Martin Dietzfelbinger, Miroslaw Kutylowski, Rüdiger Reischuk

It was shown some years ago that the computation time for many important

Boolean functions of n arguments on concurrent-read exclusive-write

parallel random-access machines

(CREW PRAMs) of unlimited size is at least f(n) = 0.72 log n.

On the other hand, it ...
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Vince Grolmusz, Gábor Tardos

Modular gates are known to be immune for the random

restriction techniques of Ajtai; Furst, Saxe, Sipser; and Yao and

Hastad. We demonstrate here a random clustering technique which

overcomes this difficulty and is capable to prove generalizations of

several known modular circuit lower bounds of Barrington, Straubing,

Therien; Krause ...
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Stasys Jukna

We consider a general model of monotone circuits, which

we call d-local. In these circuits we allow as gates:

(i) arbitrary monotone Boolean functions whose minterms or

maxterms (or both) have length at most <i>d</i>, and

(ii) arbitrary real-valued non-decreasing functions on ...
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Marco Cadoli, Francesco Donini, Paolo Liberatore, Marco Schaerf

In this paper we study the problem of approximating a boolean

function using the Hamming distance as the approximation measure.

Namely, given a boolean function f, its k-approximation is the

function f^k returning true on the same points in which f does,

plus all points whose Hamming distance from the ...
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Ilias Diakonikolas, Homin Lee, Kevin Matulef, Krzysztof Onak, Ronitt Rubinfeld, Rocco Servedio, Andrew Wan

We describe a general method for testing whether a function on n input variables has a concise representation. The approach combines ideas from the junta test of Fischer et al. with ideas from learning theory, and yields property testers that make poly(s/epsilon) queries (independent of n) for Boolean function classes ... more >>>

Stasys Jukna, Georg Schnitger

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained

by setting all *-entries to constants 0 or 1. A system of semi-linear

equations over GF(2) has the form Mx=f(x), where M is a completion of

A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate ...
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Arkadev Chattopadhyay, Avi Wigderson

We study solution sets to systems of generalized linear equations of the following form:

$\ell_i (x_1, x_2, \cdots , x_n)\, \in \,A_i \,\, (\text{mod } m)$,

where $\ell_1, \ldots ,\ell_t$ are linear forms in $n$ Boolean variables, each $A_i$ is an arbitrary subset of $\mathbb{Z}_m$, and $m$ is a composite ...
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Evgeny Demenkov, Alexander Kulikov

A Boolean function $f \colon \mathbb{F}^n_2 \rightarrow \mathbb{F}_2$ is called an affine disperser for sources of dimension $d$, if $f$ is not constant on any affine subspace of $\mathbb{F}^n_2$ of dimension at least $d$. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for $d=o(n)$. The main ... more >>>

Anna Gal, Jing-Tang Jang

Spira showed that any Boolean formula of size $s$ can be simulated in depth $O(\log s)$. We generalize Spira's theorem and show that any Boolean circuit of size $s$ with segregators of size $f(s)$ can be simulated in depth $O(f(s)\log s)$. If the segregator size is at least $s^{\varepsilon}$ for ... more >>>

Ilya Volkovich

We extend the line of research initiated by Fortnow and Klivans \cite{FortnowKlivans09} that studies the relationship between efficient learning algorithms and circuit lower bounds. In \cite{FortnowKlivans09}, it was shown that if a Boolean circuit class $\mathcal{C}$ has an efficient \emph{deterministic} exact learning algorithm, (i.e. an algorithm that uses membership and ... more >>>

Igor Carboni Oliveira, Rahul Santhanam

We consider $\cal C$-compression games, a hybrid model between computational and communication complexity. A $\cal C$-compression game for a function $f \colon \{0,1\}^n \to \{0,1\}$ is a two-party communication game, where the first party Alice knows the entire input $x$ but is restricted to use strategies computed by $\cal C$-circuits, ... more >>>

Benjamin Rossman, Rocco Servedio, Li-Yang Tan

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ ... more >>>

Xi Chen, Igor Carboni Oliveira, Rocco Servedio

Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes ... more >>>

Magnus Gausdal Find, Alexander Golovnev, Edward Hirsch, Alexander Kulikov

We consider Boolean circuits over the full binary basis. We prove a $(3+\frac{1}{86})n-o(n)$ lower bound on the size of such a circuit for an explicitly defined predicate, namely an affine disperser for sublinear dimension. This improves the $3n-o(n)$ bound of Norbert Blum (1984). The proof is based on the gate ... more >>>

Alexander Golovnev, Alexander Kulikov

In this paper we motivate the study of Boolean dispersers for quadratic varieties by showing that an explicit construction of such objects gives improved circuit lower bounds. An $(n,k,s)$-quadratic disperser is a function on $n$ variables that is not constant on any subset of $\mathbb{F}_2^n$ of size at least $s$ ... more >>>

Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have ... more >>>

Krishnamoorthy Dinesh, Samir Otiv, Jayalal Sarma

For a Boolean function $f:\{0,1\}^n \to \{0,1\}$ computed by a circuit $C$ over a finite basis $\cal{B}$, the energy complexity of $C$ (denoted by $\mathbf{EC}_{{\cal B}}(C)$) is the maximum over all inputs $\{0,1\}^n$ the numbers of gates of the circuit $C$ (excluding the inputs) that output a one. Energy Complexity ... more >>>

Jiatu Li, Tianqi Yang

Proving circuit lower bounds has been an important but extremely hard problem for decades. Although one may show that almost every function $f:\mathbb{F}_2^n\to\mathbb{F}_2$ requires circuit of size $\Omega(2^n/n)$ by a simple counting argument, it remains unknown whether there is an explicit function (for example, a function in $NP$) not computable ... more >>>

Mason DiCicco, Vladimir Podolskii, Daniel Reichman

A nearest neighbor representation of a Boolean function $f$ is a set of vectors (anchors) labeled by $0$ or $1$ such that $f(x) = 1$ if and only if the closest anchor to $x$ is labeled by $1$. This model was introduced by Hajnal, Liu, and Turán (2022), who studied ... more >>>

Sravanthi Chede, Leroy Chew, Anil Shukla

In this paper we present a new proof system framework CLIP (Cumulation Linear Induction Proposition) for propositional model counting. A CLIP proof firstly involves a circuit, calculating the cumulative function (or running count) of models counted up to a point, and secondly a propositional proof arguing for the correctness of ... more >>>