关于格值逻辑及其语言真值不确定性推理研究
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摘要
近四十年来,由于不确定性推理在控制系统中的广泛应用,关于不确定性推理的逻辑基础-非经典逻辑理论的研究更加引起了人们广泛的关注。自从Pavelka等人在二十世纪七十年代末初步建立命题模糊逻辑理论以来,非经典逻辑已经成为人工智能研究领域里的一个重要研究方向,同时也是不确定性推理研究的一个关键问题。一方面,非经典逻辑在机器自动证明理论、多智能体系统、程序验证等领域获得了广泛的应用。另一方面,它也丰富和发展了纯数学理论的研究。格值逻辑是一种重要的非经典逻辑,它不仅能刻画全序性信息,而且还能刻画非全序(即不可比较)的不确定性信息。本文的主要工作是建立基于格蕴涵代数的程度化的格值命题逻辑的语义和语法理论,并建立基于格值命题逻辑系统的语言真值不确定性推理理论和推理方法,同时构造相应的不确定性推理算法。
第一部分,本文得到了由格蕴涵代数所诱导的格蕴涵序半群的代数特性。引入了两个新的概念：格蕴涵n-序半群和格蕴涵p-序半群；证明了一个格蕴涵n-序半群是剩余半群,一个格蕴涵p-序半群是算术格序半群；定义了格蕴涵n-序半群之间的同态映射,并在此基础之上刻画了格蕴涵n-序半群和格蕴涵p-序半群中的滤子和s1理想的代数性质。本文还给出了格蕴涵n-序半群中s1理想的几类典型的扩张。这些讨论有望为进一步研究格蕴涵代数的性质和基于格蕴涵代数的格值逻辑提供了一种新的思路。本文也给出了语言真值格蕴涵代数L18的所有的子代数、滤子以及LI理想。
第二部分,本文在基于格蕴涵代数的格值命题逻辑系统Lp中建立了广义重言式理论。定义了系统Lp中的L-型重言式、L-型矛盾式以及α-重言式等概念,给出了几类广义重言式之间的关系定理。本文定义了L-型模糊逻辑公式集的可满足性概念,在此基础之上,给出了语义闭包算子的概念,并基于语义结论算子定义了信息的相容性、理论等概念。给出了语义闭包算子的紧致性、逻辑紧致性定理以及相应的闭包系统,同时也建立由某个给定的信息所诱导的Fp上的同余关系,并建立了相应的商代数理论。结合模糊集和L-模糊集合理论,本文也得到了由语义闭包算子所诱导的P(Fp)上的闭包算子的性质定理。
第三部分,本文建立了一种程度化的基于格蕴涵代数的格值命题逻辑系统Lp的语法理论。定义了程度化的形式证明和语法结论算子,证明了一些常用的定理,定义了基于某种信息的可证等价关系并证明了关于可证等价关系的几个重要定理。研究了所建立的语法理论与第三章所建立的语义理论的协调性问题,建立了广义演绎定理和某些特殊情况下的完备性定理。这些研究拟为构造基于格值逻辑系统的不确定性推理方法提供必要的理论准备。
第四部分,本文刻画了基于语言真值格蕴涵代数L18的格值命题逻辑系统的一些性质,并针对三类典型的不确定性推理模型,建立了基于程度化的格值命题逻辑系统的语言真值不确定性推理理论和推理方法,同时,也构造了相应的不确定性推理算法。从逻辑语义和语法上,本文也详细分析了所建立的不确定性推理方法和推理算法的合理性。
In the recent forty years, since the theory of uncertain reasoning has been applied broadly in control systems, the logical foundation of it-non-classical logic have attracted a considerable deal of attention much more, In the late of 1970's, Pavelka etc. established elementarily fuzzy propositional logic, hereafter non-classical logics have been developed into an important research direction in Artificial Intelligence field. On the one hand, non-classical logics have been applied broadly in machine automatic prove, multi-agent system and program validation and so forth research fields. On the other hand, non-classical logics have also enriched and developed the theory of pure mathematics. Lattice-valued logic is a kind of important non-classical logic, it not only can characterizes the information with linearly ordered, but also the information with nonlinearly ordered, that is to say, the incomparable information. The main aim of this paper is to establish the lattice-valued propositional logic system with degree based on lattice implication algebras, which includes the semantic theory and syntactic theory, and further establish the theory and methods of uncertain reasoning with linguistic truth-valued, and set up the corresponding algorithms.
In section one, the algebraic properties of lattice implication ordered semigroups induced by lattice implication algebras is obtained. We introduce two new concepts, i.e. lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice-ordered semigroup. We also define the notion of lattice implication n-ordered semigroup homomorphism, and based on it, we characterize the algebraic properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups. At one time, we present several typical expansions of sl ideals in lattice implication n-ordered semigroups. It should be hopeful that these investigations can provide a kind of new train of thought investigations for further researching into the properties of lattice implication algebra and the theory of lattice-valued logic based on lattice implication algebras. Likewise we find out all subalgebras, filters and LI-ideals in linguistic truth-valued lattice implication algebra L18.
In section two, the theory of generalized tautology is established in lattice-valued propositional system (?)P based on lattice implication algebras. We define several notions of L-type tautology, L-type contradiction andα-tautology etc., present several theorems about the relations among these generalized tautologies. We define the notion of satisfiability for L-fuzzy logic formulas set, based on it, define the semantic closure operation, and further define the notion of consistency of information and theory based on semantic closure operation. We also present several theorems about the compact property, logic compact property of semantic closure operation and the corresponding closure systems, establish the congruence relation on (?)P induced by certain given information and the corresponding quotient lattice implication algebra. Combining with the theory of fuzzy set and L-fuzzy set, we also obtained theorems about the properties of closure operation on P((?)p) induced by the semantic closure operation.
In section three, we establish a kind of syntactic theory with some degree in the lattice-valued propositional logic system based on lattice implication algebras. We define form proof with some degree and syntactic closure operation, prove some theorems used often, and define the notion of provable equivalent relation, prove several important theorems about provable equivalent relations. We also investigate the consistency of the syntactic theory and the semantic theory established in above chapter, and establish generalized deduction theorem and completeness theorem. These results are useful to provide necessary academic preparation for establishing uncertain reasoning methods based on lattice-valued logic system.
