71 research outputs found

    Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

    Get PDF
    We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an ε\varepsilon-PRG for the class of size-MM depth-dd AC0\mathsf{AC}^0 circuits with seed length log(M)d+O(1)log(1/ε)\log(M)^{d+O(1)}\cdot \log(1/\varepsilon), and an ε\varepsilon-PRG for the class of SS-sparse F2\mathbb{F}_2 polynomials with seed length 2O(logS)log(1/ε)2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon). 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 AC0\mathsf{AC}^0 and sparse F2\mathbb{F}_2 polynomials

    Affine Extractors and AC0-Parity

    Get PDF
    We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem

    On the Probabilistic Degree of OR over the Reals

    Full text link
    We study the probabilistic degree over reals of the OR function on nn variables. For an error parameter ϵ\epsilon in (0,1/3), the ϵ\epsilon-error probabilistic degree of any Boolean function ff over reals is the smallest non-negative integer dd such that the following holds: there exists a distribution DD of polynomials entirely supported on polynomials of degree at most dd such that for all z{0,1}nz \in \{0,1\}^n, we have PrPD[P(z)=f(z)]1ϵPr_{P \sim D} [P(z) = f(z) ] \geq 1- \epsilon. It is known from the works of Tarui ({Theoret. Comput. Sci.} 1993) and Beigel, Reingold, and Spielman ({ Proc. 6th CCC} 1991), that the ϵ\epsilon-error probabilistic degree of the OR function is at most O(logn.log1/ϵ)O(\log n.\log 1/\epsilon). Our first observation is that this can be improved to Olog(nlog1/ϵ)O{\log {{n}\choose{\leq \log 1/\epsilon}}}, which is better for small values of ϵ\epsilon. In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials PP in the support of the distribution DD have the following special structure:P=1(1L1).(1L2)...(1Lt)P = 1 - (1-L_1).(1-L_2)...(1-L_t), where each Li(x1,...,xn)L_i(x_1,..., x_n) is a linear form in the variables x1,...,xnx_1,...,x_n, i.e., the polynomial 1P(x1,...,xn)1-P(x_1,...,x_n) is a product of affine forms. We show that the ϵ\epsilon-error probabilistic degree of OR when restricted to polynomials of the above form is Ω(loga/log2a)\Omega ( \log a/\log^2 a ) where a=log(nlog1/ϵ)a = \log {{n}\choose{\leq \log 1/\epsilon}}. Thus matching the above upper bound (up to poly-logarithmic factors)

    Pseudorandom Generators for Low Sensitivity Functions

    Get PDF
    A Boolean function is said to have maximal sensitivity s if s is the largest number of Hamming neighbors of a point which differ from it in function value. We initiate the study of pseudorandom generators fooling low-sensitivity functions as an intermediate step towards settling the sensitivity conjecture. We construct a pseudorandom generator with seed-length 2^{O(s^{1/2})} log(n) that fools Boolean functions on n variables with maximal sensitivity at most s. Prior to our work, the (implicitly) best pseudorandom generators for this class of functions required seed-length 2^{O(s)} log(n)

    Bounded Indistinguishability for Simple Sources

    Get PDF

    On the Probabilistic Degree of OR over the Reals

    Get PDF
    We study the probabilistic degree over R of the OR function on n variables. For epsilon in (0,1/3), the epsilon-error probabilistic degree of any Boolean function f:{0,1}^n -> {0,1} over R is the smallest non-negative integer d such that the following holds: there exists a distribution of polynomials Pol in R[x_1,...,x_n] entirely supported on polynomials of degree at most d such that for all z in {0,1}^n, we have Pr_{P ~ Pol}[P(z) = f(z)] >= 1- epsilon. It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the epsilon-error probabilistic degree of the OR function is at most O(log n * log(1/epsilon)). Our first observation is that this can be improved to O{log (n atop <= log(1/epsilon))}, which is better for small values of epsilon. In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution Pol have the following special structure: P(x_1,...,x_n) = 1 - prod_{i in [t]} (1- L_i(x_1,...,x_n)), where each L_i(x_1,..., x_n) is a linear form in the variables x_1,...,x_n, i.e., the polynomial 1-P(bar{x}) is a product of affine forms. We show that the epsilon-error probabilistic degree of OR when restricted to polynomials of the above form is Omega(log (n over <= log(1/epsilon))/log^2 (log (n over <= log(1/epsilon))})), thus matching the above upper bound (up to polylogarithmic factors)

    Tight Bounds on the Fourier Spectrum of AC0

    Get PDF
    We show that AC^0 circuits on n variables with depth d and size m have at most 2^{-Omega(k/log^{d-1} m)} of their Fourier mass at level k or above. Our proof builds on a previous result by Hastad (SICOMP, 2014) who proved this bound for the special case k=n. Our result improves the seminal result of Linial, Mansour and Nisan (JACM, 1993) and is tight up to the constants hidden in the Omega notation. As an application, we improve Braverman\u27s celebrated result (JACM, 2010). Braverman showed that any r(m,d,epsilon)-wise independent distribution epsilon-fools AC^0 circuits of size m and depth d, for r(m,d,epsilon) = O(log(m/epsilon))^{2d^2+7d+3}. Our improved bounds on the Fourier tails of AC^0 circuits allows us to improve this estimate to r(m,d,epsilon) = O(log(m/epsilon))^{3d+3}. In contrast, an example by Mansour (appearing in Luby and Velickovic\u27s paper - Algorithmica, 1996) shows that there is a log^{d-1}(m)log(1/epsilon)-wise independent distribution that does not epsilon-fool AC^0 circuits of size m and depth d. Hence, our result is tight up to the factor 33 in the exponent
    corecore