Determining the Multiplicative Complexity of Boolean Functions using SAT

Fast LTL Satisfiability Checking by SAT Solvers

Satisfiability checking for Linear Temporal Logic (LTL) is a fundamental step in checking for possible errors in LTL assertions. Extant LTL satisfiability checkers use a variety of different search procedures. With the sole exception of LTL satisfiability checking based on bounded model checking, which does not provide a complete decision procedure, LTL satisfiability checkers have not taken advantage of the remarkable progress over the past 20 years in Boolean satisfiability solving. In this paper, we propose a new LTL satisfiability-checking framework that is accelerated using a Boolean SAT …

A Polynomial Time Bounded-error Quantum Algorithm for Boolean Satisfiability

The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in $O(m^2)$ for an E3-CNF Boolean formula with $m$ clauses and in the worst case runs in $O(n^6)$ for an E3-CNF Boolean formula with $n$ inputs. The proposed algorithm maximizes …

Generalized cofactors and decomposition of Boolean satisfiability problems

We propose an approach for decomposing Boolean satisfiability problems while extending recent results of \cite{sul2} on solving Boolean systems of equations. Developments in \cite{sul2} were aimed at the expansion of functions $f$ in orthonormal (ON) sets of base functions as a generalization of the Boole-Shannon expansion and the derivation of the consistency condition for the equation $f=0$ in terms of the expansion co-efficients. In this paper, we further extend the Boole-Shannon expansion over an arbitrary set of base functions and derive the consistency condition for $f=1$. The generaliza…

An algorithm for Boolean satisfiability based on generalized orthonormal expansion

This paper proposes an algorithm for deciding consistency of systems of Boolean equations in several variables with co-efficients in the two element Boolean algebra $B_{0}=\{0,1\}$ and find all satisfying assignments. The algorithm is based on the application of a well known generalized Boole-Shannon orthonormal (ON) expansion of Boolean functions. A necessary and sufficient consistency condition for a special class of functions was developed in \cite{sule} using such an expansion. Paper \cite{sule} develops a condition for consistency of the equation $f(X)=0$ for the special classes of Boolea…

Complexity and Stop Conditions for NP as General Assignment Problems, the Travel Salesman Problem in $\mathbb{R}^2$, Knight Tour Problem and Boolean Satisfiability Problem

This paper presents stop conditions for solving General Assignment Problems (GAP), in particular for Travel Salesman Problem in an Euclidian 2D space the well known condition Jordan's simple curve and opposite condition for the Knight Tour Problem. The Jordan's simple curve condition means that a optimal trajectory must be simple curve, i.e., without crossing but for Knight Tour Problem we use the contrary, the feasible trajectory must have crossing in all cities of the tour. The paper presents the algorithms, examples and some results come from Concorde's Home page. Several problem are studie…

Boolean Satisfiability using Noise Based Logic

In this paper, we present a novel algorithm to solve the Boolean Satisfiability (SAT) problem, using noise-based logic (NBL). Contrary to what the name may suggest, NBL is not a random/fuzzy logic system. In fact, it is a completely deterministic logic system. A key property of NBL is that it allows us to apply a superposition of many input vectors to a SAT instance at the same time, circumventing a key restriction and assumption in the traditional approach to solving SAT. By exploiting the superposition property of NBL, our NBL-based SAT algorithm can determine whether an instance is SAT or n…

伪布尔编码中比较器网络的复用

Reusing Comparator Networks in Pseudo-Boolean Encodings

针对伪布尔约束转SAT编码的规模优化问题，提出基于贪心集合覆盖算法变体的比较器网络复用方法，可减小SAT实例规模并提升可求解实例数量。

Approximating the Influence of a monotone Boolean function in O(\sqrt{n}) query complexity

The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n \longrightarrow \{\pm1\}$, \else $f: \bitset^n \to \bitset$, \fi which we denote by $I[f]$. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of $(1\pm \eps)$ by performing $O(\frac{\sqrt{n}\log n}{I[f]} \poly(1/\eps)) $ queries. % \mnote{D: say something about technique?} We also prove a lower b…

Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $Γ$ and a degree bound $Δ$, we study the complexity of #CSP$_Δ(Γ)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $Γ$ and whose variables can appear at most $Δ$ times. Our main result shows that: (i) if every function in $Γ$ is affine, then #CSP$_Δ(Γ)$ is in FP for all $Δ$, (ii) otherwise, if every function in $Γ$ is in a class called IM$_2$, then for all sufficiently large $Δ$, #…

Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-grained Perspective into Boolean Constraint Satisfaction

To study the question under which circumstances small solutions can be found faster than by exhaustive search (and by how much), we study the fine-grained complexity of Boolean constraint satisfaction with size constraint exactly $k$. More precisely, we aim to determine, for any finite constraint family, the optimal running time $f(k)n^{g(k)}$ required to find satisfying assignments that set precisely $k$ of the $n$ variables to $1$. Under central hardness assumptions on detecting cliques in graphs and 3-uniform hypergraphs, we give an almost tight characterization of $g(k)$ into four regimes:…