Two fundamental problems on directed graphs are to decide $s$-$t$ connectivity, and to estimate the behavior of random walks. Currently, there is no known algorithm for $s$-$t$ connectivity running in polynomial time and $n^{o(1)}$ space, and no known algorithm for estimating the $n$-step random walk matrix running in non-deterministic logspace. ... more >>>

A catalytic machine is a space-bounded Turing machine with additional access to a second, much larger work tape, with the caveat that this tape is full, and its contents must be preserved by the computation. Catalytic machines were defined by Buhrman et al. (STOC 2014), who, alongside many follow-up works, ... more >>>

The study of space-bounded computation has drawn extensively from ideas and results in the field of communication complexity. Catalytic Computation (Buhrman, Cleve, Koucký, Loff and Speelman, STOC 2013) studies the power of bounded space augmented with a pre-filled hard drive that can be used non-destructively during the computation. Presently, many ... more >>>

This paper revisits the study of two classical technical tools in theoretical computer science: Yao's transformation of distinguishers to next-bit predictors (FOCS 1982), and the ``reconstruction paradigm'' in pseudorandomness (e.g., as in Nisan and Wigderson, JCSS 1994). Recent works of Pyne, Raz, and Zhan (FOCS 2023) and Doron, Pyne, and ... more >>>

In the catalytic logspace ($CL$) model of (Buhrman et.~al.~STOC 2013), we are given a small work tape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. This ... more >>>

We provide compelling evidence for the potential of hardness-vs.-randomness approaches to make progress on the long-standing problem of derandomizing space-bounded computation.

Our first contribution is a derandomization of bounded-space machines from hardness assumptions for classes of uniform deterministic algorithms, for which strong (but non-matching) lower bounds can be unconditionally proved. ... more >>>

We prove that for every $\alpha \in [1,1.5]$,
$$
\text{BPSPACE}[S]\subseteq \text{TISP}[2^{S^{\alpha}},S^{3-\alpha}]
$$
where $\text{BPSPACE}[S]$ corresponds to randomized space $O(S)$ computation, and $\text{TISP}[T,S]$ to time $poly(T)$, space $O(S)$ computation. Our result smoothly interpolates between the results of (Nisan STOC 1992) and (Saks and Zhou FOCS 1995), which prove $\text{BPSPACE}[S]$ is contained ... more >>>

We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation.

Regular SOBPs of length $n$ and width $\lfloor w(n+1)/2\rfloor$ can exactly simulate general ... more >>>

Let $\mathcal{L}$ be a language that can be decided in linear space and let $\epsilon >0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$ cannot be decided by circuits of size smaller than $2^{\epsilon n}$.
We ... more >>>

We give a deterministic white-box algorithm to estimate the expectation of a read-once branching program of length $n$ and width $w$ in space
$$\tilde{O}\left(\log n+\sqrt{\log n}\cdot\log w\right).$$
In particular, we obtain an almost optimal space $\tilde{O}(\log n)$ derandomization of programs up to width $w=2^{\sqrt{\log n}}$.
Previously, ... more >>>

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of read-once regular branching programs.
We prove that every read-once regular branching program $B$ of width $w \in [1,\infty]$ with $s$ accepting states on $n$-bit inputs must have its $L_{1,k}$ bounded by
$$
\min\left\{ ... more >>>

We construct an explicit $\varepsilon$-hitting set generator (HSG) for regular ordered branching programs of length $n$ and $\textit{unbounded width}$ with a single accept state that has seed length
\[O(\log(n)(\log\log(n)+\log(1/\varepsilon))).\]
Previously, the best known seed length for regular branching programs of width $w$ with a single accept state was ... more >>>

The classic Impagliazzo--Nisan--Wigderson (INW) psesudorandom generator (PRG) (STOC `94) for space-bounded computation uses a seed of length $O(\log n \cdot \log(nwd/\varepsilon))$ to fool ordered branching programs of length $n$, width $w$, and alphabet size $d$ to within error $\varepsilon$. A series of works have shown that the analysis of the ... more >>>

A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ... more >>>

We prove that the Impagliazzo-Nisan-Wigderson (STOC 1994) pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of $\widetilde{O}(\log d + \log n \cdot \log(1/\varepsilon))$, assuming the program has only one accepting vertex in the final layer. Here, $n$ is the length of the ... more >>>