constant such that , we give an explicit family of polynomials , where
is of degree in variables, such that any
representation of the above type for with requires . This strengthens a recent result of Kayal and Saha
[KS14a] which showed similar lower bounds for the model of sums of products of
linear forms in few variables. It is known that any asymptotic improvement in
the exponent of the lower bounds (even for ) would separate VP
and VNP[KS14a].
2. We obtain a deterministic subexponential time blackbox polynomial identity
testing (PIT) algorithm for circuits computed by the above model when and
the individual degree of each variable in are at most and
for any constant . We get quasipolynomial running
time when . The PIT algorithm is obtained by combining our
lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04].
To the best of our knowledge, this is the first nontrivial PIT algorithm for
this model (even for the case ), and the first nontrivial PIT algorithm
obtained from lower bounds for small depth circuits
Circuits with arbitrary gates for random operators
We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As
gates we allow arbitrary boolean functions; neither fanin nor fanout of gates
is restricted. An operator is linear if it computes n linear forms, that is,
computes a matrix-vector product y=Ax over GF(2). We prove the existence of
n-operators requiring about n^2 wires in any circuit, and linear n-operators
requiring about n^2/\log n wires in depth-2 circuits, if either all output
gates or all gates on the middle layer are linear.Comment: 7 page
Arithmetic circuits: the chasm at depth four gets wider
In their paper on the "chasm at depth four", Agrawal and Vinay have shown
that polynomials in m variables of degree O(m) which admit arithmetic circuits
of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m).
This theorem shows that for problems such as arithmetic circuit lower bounds or
black-box derandomization of identity testing, the case of depth four circuits
is in a certain sense the general case. In this paper we show that smaller
depth four circuits can be obtained if we start from polynomial size arithmetic
circuits. For instance, we show that if the permanent of n*n matrices has
circuits of size polynomial in n, then it also has depth 4 circuits of size
n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of
polynomial size. These results have potential applications to lower bounds and
deterministic identity testing, in particular for sums of products of sparse
univariate polynomials. We also give an application to boolean circuit
complexity, and a simple (but suboptimal) reduction to polylogarithmic depth
for arithmetic circuits of polynomial size and polynomially bounded degree
Quasi-polynomial Hitting-set for Set-depth-Delta Formulas
We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition
(X_1,...,X_d) of the variable indices [n] that the top product layer respects,
i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a
sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we
call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi
gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in
C respects a (unknown) partition on the variables; if Delta is even then the
product gates of the bottom-most Pi-layer are allowed to compute arbitrary
monomials.
In this work, we give a hitting-set generator for set-depth-Delta formulas
(over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta -
1}), where s is the size bound on the input set-depth-Delta formula. In other
words, we give a quasi-polynomial time blackbox polynomial identity test for
such constant-depth formulas. Previously, the very special case of Delta=3
(also known as set-multilinear depth-3 circuits) had no known sub-exponential
time hitting-set generator. This was declared as an open problem by Shpilka &
Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson
(FOCS 1995). Our work settles this question, not only for depth-3 but, up to
depth epsilon.log s / loglog s, for a fixed constant epsilon < 1.
The technique is to investigate depth-Delta formulas via depth-(Delta-1)
formulas over a Hadamard algebra, after applying a `shift' on the variables. We
propose a new algebraic conjecture about the low-support rank-concentration in
the latter formulas, and manage to prove it in the case of set-depth-Delta
formulas.Comment: 22 page
On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
Recently, Gupta et.al. [GKKS2013] proved that over Q any -variate
and -degree polynomial in VP can also be computed by a depth three
circuit of size . Over fixed-size
finite fields, Grigoriev and Karpinski proved that any
circuit that computes (or ) must be of size
[GK1998]. In this paper, we prove that over fixed-size finite fields, any
circuit for computing the iterated matrix multiplication
polynomial of generic matrices of size , must be of size
. The importance of this result is that over fixed-size
fields there is no depth reduction technique that can be used to compute all
the -variate and -degree polynomials in VP by depth 3 circuits of
size . The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size .
We also give an example of an explicit polynomial () in
VNP (not known to be in VP), for which any circuit computing
it (over fixed-size fields) must be of size . The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
circuit computing
(over any field) must be of size . To the best of our knowledge, the polynomial is the
first example of an explicit polynomial in VNP such that it requires
size depth four circuits, but no known matching
upper bound
- …