A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL=L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan, pseudorandom generators with seed-length $O(\log^2 n)$ were constructed. Unfortunately, improving on this seed-length in general has proven challenging and seems to require new ideas.
A recent line of inquiry has suggested focusing on a particular limitation of the existing PRGs, which is that they only fool roBPs when the variables are read in a particular known order, such as $x_1<\cdots<x_n$. In comparison, existentially one can obtain logarithmic seed-length for fooling the set of polynomial-size roBPs that read the variables under any fixed unknown permutation $x_{π(1)}<\cdots<x_{π(n)}$. While recent works have established novel PRGs in this setting for subclasses of roBPs, there were no known $n^{o(1)}$ seed-length explicit PRGs for general polynomial-size roBPs in this setting.
In this work, we follow the "bounded independence plus noise" paradigm of Haramaty, Lee and Viola, and give an improved analysis in the general roBP unknown-order setting. With this analysis we obtain an explicit PRG with seed-length $O(\log^3 n)$ for polynomial-size roBPs reading their bits in an unknown order. Plugging in a recent Fourier tail bound of Chattopadhyay, Hatami, Reingold, and Tal, we can obtain a $\widetilde{O}(\log^2 n)$ seed-length when the roBP is of constant width.
In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$ and width $w$ read-once branching programs with seed length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error $\varepsilon$. It remains a central question to reduce the seed length to $O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$. However, there has been no improvement on Nisan's construction for the case $n=w$, which is most relevant to space-bounded derandomization. Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the not…
A sliding-window algorithm of window size $t$ is an algorithm whose current operation depends solely on the last $t$ symbols read. We construct pseudorandom generators (PRGs) for low-space randomized sliding-window algorithms that have access to a binary randomness source. More specifically, we lift these algorithms to the non-uniform setting of branching programs and study them as a subclass thereof that we call sliding-window branching programs (SWBPs), accordingly. For general SWBPs, given a base PRG $G_\mathrm{base}$ with seed length $d_\mathrm{base}$ that $\varepsilon_\mathrm{base}$-fools…
We show that quantum algorithms of time $T$ and space $S\ge \log T$ with unitary operations and intermediate measurements can be simulated by quantum algorithms of time $T \cdot \mathrm{poly} (S)$ and space $ {O}(S\cdot \log T)$ with unitary operations and without intermediate measurements. The best results prior to this work required either $Ω(T)$ space (by the deferred measurement principle) or $\mathrm{poly}(2^S)$ time [FR21,GRZ21]. Our result is thus a time-efficient and space-efficient simulation of algorithms with unitary operations and intermediate measurements by algorithms with unitar…
A natural model of read-once linear branching programs is a branching program where queries are $\mathbb{F}_2$ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows. - Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then pr…
Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove: (1) An exponential separation between sum of ROFs and read-$k$ formulas for some constant $k$. (2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs. Our results are based on analysis of the partial derivati…
In this paper we study syntactic branching programs of bounded repetition representing CNFs of bounded treewidth. For this purpose we introduce two new structural graph parameters $d$-pathwidth and clique preserving $d$-pathwidth denoted by $d-pw(G)$ and $d-cpw(G)$ where $G$ is a graph. We show that $2-cpw(G) \leq O(tw(G) Δ(G))$ where $tw(G)$ and $Δ(G)$ are, respectively the treewidth and maximal degree of $G$. Using this upper bound, we demonstrate that each CNF $ψ$ can be represented as a conjunction of two OBDDs of size $2^{O(Δ(ψ)*tw(ψ)^2)}$ where $tw(ψ)$ is the treewidth of the primal grap…
Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, …