In section four, we characterized some properties of lattice-valued propositional logic system based on linguistic truth-valued lattice implication algebra L18, and then establish the theory and methods of uncertain reasoning with linguistic truth-valued, at one time, set up the corresponding reasoning algorithms. From the perspective of logic semantics and syntax, we also analyze the rationality of uncertain reasoning methods and algorithms which have been established ahead.
第一部分,本文得到了由格蕴涵代数所诱导的格蕴涵序半群的代数特性。引入了两个新的概念：格蕴涵n-序半群和格蕴涵p-序半群；证明了一个格蕴涵n-序半群是剩余半群,一个格蕴涵p-序半群是算术格序半群；定义了格蕴涵n-序半群之间的同态映射,并在此基础之上刻画了格蕴涵n-序半群和格蕴涵p-序半群中的滤子和s1理想的代数性质。本文还给出了格蕴涵n-序半群中s1理想的几类典型的扩张。这些讨论有望为进一步研究格蕴涵代数的性质和基于格蕴涵代数的格值逻辑提供了一种新的思路。本文也给出了语言真值格蕴涵代数L18的所有的子代数、滤子以及LI理想。
第二部分,本文在基于格蕴涵代数的格值命题逻辑系统Lp中建立了广义重言式理论。定义了系统Lp中的L-型重言式、L-型矛盾式以及α-重言式等概念,给出了几类广义重言式之间的关系定理。本文定义了L-型模糊逻辑公式集的可满足性概念,在此基础之上,给出了语义闭包算子的概念,并基于语义结论算子定义了信息的相容性、理论等概念。给出了语义闭包算子的紧致性、逻辑紧致性定理以及相应的闭包系统,同时也建立由某个给定的信息所诱导的Fp上的同余关系,并建立了相应的商代数理论。结合模糊集和L-模糊集合理论,本文也得到了由语义闭包算子所诱导的P(Fp)上的闭包算子的性质定理。
第三部分,本文建立了一种程度化的基于格蕴涵代数的格值命题逻辑系统Lp的语法理论。定义了程度化的形式证明和语法结论算子,证明了一些常用的定理,定义了基于某种信息的可证等价关系并证明了关于可证等价关系的几个重要定理。研究了所建立的语法理论与第三章所建立的语义理论的协调性问题,建立了广义演绎定理和某些特殊情况下的完备性定理。这些研究拟为构造基于格值逻辑系统的不确定性推理方法提供必要的理论准备。
第四部分,本文刻画了基于语言真值格蕴涵代数L18的格值命题逻辑系统的一些性质,并针对三类典型的不确定性推理模型,建立了基于程度化的格值命题逻辑系统的语言真值不确定性推理理论和推理方法,同时,也构造了相应的不确定性推理算法。从逻辑语义和语法上,本文也详细分析了所建立的不确定性推理方法和推理算法的合理性。
In the recent forty years, since the theory of uncertain reasoning has been applied broadly in control systems, the logical foundation of it-non-classical logic have attracted a considerable deal of attention much more, In the late of 1970's, Pavelka etc. established elementarily fuzzy propositional logic, hereafter non-classical logics have been developed into an important research direction in Artificial Intelligence field. On the one hand, non-classical logics have been applied broadly in machine automatic prove, multi-agent system and program validation and so forth research fields. On the other hand, non-classical logics have also enriched and developed the theory of pure mathematics. Lattice-valued logic is a kind of important non-classical logic, it not only can characterizes the information with linearly ordered, but also the information with nonlinearly ordered, that is to say, the incomparable information. The main aim of this paper is to establish the lattice-valued propositional logic system with degree based on lattice implication algebras, which includes the semantic theory and syntactic theory, and further establish the theory and methods of uncertain reasoning with linguistic truth-valued, and set up the corresponding algorithms.
In section one, the algebraic properties of lattice implication ordered semigroups induced by lattice implication algebras is obtained. We introduce two new concepts, i.e. lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice-ordered semigroup. We also define the notion of lattice implication n-ordered semigroup homomorphism, and based on it, we characterize the algebraic properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups. At one time, we present several typical expansions of sl ideals in lattice implication n-ordered semigroups. It should be hopeful that these investigations can provide a kind of new train of thought investigations for further researching into the properties of lattice implication algebra and the theory of lattice-valued logic based on lattice implication algebras. Likewise we find out all subalgebras, filters and LI-ideals in linguistic truth-valued lattice implication algebra L18.
In section two, the theory of generalized tautology is established in lattice-valued propositional system (?)P based on lattice implication algebras. We define several notions of L-type tautology, L-type contradiction andα-tautology etc., present several theorems about the relations among these generalized tautologies. We define the notion of satisfiability for L-fuzzy logic formulas set, based on it, define the semantic closure operation, and further define the notion of consistency of information and theory based on semantic closure operation. We also present several theorems about the compact property, logic compact property of semantic closure operation and the corresponding closure systems, establish the congruence relation on (?)P induced by certain given information and the corresponding quotient lattice implication algebra. Combining with the theory of fuzzy set and L-fuzzy set, we also obtained theorems about the properties of closure operation on P((?)p) induced by the semantic closure operation.
In section three, we establish a kind of syntactic theory with some degree in the lattice-valued propositional logic system based on lattice implication algebras. We define form proof with some degree and syntactic closure operation, prove some theorems used often, and define the notion of provable equivalent relation, prove several important theorems about provable equivalent relations. We also investigate the consistency of the syntactic theory and the semantic theory established in above chapter, and establish generalized deduction theorem and completeness theorem. These results are useful to provide necessary academic preparation for establishing uncertain reasoning methods based on lattice-valued logic system.
In section four, we characterized some properties of lattice-valued propositional logic system based on linguistic truth-valued lattice implication algebra L18, and then establish the theory and methods of uncertain reasoning with linguistic truth-valued, at one time, set up the corresponding reasoning algorithms. From the perspective of logic semantics and syntax, we also analyze the rationality of uncertain reasoning methods and algorithms which have been established ahead.
