Related papers: Limitations on counting in Boolean circuits and se…
Boolean automata networks (BANs) are a generalisation of Boolean cellular automata. In such, any theorem describing the way BANs compute information is a strong tool that can be applied to a wide range of models of computation. In this…
We propose a method for exact circuit synthesis using a discrete gate set, as required for fault-tolerant quantum computing. Our approach translates the problem of synthesizing a gate specified by its unitary matrix into a boolean…
We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be…
Any monotone Boolean circuit computing the $n$-dimensional Boolean convolution requires at least $n^2$ and-gates. This precisely matches the obvious upper bound.
Neural networks are dynamical systems that compute with their dynamics. One example is the Hopfield model, forming an associative memory which stores patterns as global attractors of the network dynamics. From studies of dynamical networks…
Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for…
The problem of rectangle tiling binary arrays is defined as follows. Given an $n \times n$ array $A$ of zeros and ones and a natural number $p$, our task is to partition $A$ into at most $p$ rectangular tiles, so that the maximal weight of…
Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen…
In this paper we show that passive self-assembly in the context of the tile self-assembly model is capable of performing fuel efficient, universal computation. The tile self-assembly model is a premiere model of self-assembly in which…
Tile-based self-assembly systems are capable of universal computation and algorithmically-directed growth. Systems capable of such behavior typically make use of "glue cooperation" in which the glues on at least $2$ sides of a tile must…
We prove a result which strongly hints at the computational weakness of a model of tile assembly that has so far resisted many attempts of formal analysis or positive constructions. Specifically, we prove that, in Winfree's abstract Tile…
We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an…
We consider the fundamental problem of constructing fast circuits for the carry bit computation in binary addition. Up to a small additive constant, the carry bit computation reduces to computing an \aop, i.e., a formula of type $t_0 \land…
Boolean networks are dynamical models of disease development in which the activation levels of genes are represented by binary variables. Given a Boolean network, controls represent mutations or medical treatments that fix the activation…
We investigate the role of nondeterminism in Winfree's abstract Tile Assembly Model (aTAM), which was conceived to model artificial molecular self-assembling systems constructed from DNA. Of particular practical importance is to find tile…
Quantum annealers manufactured by D-Wave Systems, Inc., are computational devices capable of finding high-quality solutions of NP-hard problems. In this contribution, we explore the potential and effectiveness of such quantum annealers for…
(Abridged) Designing computationally efficient algorithms in the agnostic learning model (Haussler, 1992; Kearns et al., 1994) is notoriously difficult. In this work, we consider agnostic learning with membership queries for touchstone…
We study the gap between the minimum size of a Boolean circuit (DAG) and the minimum size of a formula (tree circuit) over the And-Inverter Graph (AIG) basis {AND, NOT} with free inversions. We prove that this gap is always 0 or 1 (Unit Gap…
A common lens to theoretically study neural net architectures is to analyze the functions they can approximate. However, constructions from approximation theory may be unrealistic and therefore less meaningful. For example, a common…
Model counting, a fundamental task in computer science, involves determining the number of satisfying assignments to a Boolean formula, typically represented in conjunctive normal form (CNF). While model counting for CNF formulas has…