计算机科学 ›› 2020, Vol. 47 ›› Issue (5): 32-37.doi: 10.11896/jsjkx.190400018

• 理论计算机科学 • 上一篇    下一篇

一种布尔公式的代数逻辑约化新方法

刘江, 周鸿昊   

  1. 中国科学院重庆绿色智能技术研究院高性能计算应用研究中心 重庆400714
    中国科学院大学 北京100049
  • 收稿日期:2019-03-02 出版日期:2020-05-15 发布日期:2020-05-19
  • 通讯作者: 刘江(liujiang@cigit.ac.cn)
  • 基金资助:
    国家自然科学基金(61672488)

New Algebraic Logic Reduction Method for Boolean Formula

LIU Jiang, ZHOU Hong-hao   

  1. High Performance Computing Application Research Center,Chongqing Institute of Green and Intelligent Technology,Chinese Academy of ences,Chongqing 400714,China
    University of Chinese Academy of Sciences,Beijing 100049,China
  • Received:2019-03-02 Online:2020-05-15 Published:2020-05-19
  • About author:LIU Jiang,born in 1979,Ph.D,associate professor,is a member of China Computer Federation.His main research interests include computability theory,formal methods and computer algorithms.
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (61672488)

摘要: 布尔可满足问题是最早被证明的NP完全问题之一,1-in-3-SAT问题是一个NP完全的布尔可满足子类问题。1-in-3-SAT的计算复杂度取决于对应公式的变量以及子句的个数。将1-in-3公式归约为一个变量数或者子句数更少的1-in-3公式,是提高1-in-3-SAT问题求解效率的一个关键。基于一个新的范式形式——XCNF,针对1-in-3-SAT问题提出一种新的代数逻辑约化方法,用于在多项式时间内约减一个1-in-3公式的变量数和子句数。所提算法的主要思想为:首先将1-in-3公式转化为XCNF公式,然后尝试找出XCNF公式中的X-纯文字,并利用X-纯文字法则对1-in-3公式中相应的布尔变量赋值,最后得到一个约减公式,该约减公式与原公式的1-in-3可满足性等价。

关键词: NP完全问题, 布尔可满足性问题, 1-in-3-SAT, XCNF, X-纯文字

Abstract: Boolean satisfiability problem is one of the earliest proven NP complete problem.1-in-3-SAT problem is an NP complete subclass of Boolean satisfiability problem.The computational complexity of 1-in-3-SAT depends on the number of the variables and clauses in the formula.How to reduce the 1-in-3 formula to one with less variables or clauses is the key to improve the efficiency of solving 1-in-3-SAT.Based on a new type of normal form-XCNF,this paper proposes a new algebraic logic reduction method to reduce the number of variables and clauses in polynomial time.The main idea is as follows.First,the method transforms the 1-in-3 formula into a XCNF formula,then tries to find out the X pure literal in the XCNF formula and assign the corresponding Boolean variable in the 1-in-3 formula with X pure literal rule.At last,a reduced formula which has the same 1-in-3 satisfiability with the original one can be obtained.

Key words: NP complete problem, Boolean satisfiability problem, 1-in-3-SAT, XCNF, X pure literal

中图分类号: 