引文
[1]Baldwin, J. F. Fuzzy logic and fuzzy reasoning. Internat. J. Man-Machine Stud..1979,11: 456-480
[2]Baldwin, J.F. Anew approach to approximate reasoning using a fuzzy logic, Department of Engineering Mathematics, University of Bristol, UK,1978
[3]Baldwin, J.F. A model of fuzzy reasoning and fuzzy logic, Department of Engineering Mathematics, University of Bristol, UK,1978.
[4]Baldwin, J.F., Guild, N.C.F. Feasible algorithms for approximate reasoning using a fuzzy logic, Department of Engineering Mathematics, University of Bristol, UK,1978.
[5]Baldwin, J.F., Pilsworth, B.W. A model of fuzzy reasoning through multi-valued logic and set theory, Department of Engineering Mathematics, University of Bristol, UK, Internat. J. Man-Mach. Studies,1979
[6]Benferhat, S., Lagruea, S., Papinic, O. Reasoning with partially ordered information in a possibilistic logic framework. Fuzzy Sets and Systems.2004,144:25-41
[7]Chen, S.M, Hsiao, W.H. Bidirectional uncertainty reasoning for rule-based systems using interval-valued fuzzy sets. Fuzzy Sets and Systems.2000,113:185-203
[8]Chen, S.W., Xu, Y. Uncertainty Reasoning Based on Lattice-Valued First-Order Logic Lvfl. Proc. of 2004 IEEE International Conference on Systems, Man and Cybernetics, The Hague, The Netherlands,2004:2237-2242
[9]Chang, C.C. Algebraic analysis of many valued logics, Trans. Am. Math. Soc.88:476-490, 1958
[10]Cignoli, R., Esteva, F., Godo, L., Torrens, A. Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Comput.4:106-112,2000
[11]Cignoli, R., Ottaviano, I. D, Mundici, D. Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht,2000.
[12]P. Cintula, From fuzzy logic to fuzzy mathematics, Ph.D. Thesis, Technical University, Prague,2005
[13]P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, C. Noguera, Distinguished algebraic semantics for t-norm based fuzzy logics:methods and algebraic equivalencies, Annals of Pure and Applied Logic,2009,160 (1):53-81.
[14]P. Cintula, P. Hajek, Complexity issues in axiomatic extensions of Lukasiewicz logic, Journal of Logic and Computation,2009,19 (2):245-260.
[15]P. Cintula, P. Hajek, Triangular norm based predicate fuzzy logics,161 (2010) 311-346
[16]Ciftcibasi, T., Altunay, D. Two-sided (intuitionistic) fuzzy reasoning. IEEE Trans. Systems, Man, and Cybernetics-Part A:Systems and Humans.1998,28 (5):662-677
[17]Czogala, E, Leski, J. On equivalence of uncertainty reasoning results using different interpretations of fuzzy if-then rules. Fuzzy Sets and Systems.2001,117:279-296
[18]De-Gang Wang, Yan-Ping Meng, Hong-Xing Li. A fuzzy similarity inference method for fuzzy reasoning. Computers and Mathematics with Applications.2008,56:2445-2454
[19]Dubois, D, Prade, H. Fuzzy relation equations and causal reasoning. Fuzzy Sets and Systems.1995,75:119-134
[20]Emami, M.R, Turksen, I. B, Goldenberg, A.A. A unified parameterized formulation of reasoning in fuzzy modeling and control. Fuzzy Sets and Systems.1999,108:59-81
[21]Epstein, G.. Multiple-Valued Logic Design. IOP Publishing Ltd.,1993, London
[22]Esteva, F., Garcia, C.P., Godo, L. Enriched interval bilattices and partial many valued logics:An approach to deal with graded truth and imprecision. Internat. J. Uncertain Fuzziness Knowledge Based System.1994,2:37-64
[23]Esteva, F., Godo, L., Hajek, P., Navara, M. Residuated fuzzy logic with an involutive negation. Arch. Math. Logic.2000,39:103-124
[24]Esteva, F. Godo, L., Monoidal t-norm based logic:towards a logic for left-continuous t-norms, Fuzzy Sets Systems 124:271-288,2001
[25]Esteva, F. Godo, L., Hajek, P., Montagna, F. Hoops and fuzzy logic, J. Logic Comput.13: 531-555,2003
[26]Esteva, F. Godo, L., Montagna, F. Equational characterization of the subvarieties of BL generated by t-norm algebras, Studia Logica,76:161-200,2004
[27]F. Esteva, L.Godo, C.Noguera, First-order t-norm based fuzzy logics with truth-constants:distinguished semantics and completeness properties, Annals of Pure and Applied Logic 161(2) (2009) 185-202.
[28]F. Esteva, L.Godo, C.Noguera, On expansions of WNM t-norm based logics with truth-constants. Fuzzy Sets and Systems (2009), to appear, doi:10.1016/j.fss.2009.09.002.
[29]Ewa Orlowska, Anna Maria Radzikowska. Representation theorems for some fuzzy logics based on residuated non-distributive lattices. Fuzzy Sets and Systems.2008,159: 1247-1259
[30]Goguen, J.A. L-type Fuzzy Sets. J. Math. An. Appl..1967,18:145-174
[31]P. Hajek, On arithmetical complexity of fragments of prominent fuzzy predicate logics, Soft Computing,2008,12 (4):335-340.
[32]Hajek, P., Godo, L., Esteva, F. A complete many-valued logic with product conjunction. Arch. Math. Logic.1996,35:191-208
[33]Hajek, P. Meta-Mathematics of Fuzzy Logic. Kluwer Academic Publishers-Dordrecht, 2000
[34]Hajek, P. On very true. Fuzzy Sets and Systems.2001,124:329-333
[35]P. Hajek, Arithmetical complexity of fuzzy predicate logics-a survey Ⅱ, Annals of Pure and Applied Logic,2009,161 (2):212-219.
