72 research outputs found
DNF Sparsification and a Faster Deterministic Counting Algorithm
Given a DNF formula on n variables, the two natural size measures are the
number of terms or size s(f), and the maximum width of a term w(f). It is
folklore that short DNF formulas can be made narrow. We prove a converse,
showing that narrow formulas can be sparsified. More precisely, any width w DNF
irrespective of its size can be -approximated by a width DNF with
at most terms.
We combine our sparsification result with the work of Luby and Velikovic to
give a faster deterministic algorithm for approximately counting the number of
satisfying solutions to a DNF. Given a formula on n variables with poly(n)
terms, we give a deterministic time algorithm
that computes an additive approximation to the fraction of
satisfying assignments of f for \epsilon = 1/\poly(\log n). The previous best
result due to Luby and Velickovic from nearly two decades ago had a run-time of
.Comment: To appear in the IEEE Conference on Computational Complexity, 201
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs
We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length O~( log^3 n ).
The previously best known seed length for this model is n^{1/2+o(1)} due to Impagliazzo, Meka, and Zuckerman (FOCS\u2712). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM\u2713) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any f : {0,1}^n -> {0,1} computed by such a branching program, and k in [n], sum_{|s|=k} |hat{f}(s)| < n^2 * (O(log n))^k,
where f(x) = sum_s hat{f}(s) (-1)^ is the standard Fourier transform over Z_2^n. The base O(log n) of the Fourier growth is tight up to a factor of log log n
Better Pseudorandom Generators from Milder Pseudorandom Restrictions
We present an iterative approach to constructing pseudorandom generators,
based on the repeated application of mild pseudorandom restrictions. We use
this template to construct pseudorandom generators for combinatorial rectangles
and read-once CNFs and a hitting set generator for width-3 branching programs,
all of which achieve near-optimal seed-length even in the low-error regime: We
get seed-length O(log (n/epsilon)) for error epsilon. Previously, only
constructions with seed-length O(\log^{3/2} n) or O(\log^2 n) were known for
these classes with polynomially small error.
The (pseudo)random restrictions we use are milder than those typically used
for proving circuit lower bounds in that we only set a constant fraction of the
bits at a time. While such restrictions do not simplify the functions
drastically, we show that they can be derandomized using small-bias spaces.Comment: To appear in FOCS 201
Non-Malleable Codes for Small-Depth Circuits
We construct efficient, unconditional non-malleable codes that are secure
against tampering functions computed by small-depth circuits. For
constant-depth circuits of polynomial size (i.e. tampering
functions), our codes have codeword length for a -bit
message. This is an exponential improvement of the previous best construction
due to Chattopadhyay and Li (STOC 2017), which had codeword length
. Our construction remains efficient for circuit depths as
large as (indeed, our codeword length remains
, and extending our result beyond this would require
separating from .
We obtain our codes via a new efficient non-malleable reduction from
small-depth tampering to split-state tampering. A novel aspect of our work is
the incorporation of techniques from unconditional derandomization into the
framework of non-malleable reductions. In particular, a key ingredient in our
analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC
2013), a derandomization of the influential switching lemma from circuit
complexity; the randomness-efficiency of this switching lemma translates into
the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure
Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas
We give an explicit pseudorandom generator (PRG) for read-once AC^0, i.e., constant-depth read-once formulas over the basis {wedge, vee, neg} with unbounded fan-in. The seed length of our PRG is O~(log(n/epsilon)). Previously, PRGs with near-optimal seed length were known only for the depth-2 case [Gopalan et al., 2012]. For a constant depth d > 2, the best prior PRG is a recent construction by Forbes and Kelley with seed length O~(log^2 n + log n log(1/epsilon)) for the more general model of constant-width read-once branching programs with arbitrary variable order [Michael A. Forbes and Zander Kelley, 2018]. Looking beyond read-once AC^0, we also show that our PRG fools read-once AC^0[oplus] with seed length O~(t + log(n/epsilon)), where t is the number of parity gates in the formula.
Our construction follows Ajtai and Wigderson\u27s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989]. We assume by recursion that we already have a PRG for depth-d AC^0 formulas. To fool depth-(d + 1) AC^0 formulas, we use the given PRG, combined with a small-bias distribution and almost k-wise independence, to sample a pseudorandom restriction. The analysis of Forbes and Kelley [Michael A. Forbes and Zander Kelley, 2018] shows that our restriction approximately preserves the expectation of the formula. The crux of our work is showing that after poly(log log n) independent applications of our pseudorandom restriction, the formula simplifies in the sense that every gate other than the output has only polylog n remaining children. Finally, as the last step, we use a recent PRG by Meka, Reingold, and Tal [Meka et al., 2019] to fool this simpler formula
Fourier Growth of Structured ??-Polynomials and Applications
We analyze the Fourier growth, i.e. the L? Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F?-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators.
Our main structural results on Fourier growth are as follows:
- We show that any symmetric degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013].
- We show that any read-? degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? (k ? d)^{O(k)}.
- We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds.
Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F?-polynomials
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