Related papers: CNFs and DNFs with Exactly $k$ Solutions

We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k >=…

We determine the thresholds for the number of variables, number of clauses, number of clause intersection pairs and the maximum clause degree of a k-CNF formula that guarantees satisfiability under the assumption that every two clauses…

Propositional model counting} (#SAT), i.e., counting the number of satisfying assignments of a propositional formula, is a problem of significant theoretical and practical interest. Due to the inherent complexity of the problem, approximate…

We consider a basic question on the expressiveness of $k$-CNF formulas: How well can $k$-CNF formulas capture threshold functions? Specifically, what is the largest number of assignments (of Hamming weight $t$) accepted by a $k$-CNF formula…

Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f). It is folklore that short DNF formulas can be made narrow. We prove a converse, showing that…

A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. A CNF formula without any conflicts is satisfiable. The Lovasz Local Lemma implies that a…

Inspired by a recent article by Anthony Zaleski and Doron Zeilberger, we investigate the question of determining the largest k for which there exists boolean formulas in disjunctive normal form (DNF) with n variables, none of whose…

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version…

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the…

Given a CNF formula F on n variables, the problem of model counting or #SAT is to compute the number of satisfying assignments of F . Model counting is a fundamental but hard problem in computer science with varied applications. Recent…

Model counting is a fundamental task that involves determining the number of satisfying assignments to a logical formula, typically in conjunctive normal form (CNF). While CNF model counting has received extensive attention over recent…

$ $In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In $k$-clustering, opening $k$ facilities induces an…

We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of…

In 1992 Mansour proved that every size-$s$ DNF formula is Fourier-concentrated on $s^{O(\log\log s)}$ coefficients. We improve this to $s^{O(\log\log k)}$ where $k$ is the read number of the DNF. Since $k$ is always at most $s$, our bound…

During the last decade, deep neural networks (DNN) have demonstrated impressive performances solving a wide range of problems in various domains such as medicine, finance, law, etc. Despite their great performances, they have long been…

Given a CNF formula $\varphi$ with clauses $C_1, \dots, C_m$ over a set of variables $V$, a truth assignment $\mathbf{a} : V \to \{0, 1\}$ generates a binary sequence $\sigma_\varphi(\mathbf{a})=(C_1(\mathbf{a}), \ldots, C_m(\mathbf{a}))$,…

Circuits in deterministic decomposable negation normal form (d-DNNF) are representations of Boolean functions that enable linear-time model counting. This paper strengthens our theoretical knowledge of what classes of functions can be…

We call a CNF formula linear if any two clauses have at most one variable in common. Let m(k) be the largest integer m such that any linear k-CNF formula with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2) k^4…

Given a CNF formula and a weight for each assignment of values to variables, two natural problems are weighted model counting and distribution-aware sampling of satisfying assignments. Both problems have a wide variety of important…

Approximate model counting is the task of approximating the number of solutions to an input Boolean formula. The state-of-the-art approximate model counter for formulas in conjunctive normal form (CNF), ApproxMC, provides a scalable means…