[36]Hajek, P. Basic fuzzy logic and BL-algebras, Soft Comput.2:124-128,1998
[37]Hajek, P. Metamathematics of Fuzzy Logic, Trends in Logic, vol.4, Kluwer Academic Publishers, Dordrecht,1998
[38]P. Hajek, On vagueness, truth values and fuzzy logics, Studia Logica 91 (3) (2009) 367-382.
[39]P. Hajek, On witnessed models in fuzzy logic Ⅲ. Mathematical Logic Quarterly (2009), submitted for publication.
[40]P. Hajek, Towards metamathematics of weak arithmetic over fuzzy logic, submitted for publication.
[41]Hajek, P., Hanikova, Z. A development of set theory in fuzzy logic, in:M. Fitting, E. Orlowska (Eds.), Beyond Two. Theory and Applications of Multiple-Valued Logic, Physica-Verlag, Heidelberg,2003:273-285
[42]Hajek, P., Paris, J., Shepherdson, J. Rational Pavelka predicate logic is a conservative extension of Lukasiewicz predicate logic, J. Symbolic Logic 65:669-682,2000
[43]Herrera, F. and Herrera-Viedma, E. Aggregation operators for linguistic weighted information. IEEE Trans. System., Man, Cybernet. Part A:Systems Humans.1997,27: 646-656
[44]Herrera,F. and Martinez, L. A 2-Tuple Fuzzy Linguistic Representation Model for Computing with Words. IEEE Transactions on Fuzzy Systems.2000,8(6):746-752
[45]Herrera,F. and Martinez, L. An approach for combining linguistic and numerical information based on 2-tuple fuzzy linguistic representation model in decision-making. Int. J. Uncertainty, Fuzziness, Knowledge-Based Syst..2000,8(5):539-562
[46]Herrera,F. and Martinez, L. A Model Based on Linguistic 2-Tuples for Dealing with Multigranular Hierarchical Linguistic Contexts in Multi-Expert Decision-Making. IEEE Transactions on Systems, Man, and Cybernetics, Part B:Cybernetics.2001,31(2): 227-234
[47]Herrera,F., Lopez, E. and Rodriguez, M.A. A linguistic decision model for promotion mix management solved with genetic algorithms. Fuzzy Sets and Systems.2002,131:47-61
[48]Ho C. Nguyen, Wechler,W. Hedge algebras:an algebraic approach to structure of sets of linguistic truth values. Fuzzy Sets and Systems,1990,35:281-293
[49]Ho C. Nguyen, Wechler,W. Extended hedge algebras and their application to fuzzy logic. Fuzzy Sets and Systems.1992,52:259-281
[50]Ho C. Nguyen, Nam V. Huynh, Khang,T.D., Chau, N.H. Hedge algebras, linguistic-valued logic and their application to fuzzy reasoning. Internat. J. Uncertainty Fuzziness Knowledge-Based Systems.1999,7(4):347-361
[51]Ho C. Nguyen, Nam V. Huynh. Ordered structure-based semantics of linguistic terms of linguistic variables and approximate reasoning. AIP Conference Proceedings.2000, 517(1):98-116
[52]Ho C. Nguyen, Nam V. Huynh. An algebraic approach to linguistic hedges in Zadeh's fuzzy logic. Fuzzy Sets and Systems.2002,129(2):229-254
[53]Hongyan Xing, Daowen Qiu, Fuchun Liu. Automata theory based on complete residuated lattice-valued logic:Pushdown automata. Fuzzy Sets and Systems.2009,160:1125-1140
[54]Hongyan Xing, Daowen Qiu. Pumping lemma in context-free grammar theory based on complete residuated lattice-valued logic. Fuzzy Sets and Systems.2009,160:1141-1151
[55]Yuncheng Jiang, JuWang, PeiminDeng, SuqinTang, Reasoning within expressive fuzzy rough description logics. Fuzzy Sets and Systems,2009,160:3403-3424
[56]Kleene, S. C.. Introduction to Metamathematics. Van Nostrand,1952, Amsterdam and Princeton.中译本：莫绍揆译.元数学导引.科学出版社,1985,北京
[57]Lee Tsu-Tian, Yang Xu. The Consistency of Rule-Bases in Lattice-Valued First-Order Logic LF(X). Proceedings of 2003 IEEE International Conference on Systems, Man & Cybernetics. Washington, D.C., U.S.A. October 5-8,2003:4968-4973
[58]Lewis, D.K. Counterfactuals. Cambridge, Harvard University Press, MA,1973
[59]李顺琴,王国俊.修正的Godel逻辑系统中子代数的广义重言式理论.计算机工程与应用.2008,44(36)：58-60
[60]刘保翠,王国俊.二值命题逻辑中的三种Γ近似推理模式及其等价性.模糊系统与数学.2008,22(2)：10-17
[61]Liu, J., Xu, Y., Ruan, D., Martinez, L. A lattice-valued linguistic-based decision-making method. Proc. of 2005 IEEE International Conference on Granular Computing. Beijing, China, July 25-27,2005:199-202
[62]刘叙华.广义模糊逻辑和锁语义归结原理.计算机学报.1980,2：97-111
[63]刘叙华.狭义模糊逻辑中的锁语义归结.吉林大学学报.1980,4：56-60
[64]Ma, J., Xu Y., Li T., Li W. Uncertainty reasoning based on filter of lattice implication algebra. Proc. of 2003 IEEE Conference on SMC. Washington D.C., U.S.A, October 5-8, 2003:4980-4985
[65]Ma, J., Li, W., Xu, Y., Song, Z. A Model for handling linguistic terms in the framework of lattice-valued logic LF(X). Proc. Of 2004 IEEE International Conference on Systems, Man and Cybernetics. The Hague, Netherlands, October 10-13,2004:1504-1509
[66]Ma, J. Y. Xu, "Study on struture of lattice implication algebra", International Journal of Fuzzy Mathematics,2006,14(1):165-173
[67]Ma, J., Chen, S., Xu, Y. Fuzzy logic from the viewpoint of machine intelligence. Fuzzy Sets and Systems.2006,157(5):628-634
[68]Melek, W. W., Goldenberg, A. A.. The development of a robust fuzzy inference mechanism. Int. J. Approximate Reasoning.2005(39):29-47
