Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1406.4712

An algorithm for Boolean satisfiability based on generalized orthonormal expansion

Virendra Sule
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 Boolean functions 1) $f$ in $B(Φ(X))$ for an ON set $Φ$ of Boolean functions in $X$ over a general Boolean algebra $B$ and 2) $f$ in $B(X_{2})(Φ(X_{1}))$. The present paper addresses the problem of obtaining the consistency conditions for arbitrary Boolean functions in $B_{0}(X)$. Next, the consistency for a single equation is shown equivalent to another system of Boolean equations which involves the ON functions and characterizes all solutions. This result is then extended for Boolean systems in several variables over the algebra $B_{0}=\{0,1\}$ which does not convert the system into a single equation. This condition leads to the algorithm for computing all solutions of the Boolean system without using analogous resolution and determine satisfiability. For special systems defined by CNF formulas this algorithm results into an extension of the DPLL algorithm in which the \emph{splitting rule} is generalized to several variables in terms of ON terms in the sense that splitting of CNF set in a single variable $x$ is equivalent to ON terms $x,x'$.

Explore related subjects

cs.DSmath.LO

Keep this discovery

BibTeXRIS

Virendra Sule. 2014-07-15. An algorithm for Boolean satisfiability based on generalized orthonormal expansion. https://arxiv.org/abs/1406.4712

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