多值逻辑中基于Camberra模糊距离的计量化方法
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摘要
本文以模糊集间的Camberra距离为工具,给出了多值Lukasiewicz逻辑系统中公式间的Camberra-距离,Camberra-相似度与Camberra-真度的概念,讨论了Camberra-相似度与Camberra-真度的性质,证明了每一个公式φ的Camberra-真度都等于一些互不相容的公式的Camberra-真度之和.然后以Camberra-真度为依托,研究了Lukasiewicz逻辑度量空间的一些性质,证明了三值Lukasiewicz逻辑度量空间没有孤立点,以及每一个球形领域都是不相容理论等结论.为在公式集F(S)上展开程度化推理提供了一种新的方法.
In this paper,by using Camberra fuzzy distance on fuzzy set,we propose the notions of Camberra-distance,Camberra-similarity degree between two formulas and Camberra-truth degree of one formula in multiple-valued Lukasiewicz logic system,and discuss the properties of Camberra-similarity degree and Camberra-truth degree,prove that the Camberratruth degree of formula φ is equal to the sum of Camberra-truth degrees of some incompatible formulas. And then,based on Camberra-truth degree,we study some properties of Lukasiewicz logic metric space,prove that there is no isolated point,and arbitrarily sphere neighbourhood is an inconsistent theory in three valued Lukasiewicz logic metric space and so on. Which provides a newmethod for expand the grade reasoning on formula set F(S).
In this paper,by using Camberra fuzzy distance on fuzzy set,we propose the notions of Camberra-distance,Camberra-similarity degree between two formulas and Camberra-truth degree of one formula in multiple-valued Lukasiewicz logic system,and discuss the properties of Camberra-similarity degree and Camberra-truth degree,prove that the Camberratruth degree of formula φ is equal to the sum of Camberra-truth degrees of some incompatible formulas. And then,based on Camberra-truth degree,we study some properties of Lukasiewicz logic metric space,prove that there is no isolated point,and arbitrarily sphere neighbourhood is an inconsistent theory in three valued Lukasiewicz logic metric space and so on. Which provides a newmethod for expand the grade reasoning on formula set F(S).
引文
[1]周红军,王国俊. Borel型概率计量逻辑[J].中国科学:信息科学,2011,41(11):1328-1342.ZHOU Hong-jun,WANG Guo-jun. Borel probabilistic quantitative logic[J]. Science China:Information Science,2011,41(11):1328-1342.(in Chinese)
[2]周红军,折延宏.ukasiewicz命题逻辑中命题的choquet积分真度理论[J].电子学报,2013,42(12):2327-2333.ZHOU Hong-jun,SHE Yan-hong. Theory of choquet integral truth degrees of propositions inukasiew icz propositional logic[J]. Acta Electronica Sinica,2013,42(12):2327-2333.(in Chinese)
[3] WANG Guo-jun,FU Li,SONG Jian-she. Theory of truth degrees of propositions in tw o-valued logic[J]. Science in China(Series A),2002,45(9):1106-1116.
[4]LI Bi-jing,WANG Guo-jun. Theory of truth degrees of formulas inukasiew icz n-valued propositional logic and a limit theorem[J]. Science in China(Series F),2005,48(6):727-738.
[5]李俊,王国俊.逻辑系统L*n中命题的真度理论[J].中国科学(E辑),2006,36(6):631-643.LI Jun,WANG Guo-jun. Theory of truth degrees of proposition in logic sysem L*n[J]. Sciencein China(E),2006,36(6):631-643.(in Chinese)
[6]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):3083-3087.SHI Hui-xian,WANG Guo-jun. Quantitative method for multi-value modal logics[J]. Journal of Softw are,2012,23(12):3083-3087.(in Chinese)
[7]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(F),2012,42(5):648-662.WANG Guo-jun. Axiomatic theory of truth degree for a class of first-order formulas and its application[J]. Science in China(F),2012,42(5):648-662.(in Chinese)
[8]WANG Guo-jun,ZHOU Hong-jun. Quantitative logic[J].Information Science,2009,179(3):226-247.
[9]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.LI Jun,DENG Fu-gui. Unified theory of truth degrees in nvalueds-M TL propositional logic[J]. Acta Electronica Sinica,2011,39(8):1864-1868.(in Chinese)
[10]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.LI Jun,YAO Jin-tao. Theory of integral truth degrees of formula in SM TL propositional logic[J]. Acta Electronica Sinica,2013,41(5):878-883.(in Chinese)
[11]周红军.概率计量逻辑及其应用[M].北京:科学出版社,2015.
[12]雷英杰,赵杰,等.直觉模糊集理论及其应用[M].北京:科学出版社,2014.
[13]陈水利,李敬功,王向公.模糊集理论及其应用[M].北京:科学出版社,2005.
[14]Roser J B,Turquette A R. Many-Valued Logic[M]. Amsterdam:North-Holland,1952.
[15]Pavelka J. logic:I-III[J]. On Fuzzy Zeitschrift Für Mathematische Logic and Grundlagen M athematik,1979,25(2):45-52; 119-134; 447-464.