[69]Mizumoto, M. Extended fuzzy reasoning. In:Gupta, M. M., Kandel, A., Bandler, W., Kiszka, J.B. (Eds.) Approximate Reasoning in Expert Systems. North-Holland, Amsterdam,67-76
[70]Niittymaki, J, Turunen, E. Traffic signal control on similarity logic reasoning. Fuzzy Sets and Systems.2003,133(1):109-131
[71]Novak, V. On the syntactico-semantical completeness of first-order fuzzy logic. Part I syntactical aspects; Part Ⅱ-main results. Kybernetika.1990,26:47-66; 134-154
[72]Novak, V., Perfilieva, Ⅰ., Mojckojr, J. Mathematical Principles of Fuzzy Logic. Kluwer, 1999
[73]Novak, V. On fuzzy type theory. Fuzzy Sets and Systems.2005,149:235-273
[74]Novak, V. Logical structure of fuzzy IF-THEN rules. Fuzzy Sets and Systems.2006,157: 2003-2029
[75]Novak, V. Which logic is the real fuzzy logic?, Fuzzy Sets and Systems.2006,157: 635-641
[76]Novak, V. Fuzzy logic with countable evaluated syntax revisited. Fuzzy Sets and Systems. 2007,158:929-936
[77]Novak, V. A formal theory of intermediate quantifiers. Fuzzy Sets and Systems.2008,159: 1229-1246
[78]Novak, V. A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets and Systems.2008,159:2939-2969
[79]Xiaodong Pan, Yang Xu. Lattice implication ordered semigroups, Information Science. 2008,178:403-413
[80]潘小东.关于格蕴涵代数方程的研究.西南交通大学硕士研究生学位论文,2005
[81]Pavelka, J. On fuzzy logic I:Many-valued rules of inference, II:Enriched residuated lattices and semantics of propositional calculi, III:Semantical completeness of some many-valued propositional calculi. Zeitschr. F. Math. Logik und Grundlagend. Math.. 1979,25:45-52,119-134,447-464
[82]Z. Pei, D. Ruan, J. Liu, Y. Xu, Linguistic Values based Intelligent Information Processing: Theory, Methods, and Application, Atlantis Computational Intelligence Systems-Vol.1, Atlantis press & World Scientic,2009.
[83]Perfilieva, I. Correct models of fuzzy IF-THEN rules are continuous. Fuzzy Sets and Systems.2006,157:3188-3197
[84]P. P. Wang (ed.). Computing with words. John Wiley and Sons, Inc,2001:347-366
[85]Qin, K.Y., Xu, Y. Lattice-valued proposition logic (Ⅱ). J. Southwest Jiaotong University. 1994,2(1):22-27
[86]Qiu, D., Wang, H. A probabilistic model of computing with words. Journal of Computer and System Sciences.2005,70:176-200
[87]Rasiowa, H. An Algebraic Approach to Non-Classical Logics. Stud. In Logic. North Holland, Amsterdam,1974
[88]Rescher, N. Many-Valued Logic. McGraw-Hill,1969, New York
[89]R. Horcik, P. Cintula, Product Lukasiewicz logic, Arch. Math. Logic 43:477-503,2004
[90]Ruan, D, Kerre, E.E. (Eds.) Fuzzy Sets Theory and Applications. Kluwer Academic Publishers,2000:81-105
[91]宋士吉,吴澄.模糊推理的反向三I算法.中国科学E.2002,32(2)：230-246
[92]Terano, T., Sugeno, M., Mukaidono, K. Shigematsu (eds.) Fuzzy Engineering toward Human Friendly System. Ohmusha, Japan,1991:60-69
[93]Turksen, I. B. Type 2 representation and reasoning for CWW. Fuzzy Sets and Systems. 2002,127(1):17-36
[94]Turunen, E. A note on Pavelka's fuzzy logic. Zeitschr. f. Math. Logik und Grundlagen d. Math.1991,37:39-40
[95]Turunen, E. Well-defined Fuzzy Sentential Logic. Mathematical Logic Quarterly.1995,41: 236-248
[96]Wang, H., Qiu, D. Computing with words via Turing machines:a formal approach. IEEE Trans. Fuzzy Systems.2003,11(6):742-753
[97]王国俊.模糊逻辑与模糊推理.全国第七届多值逻辑与模糊逻辑学术会议论文集.西安,1996：82-96
[98]王国俊.一类代数上的逻辑学(Ⅰ).陕西师范大学学报.1997,25(1)：1-8
[99]王国俊.一类代数上的逻辑学(Ⅱ).陕西师范大学学报.1997,25(3)：1-8
[100]王国俊.Fuzzy命题演算的一种形式演绎系统.科学通报.1997,42(10)：1041-1045
[101]Wang, G. J. On the logic foundation of fuzzy reasoning. Information Sciences.1999, 117:47-88
[102]王国俊.模糊推理的全蕴涵三Ⅰ算法.中国科学E.1999,29(1)：43-53
[103]王国俊.非经典数理逻辑与近似推理.科学出版社.北京,2000
[104]王国俊,任燕.Lukasiewicz命题集的发散性与相容性.工程数学学报.2003,20(3)：13-18
[105]Wang, G. J. Formalized theory of general fuzzy reasoning. Information Sciences.2004, 160:251-266
[106]王国俊.计量逻辑学(Ⅰ).工程数学学报.2006,23(2)：191-215
[107]Guojun Wang, Hongjun Zhou, Quantitative logic.179 (2009) 226-247
[108]H. Wu, The generalized truth degree of quantitative logic in the logic system Ln* (n-valued NM-logic system), Computers and Mathematics with Applications (2010), doi:10.1016/j.camwa.2010.01.024
[109]Guo-Jun Wang, Xiao-Jing Hui, Jian-She Song. The Ro-type fuzzy logic metric space and an algorithm for solving fuzzy modus ponens. Computers and Mathematics with Applications.2008,55:1974-1987
[110]Wang, P. P. Computing with words. John Wiley and Sons, Inc.,2001
[111]WANG Xue-fang, ZHENG Feng-bin, XU Yang. Closure Operators of Lattice-valued Propositional Logic LP(X). Chin. Quart. J. of Math.2005,20 (3):301-308
[112]吴望名.模糊推理的原理与方法.贵州科技出版社.贵阳,1994
[113]谢祥云.序半群引论.科学出版社,北京.2001
[114]徐扬,乔全喜,陈超平,秦克云.不确定性推理.西南交通大学出版社.成都,1994
[115]徐扬.格蕴涵代数.西南交通大学学报.1993,89(1)：20-27
[116]Xu, Y., Qin, K.Y. Lattice-valued propositional logic (Ⅰ). J. Southwest Jiaotong University.1993,1(2):123-128
[117]Xu Y.,Qin K.Y., Song, Z.M. Syntax of First Order Lattice Valued Logic System FM. Chinese Science Bulletin.1997,42(16):1337-1340
