We introduce and study the following natural total search problem, which we call the {\it heavy element avoidance} (Heavy Avoid) problem: for a distribution on $N$ bits specified by a Boolean circuit sampling it, and for some parameter $\delta(N) \ge 1/\poly(N)$ fixed in advance, output an $N$-bit string that has ... more >>>

The classical coding theorem in Kolmogorov complexity states that if an $n$-bit string $x$ is sampled with probability $\delta$ by an algorithm with prefix-free domain then K$(x) \leq \log(1/\delta) + O(1)$. In a recent work, Lu and Oliveira [LO21] established an unconditional time-bounded version of this result, by showing that ... more >>>

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first superlinear lower bound against comparator circuits was proved only recently by Gál and Robere (ITCS 2020), who established ... more >>>

A probabilistic representation of a string $x \in \{0,1\}^n$ is given by the code of a randomized algorithm that outputs $x$ with high probability [Oliveira, ICALP 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ... more >>>

We develop a general framework that characterizes strong average-case lower bounds against circuit classes $\mathcal{C}$ contained in $\mathrm{NC}^1$, such as $\mathrm{AC}^0[\oplus]$ and $\mathrm{ACC}^0$. We apply this framework to show:

- Generic seed reduction: Pseudorandom generators (PRGs) against $\mathcal{C}$ of seed length $\leq n -1$ and error $\varepsilon(n) = n^{-\omega(1)}$ can ... more >>>

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.

1. A new pseudorandom generator and its consequences: We build on techniques developed to ... more >>>

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be ... more >>>

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function $f$ can be computed by a Boolean circuit of size at most $\theta$, for a given parameter $\theta$. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a ... more >>>

A polynomial threshold function (PTF) is defined as the sign of a polynomial $p\colon\bool^n\to\mathbb{R}$. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.

Satisfiability (#SAT). We give the first zero-error randomized algorithm ... more >>>

We show how the classical Nisan-Wigderson (NW) generator [Nisan & Wigderson, 1994] yields a nontrivial pseudorandom generator (PRG) for circuits with sublinearly many polynomial threshold function (PTF) gates. For the special case of a single PTF of degree $d$ on $n$ inputs, our PRG for error $\epsilon$ has the seed ... more >>>

A polynomial threshold function (PTF) of degree $d$ is a boolean function of the form $f=\mathrm{sgn}(p)$, where $p$ is a degree-$d$ polynomial, and $\mathrm{sgn}$ is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is ... more >>>