Related papers: Polynomial Bounds of CFLOBDDs against BDDs

A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined…

In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when…

Selman and Kautz's work on ``knowledge compilation'' established how approximation (strengthening and/or weakening) of a propositional knowledge-base can be used to speed up query processing, at the expense of completeness. In this…

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…

Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x_1,...,x_n) that approximates or sign-represents a given Boolean…

We investigate the width complexity of nondeterministic unitary OBDDs (NUOBDDs). Firstly, we present a generic lower bound on their widths based on the size of strong 1-fooling sets. Then, we present classically cheap functions that are…

A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the…

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection $\B\subset 2^{[n]}$ forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps…

This paper extends the work of Gottlob, Lee, and Valiant (PODS 2009)[GLV], and considers worst-case bounds for the size of the result Q(D) of a conjunctive query Q to a database D given an arbitrary set of functional dependencies. The…

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow…

In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…

Idempotent Boolean functions form a highly structured subclass of Boolean functions that is closely related to rotation symmetry under a normal-basis representation and to invariance under a fixed linear map in a polynomial basis. These…

A family $\mbox{$\cal F$}=\{F_1,\ldots,F_m\}$ of subsets of $[n]$ is said to be ordered, if there exists an $1\leq r\leq m$ index such that $n\in F_i$ for each $1\leq i\leq r$, $n\notin F_i$ for each $i>r$ and $|F_i|\leq |F_j|$ for each…

Let $\Lambda$ be the limit set of an infinite conformal iterated function system and let $F$ denote the set of fixed points of the maps. We prove that the box dimension of $\Lambda$ exists if and only if \[ \overline{\dim}_{\mathrm B} F\leq…

The size and complexity of software and hardware systems have significantly increased in the past years. As a result, it is harder to guarantee their correct behavior. One of the most successful methods for automated verification of…

We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$.…

Advances in neural architecture search, as well as explainability and interpretability of connectionist architectures, have been reported in the recent literature. However, our understanding of how to design Bayesian Deep Learning (BDL)…

Wavefunction extrapolation greatly reduces the number of self-consistent field (SCF) iterations and thus the overall computational cost of Born-Oppenheimer molecular dynamics (BOMD) that is based on the Kohn-Sham density functional theory.…

For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial…

Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field $\mathbb{F}$. The polynomial is said to be unbiased if the distribution of $f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform distribution over…