[118]Xu, Y., Qin, K.Y, Liu, J., Song, Z.M. L-valued propositional logic Lvpl.Inform. Sci.. 1999,114:205-235
[119]Xu, Y, Liu, J., Song, Z.M., Qin, K.Y. On semantics of L-valued first-order logic Lvfl. Internat. J. Gen. Systems.2000,29(1):53-79
[120]Xu, Y., Kerre, E.E., Ruan, D., Song, Z. M. Fuzzy Reasoning Based on the Extension Principle. Int. J. of Intelligent Systems.2001,16(4):469-495
[121]Xu, Y., Song, Z.M., Qin, K.Y., Liu, J. Syntax of L-valued first-order logic Lvfl. Int. J. Multiple-Valued Logic.2001,7:213-257
[122]Xu, Y, Liu, J., Ruan, D., LI, W. J. Fuzzy Reasoning Based on Generalized Fuzzy If-Then Rules. Int. J. of Intelligent Systems.2002,17(10):977-1006
[123]Xu, Y., Ruan, D., Qin, K.Y., Liu, J. Lattice-Valued Logic-An alternative approach to treat fuzziness and incomparability. Springer-Verlag,2003
[124]Xu, Y, Liu, J., Ruan, D., Lee, T. T. On the consistency of rule bases based on lattice-valued first-order logic LF(X). International Journal of Intelligent Systems.2006, 21(4):399-424
[125]Xu Y, Chen Shuwei, Jun Liu. Linguistic Truth-Valued Lattice Implication Algebra and Its Properties. Proc. IMACS Multiconference on "Computational Engineering in Systems Applications" (CESA2006), October 4-6,2006, Beijing, China,1413-1418
[126]Yager, R.R. Using approximate reasoning to represent default knowledge. Artificial Intelligence.1987,31:99-112
[127]Yager, R.R. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Systems, Man and Cybernet.1988,18:183-190
[128]Yager, R.R. On weighted median aggregation, International Journal of Uncertainty. Fuzziness and Knowledge-Based Systems.1994,2:101-113
[129]Yager, R.R. Uncertainty reasoning and conflict resolution. Internat. J. Uncertainty Reason.2000,25:15-42
[130]Yager, R. R. On the retranslation process in Zadeh's paradigm of computing with words. IEEE Trans. Systems, Man, and Cybernetics, Part B:Cybernetics.2004,34(2): 1184-1195
[131]Yeung, D.S, Tsang, E.C.C. Weighted fuzzy production rules. Fuzzy Sets and Systems. 1997,88(3):299-313
[132]Ying, M.S. Fuzzy topology based complete residuated lattice-valued logic (Ⅰ). Fuzzy Sets and Systems.1993,56:337-373
[133]Ying, M.S. Fuzzy topology based on residuated lattice-valued logic. Acta Math. Sinica (English Series).2001,17:89-102
[134]Ying, M. A formal model of computing with words. IEEE Trans. Fuzzy Systems. 2002,10(5):640-652
[135]Zadeh, L. A. The concept of a linguistic variable and its applications to approximate reasoning, Part Ⅰ, Part Ⅱ, Part Ⅲ. Information Sciences.1975,8:199-249,301-357,9: 43-80
[136]Zadeh, L.A. Fuzzy logic and approximate reasoning. Syntheses.1975,30:407-428
[137]Zadeh, L. A. Fuzzy logic=computing with words. IEEE Trans. Fuzzy Syst.,1996,4: 103-111
[138]Zadeh, L. A. Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems.1997,90:111-127
[139]Zadeh, L. A. From computing with numbers to computing with words-from manipulation of measurements to manipulation of perceptions. IEEE Trans. Circuits Systems.1999,45:105-119
[140]Zadeh, L. A. Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. Journal of Statistical Planning and Inference.2002,105(1): 233-264
[141]张文修,梁怡.不确定性推理.西安交通大学出版社.西安,1996
[142]Zhou, C., Ruan, D. Fuzzy control rules extraction from perception-based information using computing with words. Information Sciences.2002,142:275-290
[143]周红军,王国俊.系统L*中极大相容理论的结构刻画和紧致性定理.模糊系统与数学.2008,22(4)：8-14
[2]Baldwin, J.F. Anew approach to approximate reasoning using a fuzzy logic, Department of Engineering Mathematics, University of Bristol, UK,1978
[3]Baldwin, J.F. A model of fuzzy reasoning and fuzzy logic, Department of Engineering Mathematics, University of Bristol, UK,1978.
[4]Baldwin, J.F., Guild, N.C.F. Feasible algorithms for approximate reasoning using a fuzzy logic, Department of Engineering Mathematics, University of Bristol, UK,1978.
[5]Baldwin, J.F., Pilsworth, B.W. A model of fuzzy reasoning through multi-valued logic and set theory, Department of Engineering Mathematics, University of Bristol, UK, Internat. J. Man-Mach. Studies,1979
[6]Benferhat, S., Lagruea, S., Papinic, O. Reasoning with partially ordered information in a possibilistic logic framework. Fuzzy Sets and Systems.2004,144:25-41
[7]Chen, S.M, Hsiao, W.H. Bidirectional uncertainty reasoning for rule-based systems using interval-valued fuzzy sets. Fuzzy Sets and Systems.2000,113:185-203
[8]Chen, S.W., Xu, Y. Uncertainty Reasoning Based on Lattice-Valued First-Order Logic Lvfl. Proc. of 2004 IEEE International Conference on Systems, Man and Cybernetics, The Hague, The Netherlands,2004:2237-2242
[9]Chang, C.C. Algebraic analysis of many valued logics, Trans. Am. Math. Soc.88:476-490, 1958
[10]Cignoli, R., Esteva, F., Godo, L., Torrens, A. Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Comput.4:106-112,2000
[11]Cignoli, R., Ottaviano, I. D, Mundici, D. Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht,2000.
