Pseudorandom Generators for Low Sensitivity Functions

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)

Pseudorandom Number Generators and the Square Site Percolation Threshold

A select collection of pseudorandom number generators is applied to a Monte Carlo study of the two dimensional square site percolation model. A generator suitable for high precision calculations is identified from an application specific test of randomness. After extended computation and analysis, an ostensibly reliable value of pc = 0.59274598(4) is obtained for the percolation threshold.Comment: 11 pages, 6 figure

Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas

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

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 $\epsilon$-approximated by a width $w$ DNF with at most $(w\log(1/\epsilon))^{O(w)}$ 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 $n^{\tilde{O}(\log \log(n))}$ time algorithm that computes an additive $\epsilon$ 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 $n^{\exp(O(\sqrt{\log \log n}))}$.Comment: To appear in the IEEE Conference on Computational Complexity, 201

Pseudorandom Generators from Polarizing Random Walks

We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1,1]^n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1,1]^n that converges to {-1,1}^n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits