The satisfiability(SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiable CNF ...The satisfiability(SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiable CNF formula. A new formulation, the Universal SAT problem model, which transforms the SAT problem on Boofean space into an optimization problem on real space has been developed. Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the Universal SAT problem. In this paper, we prove that, when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio β<1, Newton's method has a convergence ratio of order two, and the convergence ratio of the coordinate descent method is approximately (1-β/m) for the Universal SAT problem with m variables. An algorithm based on the coordinate descent method for the Universal SAT problem is also presented in this paper.展开更多
Recently algorithms for solving propositional satisfiability problem,or SAT, have aroused great illterest, and more attention has been paid to trans-formation problem solving. The commonly used transformation is repre...Recently algorithms for solving propositional satisfiability problem,or SAT, have aroused great illterest, and more attention has been paid to trans-formation problem solving. The commonly used transformation is representationtransform, but since its ifltermediate computing procedure is a black box from theviewpoint of the original problem, this aPproach has many limitations. In this paper, a new approach called algorithm transform is proposed and applied to solvingSAT by Wu's method, a general algorithm for solving polynomial equations. By es-tablishing the correspondellce between the primitive operation in Wu's method andclause resolution in SAT, it is shown that Wu's method, when used for solving SAT,is primarily a restricted clause resolution procedure. While Wu's method illtroduceselltirely new concepts, e.g. characteristic set of clauses, to resolution procedure, thecomplexity result of resolution procedure suggests an exponential lower bound toWu's method for solving general polynomial equations. Moreover, this algorithmtransform can help achieve a more efficiellt imp1ementation of Wu's method since itcan avoid the complex manipulation of polynomials and can make the best use ofdomain specific knowledge.展开更多
基金NSERC Strategic Grant MEF0045793NSERC Research Grant OGP0046423.
文摘The satisfiability(SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient optimization algorithms for finding a solution for a satisfiable CNF formula. A new formulation, the Universal SAT problem model, which transforms the SAT problem on Boofean space into an optimization problem on real space has been developed. Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the Universal SAT problem. In this paper, we prove that, when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio β<1, Newton's method has a convergence ratio of order two, and the convergence ratio of the coordinate descent method is approximately (1-β/m) for the Universal SAT problem with m variables. An algorithm based on the coordinate descent method for the Universal SAT problem is also presented in this paper.
文摘Recently algorithms for solving propositional satisfiability problem,or SAT, have aroused great illterest, and more attention has been paid to trans-formation problem solving. The commonly used transformation is representationtransform, but since its ifltermediate computing procedure is a black box from theviewpoint of the original problem, this aPproach has many limitations. In this paper, a new approach called algorithm transform is proposed and applied to solvingSAT by Wu's method, a general algorithm for solving polynomial equations. By es-tablishing the correspondellce between the primitive operation in Wu's method andclause resolution in SAT, it is shown that Wu's method, when used for solving SAT,is primarily a restricted clause resolution procedure. While Wu's method illtroduceselltirely new concepts, e.g. characteristic set of clauses, to resolution procedure, thecomplexity result of resolution procedure suggests an exponential lower bound toWu's method for solving general polynomial equations. Moreover, this algorithmtransform can help achieve a more efficiellt imp1ementation of Wu's method since itcan avoid the complex manipulation of polynomials and can make the best use ofdomain specific knowledge.