[12]P. Cintula, From fuzzy logic to fuzzy mathematics, Ph.D. Thesis, Technical University, Prague,2005
[13]P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, C. Noguera, Distinguished algebraic semantics for t-norm based fuzzy logics:methods and algebraic equivalencies, Annals of Pure and Applied Logic,2009,160 (1):53-81.
[14]P. Cintula, P. Hajek, Complexity issues in axiomatic extensions of Lukasiewicz logic, Journal of Logic and Computation,2009,19 (2):245-260.
[15]P. Cintula, P. Hajek, Triangular norm based predicate fuzzy logics,161 (2010) 311-346
[16]Ciftcibasi, T., Altunay, D. Two-sided (intuitionistic) fuzzy reasoning. IEEE Trans. Systems, Man, and Cybernetics-Part A:Systems and Humans.1998,28 (5):662-677
[17]Czogala, E, Leski, J. On equivalence of uncertainty reasoning results using different interpretations of fuzzy if-then rules. Fuzzy Sets and Systems.2001,117:279-296
[18]De-Gang Wang, Yan-Ping Meng, Hong-Xing Li. A fuzzy similarity inference method for fuzzy reasoning. Computers and Mathematics with Applications.2008,56:2445-2454
[19]Dubois, D, Prade, H. Fuzzy relation equations and causal reasoning. Fuzzy Sets and Systems.1995,75:119-134
[20]Emami, M.R, Turksen, I. B, Goldenberg, A.A. A unified parameterized formulation of reasoning in fuzzy modeling and control. Fuzzy Sets and Systems.1999,108:59-81
[21]Epstein, G.. Multiple-Valued Logic Design. IOP Publishing Ltd.,1993, London
[22]Esteva, F., Garcia, C.P., Godo, L. Enriched interval bilattices and partial many valued logics:An approach to deal with graded truth and imprecision. Internat. J. Uncertain Fuzziness Knowledge Based System.1994,2:37-64
[23]Esteva, F., Godo, L., Hajek, P., Navara, M. Residuated fuzzy logic with an involutive negation. Arch. Math. Logic.2000,39:103-124
[24]Esteva, F. Godo, L., Monoidal t-norm based logic:towards a logic for left-continuous t-norms, Fuzzy Sets Systems 124:271-288,2001
[25]Esteva, F. Godo, L., Hajek, P., Montagna, F. Hoops and fuzzy logic, J. Logic Comput.13: 531-555,2003
[26]Esteva, F. Godo, L., Montagna, F. Equational characterization of the subvarieties of BL generated by t-norm algebras, Studia Logica,76:161-200,2004
[27]F. Esteva, L.Godo, C.Noguera, First-order t-norm based fuzzy logics with truth-constants:distinguished semantics and completeness properties, Annals of Pure and Applied Logic 161(2) (2009) 185-202.
[28]F. Esteva, L.Godo, C.Noguera, On expansions of WNM t-norm based logics with truth-constants. Fuzzy Sets and Systems (2009), to appear, doi:10.1016/j.fss.2009.09.002.
[29]Ewa Orlowska, Anna Maria Radzikowska. Representation theorems for some fuzzy logics based on residuated non-distributive lattices. Fuzzy Sets and Systems.2008,159: 1247-1259
[30]Goguen, J.A. L-type Fuzzy Sets. J. Math. An. Appl..1967,18:145-174
[31]P. Hajek, On arithmetical complexity of fragments of prominent fuzzy predicate logics, Soft Computing,2008,12 (4):335-340.
[32]Hajek, P., Godo, L., Esteva, F. A complete many-valued logic with product conjunction. Arch. Math. Logic.1996,35:191-208
[33]Hajek, P. Meta-Mathematics of Fuzzy Logic. Kluwer Academic Publishers-Dordrecht, 2000
[34]Hajek, P. On very true. Fuzzy Sets and Systems.2001,124:329-333
[35]P. Hajek, Arithmetical complexity of fuzzy predicate logics-a survey Ⅱ, Annals of Pure and Applied Logic,2009,161 (2):212-219.
[36]Hajek, P. Basic fuzzy logic and BL-algebras, Soft Comput.2:124-128,1998
[37]Hajek, P. Metamathematics of Fuzzy Logic, Trends in Logic, vol.4, Kluwer Academic Publishers, Dordrecht,1998
[38]P. Hajek, On vagueness, truth values and fuzzy logics, Studia Logica 91 (3) (2009) 367-382.
[39]P. Hajek, On witnessed models in fuzzy logic Ⅲ. Mathematical Logic Quarterly (2009), submitted for publication.
[40]P. Hajek, Towards metamathematics of weak arithmetic over fuzzy logic, submitted for publication.
