Jaccard Distance of Logical Formulas and Its Application

doi:10.3778/j.issn.1673-9418.2002004

Abstract

Abstract:

Quantifying the related concepts of propositional logic is a new method of propositional logic research. It is differe.nt from the syntax and semantics method and is a powerful tool for studying the structure of propositional sets. By using the Jaccard similarity coefficient, this paper introduces the concepts of Jaccard similarity degree, Jaccard pseudo-metric and Jaccard similarity degree representation method of the truth degree of a formulae in quantitative logic, and constructs a new kind of quantitative model of logic system L. This paper investigates the basic properties of Jaccard logic metric space[(F(S),ρJ),]which is induced by Jaccard pseudo-metric. This paper gets the following results: firstly, contradictory is the only isolated point in logic metric space[(F(S),ρJ);]secondly, every non-empty sphere in [(F(S),ρJ)]is an inconsistency theory.

Key words: Jaccard similarity coefficient, approximate reasoning, quantitative logic, logic metric space

摘要：

对命题逻辑的相关概念进行量化处理，是命题逻辑研究的一种新方法，是一种有别于语构与语义的研究方法，是研究命题集结构的一个有力工具。通过运用向量间的Jaccard相似系数，在经典逻辑系统L中给出了公式间的Jaccard相似度、Jaccard距离等概念，给出了计量逻辑学中公式真度的Jaccard相似度表示形式，建立了一种新的量化模型。讨论了由Jaccard距离导出的逻辑度量空间[(F(S),ρJ)]的一些性质。得到结论：矛盾式是逻辑度量空间[(F(S),ρJ)]中唯一的孤立点；逻辑度量空间[(F(S),ρJ)]中的非空开球是不相容理论。

关键词: Jaccard相似系数, 近似推理, 计量逻辑学, 逻辑度量空间

YU Peng. Jaccard Distance of Logical Formulas and Its Application[J]. Journal of Frontiers of Computer Science and Technology, 2020, 14(11): 1975-1980.

于鹏. 逻辑公式间的Jaccard距离及其应用[J]. 计算机科学与探索, 2020, 14(11): 1975-1980.

References

[1] Wang G J, Zhou H J. Quantitative logic[J]. Chinese Journal of Engineering Mathematics, 2009, 179(3): 226-247.
[2] Wang G J, Fu L, Song J S. Theory of truth degrees of proposi-tions in two-valued logic[J]. Science in China (Mathematics Physics Astronomy), 2002, 45(9): 1106-1116.
[3] Li B J, Wang G J. Theory of truth degrees of formulas in (L)ukasiewicz n-valued propositional logic and a limit theorem[J]. Science in China (Information Sciences), 2005, 48(6): 727-738.
[4] Li J, Wang G J. Theory of truth degrees of propositions in the logic system L*n[J]. Science in China (Information Sci-ences), 2006, 49(4): 471-483.
[5] Shi H X, Wang G J. Quantitative method for multi-value modal logics[J]. Journal of Software, 2012, 23(12): 3074-3087.时慧娴, 王国俊. 多值模态逻辑的计量化方法[J]. 软件学报, 2012, 23(12): 3074-3087.
[6] Wang G J. Axiomatic theory of truth degree for a class of first-order formulas and its application[J]. Science in China(Information Sciences), 2012, 42(5): 648-662.王国俊. 一类一阶逻辑公式中的公理化真度理论及其应用[J]. 中国科学(信息科学), 2012, 42(5): 648-662.
[7] Wang G J, Hui X J. Randomization of classical inference patterns and its application[J]. Science in China (Informa-tion Sciences), 2007, 50(6): 867-877.
[8] Zhou H J, Wang G J. Borel probabilistic and quantitative logic[J]. Science China (Information Sciences), 2011, 54(9): 1843-1854.
[9] Zhao B, Yu P. A kind of quantitative method based on Cam-berra fuzzy distance in multiple-valued logic[J]. Acta Elec-tronica Sinica, 2018, 46(10): 2305-2315.赵彬, 于鹏. 多值逻辑中基于Camberra模糊距离的计量化方法[J]. 电子学报, 2018, 46(10): 2305-2315.
[10] She Y H, Wang G J, He X L. Topological characterization of consistency of logic theories in n-valued Lukasiewicz logic Luk(n)[J]. Chinese Journal of Electronics, 2010, 19(3): 427-430.
[11] Wang Q P, Wang G J. Distribution of the symmetrical logic formulas in the[L*3]-logic metric space[J]. Chinese Journal of Computers, 2011, 34(1): 105-114.王庆平, 王国俊. 对称逻辑公式在[L*3]逻辑度量空间中的分布[J]. 计算机学报, 2011, 34(1): 105-114.
[12] Li J,?He J L. Rotation symmetric logic formulas in quantita-tive logic[J]. Fuzzy Systems and Mathematics, 2015, 29(2): 62-67.李俊, 何金龙. 计量逻辑学中的旋转对称逻辑公式[J]. 模糊系统与数学, 2015, 29(2): 62-67.
[13] Hu M D, Wang G J. Reflexive transforms on a classical logic metric space[J]. Journal of Shaanxi Normal University (Na-tural Science Edition), 2009, 37(6): 1-4.胡明娣, 王国俊. 经典逻辑度量空间上的反射变换[J]. 陕西师范大学学报(自然科学报), 2009, 37(6): 1-4.
[14] Zhou X N, Wang G J. Consistency degrees of theories in some systems of propositional fuzzy logic[J]. Fuzzy Sets and Systems, 2005, 152(2): 321-331.
[15] Zhou H J, Wang G J. A new consistency index based on deduction theorems in several logic systems[J]. Fuzzy Sets and Systems, 2006, 157(3): 427-443.
[16] She Y H, He X L, Qian Y H, et al. A quantitative approach to reasoning about incomplete knowledge[J]. Information Sciences, 2018, 451: 100-111.
[17] Li J, Deng F X. Unified theory of truth degrees in n-valued S-MTL propositional logic[J]. Acta Electronica Sinica, 2011, 39(8): 1864-1868.李骏, 邓富喜. n值S-MTL命题逻辑系统中公式真度的统一理论[J]. 电子学报, 2011, 39(8): 1864-1868.
[18] Li J, Yao J T. Theory of integral truth degrees of formula in SMTL propositional logic[J]. Acta Electronica Sinica, 2013, 41(5): 878-883.李骏, 姚锦涛. 命题逻辑系统SMTL中公式的积分真度理论[J]. 电子学报, 2013, 41(5): 878-883.
[19] Zhang Y, Liu Y D, Ji Z. Vector similarity measurement me-thod[J]. Technical Acoustics, 2009, 28(4): 532-536.张宇, 刘雨东, 计钊. 向量相似度测度方法[J]. 声学技术, 2009, 28(4): 532-536.
[20] Real R, Vargas J M. The probabilistic basis of Jaccard??s index of similarity[J]. Systematic Biology, 1996, 45(3): 380-385.
[21] Yu P, Zhao B. The Hamming distance representation and decomposition theorem of formula??s truth degree[J]. Journal of Software, 2018, 29(10): 3091-3110.于鹏, 赵彬. 公式真度的Hamming距离表示形式与分解定理[J]. 软件学报, 2018, 29(10): 3091-3110.