[16]Adams E W. A Prime of Probability Logic[M]. Stanford:CSLIPublications,1998.
[17] Dubois D,Prade H. Possibility theory,probability theory and multiple-valued logics:a clarification[J]. Annals of M athematics and Artificial Intelligence,2001,32(1-4):35-66.
[18]王国俊.修正的KLeene系统中的∑-α-重言式理论[J].中国科学(E辑),1998,28(2):146-152.WANG Guo-jun. The theory of∑-α-tautologies in the revised Kleene system[J]. Science in China(Series E),1998,28(2):146-152.(in Chinese)
[19]王庆平,王国俊.多值ukasiewicz逻辑公式的范数表示和计数问题[J].软件学报,2013,24(3):433-453.WANG Qing-ping,WANG Guo-jun. Normal form ofukasiew icz logic formula and related counting problems[J]. Journal of Software,2013,24(3):433-453.(in Chinese)
[20]折延宏,王国俊.三值命题逻辑系统L*3逻辑理论性态的拓扑刻画[J].数学学报,2009,52(6):1225-1235.SHE Yan-hong,WANG Guo-jun. Topological characterizations of properties of logic theories in three-valued propositional logic system L*3[J]. 2009,52(6):1225-1235.(in Chinese)
[2]周红军,折延宏.ukasiewicz命题逻辑中命题的choquet积分真度理论[J].电子学报,2013,42(12):2327-2333.ZHOU Hong-jun,SHE Yan-hong. Theory of choquet integral truth degrees of propositions inukasiew icz propositional logic[J]. Acta Electronica Sinica,2013,42(12):2327-2333.(in Chinese)
[3] WANG Guo-jun,FU Li,SONG Jian-she. Theory of truth degrees of propositions in tw o-valued logic[J]. Science in China(Series A),2002,45(9):1106-1116.
[4]LI Bi-jing,WANG Guo-jun. Theory of truth degrees of formulas inukasiew icz n-valued propositional logic and a limit theorem[J]. Science in China(Series F),2005,48(6):727-738.
[5]李俊,王国俊.逻辑系统L*n中命题的真度理论[J].中国科学(E辑),2006,36(6):631-643.LI Jun,WANG Guo-jun. Theory of truth degrees of proposition in logic sysem L*n[J]. Sciencein China(E),2006,36(6):631-643.(in Chinese)
[6]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):3083-3087.SHI Hui-xian,WANG Guo-jun. Quantitative method for multi-value modal logics[J]. Journal of Softw are,2012,23(12):3083-3087.(in Chinese)
[7]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(F),2012,42(5):648-662.WANG Guo-jun. Axiomatic theory of truth degree for a class of first-order formulas and its application[J]. Science in China(F),2012,42(5):648-662.(in Chinese)
[8]WANG Guo-jun,ZHOU Hong-jun. Quantitative logic[J].Information Science,2009,179(3):226-247.
[9]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.LI Jun,DENG Fu-gui. Unified theory of truth degrees in nvalueds-M TL propositional logic[J]. Acta Electronica Sinica,2011,39(8):1864-1868.(in Chinese)
[10]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.LI Jun,YAO Jin-tao. Theory of integral truth degrees of formula in SM TL propositional logic[J]. Acta Electronica Sinica,2013,41(5):878-883.(in Chinese)
[11]周红军.概率计量逻辑及其应用[M].北京:科学出版社,2015.
[12]雷英杰,赵杰,等.直觉模糊集理论及其应用[M].北京:科学出版社,2014.
[13]陈水利,李敬功,王向公.模糊集理论及其应用[M].北京:科学出版社,2005.
[14]Roser J B,Turquette A R. Many-Valued Logic[M]. Amsterdam:North-Holland,1952.
[15]Pavelka J. logic:I-III[J]. On Fuzzy Zeitschrift Für Mathematische Logic and Grundlagen M athematik,1979,25(2):45-52; 119-134; 447-464.
[16]Adams E W. A Prime of Probability Logic[M]. Stanford:CSLIPublications,1998.
[17] Dubois D,Prade H. Possibility theory,probability theory and multiple-valued logics:a clarification[J]. Annals of M athematics and Artificial Intelligence,2001,32(1-4):35-66.
[18]王国俊.修正的KLeene系统中的∑-α-重言式理论[J].中国科学(E辑),1998,28(2):146-152.WANG Guo-jun. The theory of∑-α-tautologies in the revised Kleene system[J]. Science in China(Series E),1998,28(2):146-152.(in Chinese)
[19]王庆平,王国俊.多值ukasiewicz逻辑公式的范数表示和计数问题[J].软件学报,2013,24(3):433-453.WANG Qing-ping,WANG Guo-jun. Normal form ofukasiew icz logic formula and related counting problems[J]. Journal of Software,2013,24(3):433-453.(in Chinese)
[20]折延宏,王国俊.三值命题逻辑系统L*3逻辑理论性态的拓扑刻画[J].数学学报,2009,52(6):1225-1235.SHE Yan-hong,WANG Guo-jun. Topological characterizations of properties of logic theories in three-valued propositional logic system L*3[J]. 2009,52(6):1225-1235.(in Chinese)