  • TP311
[1] SCHAEFER T J.The complexity of satisfiability problems[C]//Proceedings of the Tenth Annual ACM Symposium on Theory of Computing.New York:ACM,1978:216-226.
[2] DAVIS M,PUTNAM H.A computing procedure for quantification theory [J].Journal of the ACM,1960,7(3):201-215.
[3] DAVIS M,LOGEMANN G,LOVELAND D.A machine pro-gram for theorem proving [J].Communications of the ACM,1962,5(7):394-397.
[4] MARQUES-SILVA J P,SAKALLAH K A.GRASP:A search algorithm for propositional satisfiability[J].IEEE Transactions on Computers,1999,48(5):506-521.
[5] LIANG J H,GANESH V,ZULKOSKI E,et al.Understanding VSIDS branching heuristics in conflict driven clause learning SAT solvers[C]//Haifa Verification Conference.Cham:Sprin-ger,2015:225-241.
[6] CHENG R,ZHOU C L,XU N,et al.Comprehensive analysis of restart strategies of cdcl sat solver[J].Journal of Computer-Aided Design and Computer Graphics,2018,30(6):1136-1144.
[7] HOOS H H,STUTZLE T.Local search algorithms for SAT:an empirical evaluation [J].Journal of Automated Reasoning,2000,24(4):421-481.
[8] BALINT A,FROHLICH A.Improving stochastic local searchfor SAT with a new probability distribution[C]//International Conference on Theory and Applications of Satisfiability Testing.Heidelberg:Springer,2010:10-15.
[9] HONG J K,ZHANG Z H,XU G P.SAT local search algorithm based on enhanced probability controlling strategies[J].Computer Engineering and Applications,2017,53(14):56-60,110.
[10] SHANG Y,WAH B W.A discrete Lagrangian-based global-search method for solving satisfiability problems[J].Journal of global optimization,1998,12(1):61-99.
[11] GU J.Optimization algorithms for the satisfiability (SAT)problem[C]//Advances in Optimization and Approximation.Boston:Springer,1994:72-154.
[12] BRAUNSTEIN A,ZECCHINA R.Survey and belief propagation on random k-SAT[C]//International Conference on Theory and Applications of Satisfiability Testing.Berlin:Springer,2003:519-528.
[13] WANG F,ZHOU Y R,YE L.Ant colony algorithm combined with survey propagation for satisfiability problem[J].Computer Science,2012,39(4):227-231.
[14] WANG X F,XU D Y,JIANG J L,et al.Sufficient conditions for convergence of the survey propagation algorithm[J].Science China Information Sciences,2017,47(12):1646-1661.
[15] FANG C,LIU J.A linear algebra formulation for boolean satisfiability testing[J/OL].http://arxiv.org/abs/1701.02401.
[16] LIU J,ZHOU H H.A development of laf for satisfying assignments search[C]//2019 IEEE 3rd Information Technology,Networking,Electronic and Automation Control Conference (ITNEC).IEEE,2019:719-726.
[17] PATRASCU M,WILLIAMS R.On the possibility of fasterSAT algorithms[C]//Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms.Austin:Society for Industrial and Applied Mathematics,2010:1065-1075.
[18] KARP R M.Reducibility among combinatorial problems[M]//Complexity of computer computations.Boston:Springer,1972:85-103.
[19] LAMACCHIA B,ODLYZKO A.Solving large sparse linear systems over finite fields[C]//Conference on the Theory and Application of Cryptography.Berlin:Springer,1990:109-133.
[1] 姜新文. 哈密顿图判定问题的多项式时间算法[J]. 计算机科学, 2020, 47(7): 8-20.
[2] 周 旭,李肯立,乐光学,朱开乐. 一种加群Z上离散对数问题的DNA计算算法[J]. 计算机科学, 2012, 39(4): 232-235.
[3] 张琨 王珩 刘凤玉. 一种时延约束的多点到多点组播路由启发式算法[J]. 计算机科学, 2005, 32(4): 107-109.
[4] 王元珍 裴小兵. 基于Skowron分明矩阵的快速约简算法[J]. 计算机科学, 2005, 32(4): 42-44.
[5] 张静 汤红波 李鸥 胡捍英. 单播和多播QoS路由问题研究及解决方法[J]. 计算机科学, 2005, 32(3): 36-38.
[6] 李燕 王秀峰. DNA计算方法[J]. 计算机科学, 2004, 31(5): 142-143.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
[1] 雷丽晖,王静. 可能性测度下的LTL模型检测并行化研究[J]. 计算机科学, 2018, 45(4): 71 -75 .
[2] 孙启,金燕,何琨,徐凌轩. 用于求解混合车辆路径问题的混合进化算法[J]. 计算机科学, 2018, 45(4): 76 -82 .
[3] 张佳男,肖鸣宇. 带权混合支配问题的近似算法研究[J]. 计算机科学, 2018, 45(4): 83 -88 .
[4] 伍建辉,黄中祥,李武,吴健辉,彭鑫,张生. 城市道路建设时序决策的鲁棒优化[J]. 计算机科学, 2018, 45(4): 89 -93 .
[5] 史雯隽,武继刚,罗裕春. 针对移动云计算任务迁移的快速高效调度算法[J]. 计算机科学, 2018, 45(4): 94 -99 .
[6] 周燕萍,业巧林. 基于L1-范数距离的最小二乘对支持向量机[J]. 计算机科学, 2018, 45(4): 100 -105 .
[7] 刘博艺,唐湘滟,程杰仁. 基于多生长时期模板匹配的玉米螟识别方法[J]. 计算机科学, 2018, 45(4): 106 -111 .
[8] 耿海军,施新刚,王之梁,尹霞,尹少平. 基于有向无环图的互联网域内节能路由算法[J]. 计算机科学, 2018, 45(4): 112 -116 .
[9] 崔琼,李建华,王宏,南明莉. 基于节点修复的网络化指挥信息系统弹性分析模型[J]. 计算机科学, 2018, 45(4): 117 -121 .
[10] 王振朝,侯欢欢,连蕊. 抑制CMT中乱序程度的路径优化方案[J]. 计算机科学, 2018, 45(4): 122 -125 .