[41]Hajek, P., Hanikova, Z. A development of set theory in fuzzy logic, in:M. Fitting, E. Orlowska (Eds.), Beyond Two. Theory and Applications of Multiple-Valued Logic, Physica-Verlag, Heidelberg,2003:273-285
[42]Hajek, P., Paris, J., Shepherdson, J. Rational Pavelka predicate logic is a conservative extension of Lukasiewicz predicate logic, J. Symbolic Logic 65:669-682,2000
[43]Herrera, F. and Herrera-Viedma, E. Aggregation operators for linguistic weighted information. IEEE Trans. System., Man, Cybernet. Part A:Systems Humans.1997,27: 646-656
[44]Herrera,F. and Martinez, L. A 2-Tuple Fuzzy Linguistic Representation Model for Computing with Words. IEEE Transactions on Fuzzy Systems.2000,8(6):746-752
[45]Herrera,F. and Martinez, L. An approach for combining linguistic and numerical information based on 2-tuple fuzzy linguistic representation model in decision-making. Int. J. Uncertainty, Fuzziness, Knowledge-Based Syst..2000,8(5):539-562
[46]Herrera,F. and Martinez, L. A Model Based on Linguistic 2-Tuples for Dealing with Multigranular Hierarchical Linguistic Contexts in Multi-Expert Decision-Making. IEEE Transactions on Systems, Man, and Cybernetics, Part B:Cybernetics.2001,31(2): 227-234
[47]Herrera,F., Lopez, E. and Rodriguez, M.A. A linguistic decision model for promotion mix management solved with genetic algorithms. Fuzzy Sets and Systems.2002,131:47-61
[48]Ho C. Nguyen, Wechler,W. Hedge algebras:an algebraic approach to structure of sets of linguistic truth values. Fuzzy Sets and Systems,1990,35:281-293
[49]Ho C. Nguyen, Wechler,W. Extended hedge algebras and their application to fuzzy logic. Fuzzy Sets and Systems.1992,52:259-281
[50]Ho C. Nguyen, Nam V. Huynh, Khang,T.D., Chau, N.H. Hedge algebras, linguistic-valued logic and their application to fuzzy reasoning. Internat. J. Uncertainty Fuzziness Knowledge-Based Systems.1999,7(4):347-361
[51]Ho C. Nguyen, Nam V. Huynh. Ordered structure-based semantics of linguistic terms of linguistic variables and approximate reasoning. AIP Conference Proceedings.2000, 517(1):98-116
[52]Ho C. Nguyen, Nam V. Huynh. An algebraic approach to linguistic hedges in Zadeh's fuzzy logic. Fuzzy Sets and Systems.2002,129(2):229-254
[53]Hongyan Xing, Daowen Qiu, Fuchun Liu. Automata theory based on complete residuated lattice-valued logic:Pushdown automata. Fuzzy Sets and Systems.2009,160:1125-1140
[54]Hongyan Xing, Daowen Qiu. Pumping lemma in context-free grammar theory based on complete residuated lattice-valued logic. Fuzzy Sets and Systems.2009,160:1141-1151
[55]Yuncheng Jiang, JuWang, PeiminDeng, SuqinTang, Reasoning within expressive fuzzy rough description logics. Fuzzy Sets and Systems,2009,160:3403-3424
[56]Kleene, S. C.. Introduction to Metamathematics. Van Nostrand,1952, Amsterdam and Princeton.中译本：莫绍揆译.元数学导引.科学出版社,1985,北京
[57]Lee Tsu-Tian, Yang Xu. The Consistency of Rule-Bases in Lattice-Valued First-Order Logic LF(X). Proceedings of 2003 IEEE International Conference on Systems, Man & Cybernetics. Washington, D.C., U.S.A. October 5-8,2003:4968-4973
[58]Lewis, D.K. Counterfactuals. Cambridge, Harvard University Press, MA,1973
[59]李顺琴,王国俊.修正的Godel逻辑系统中子代数的广义重言式理论.计算机工程与应用.2008,44(36)：58-60
[60]刘保翠,王国俊.二值命题逻辑中的三种Γ近似推理模式及其等价性.模糊系统与数学.2008,22(2)：10-17
[61]Liu, J., Xu, Y., Ruan, D., Martinez, L. A lattice-valued linguistic-based decision-making method. Proc. of 2005 IEEE International Conference on Granular Computing. Beijing, China, July 25-27,2005:199-202
[62]刘叙华.广义模糊逻辑和锁语义归结原理.计算机学报.1980,2：97-111
[63]刘叙华.狭义模糊逻辑中的锁语义归结.吉林大学学报.1980,4：56-60
[64]Ma, J., Xu Y., Li T., Li W. Uncertainty reasoning based on filter of lattice implication algebra. Proc. of 2003 IEEE Conference on SMC. Washington D.C., U.S.A, October 5-8, 2003:4980-4985
[65]Ma, J., Li, W., Xu, Y., Song, Z. A Model for handling linguistic terms in the framework of lattice-valued logic LF(X). Proc. Of 2004 IEEE International Conference on Systems, Man and Cybernetics. The Hague, Netherlands, October 10-13,2004:1504-1509
[66]Ma, J. Y. Xu, "Study on struture of lattice implication algebra", International Journal of Fuzzy Mathematics,2006,14(1):165-173
[67]Ma, J., Chen, S., Xu, Y. Fuzzy logic from the viewpoint of machine intelligence. Fuzzy Sets and Systems.2006,157(5):628-634
[68]Melek, W. W., Goldenberg, A. A.. The development of a robust fuzzy inference mechanism. Int. J. Approximate Reasoning.2005(39):29-47
[69]Mizumoto, M. Extended fuzzy reasoning. In:Gupta, M. M., Kandel, A., Bandler, W., Kiszka, J.B. (Eds.) Approximate Reasoning in Expert Systems. North-Holland, Amsterdam,67-76
[70]Niittymaki, J, Turunen, E. Traffic signal control on similarity logic reasoning. Fuzzy Sets and Systems.2003,133(1):109-131
[71]Novak, V. On the syntactico-semantical completeness of first-order fuzzy logic. Part I syntactical aspects; Part Ⅱ-main results. Kybernetika.1990,26:47-66; 134-154
[72]Novak, V., Perfilieva, Ⅰ., Mojckojr, J. Mathematical Principles of Fuzzy Logic. Kluwer, 1999
[73]Novak, V. On fuzzy type theory. Fuzzy Sets and Systems.2005,149:235-273
[74]Novak, V. Logical structure of fuzzy IF-THEN rules. Fuzzy Sets and Systems.2006,157: 2003-2029
[75]Novak, V. Which logic is the real fuzzy logic?, Fuzzy Sets and Systems.2006,157: 635-641
[76]Novak, V. Fuzzy logic with countable evaluated syntax revisited. Fuzzy Sets and Systems. 2007,158:929-936
[77]Novak, V. A formal theory of intermediate quantifiers. Fuzzy Sets and Systems.2008,159: 1229-1246
[78]Novak, V. A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets and Systems.2008,159:2939-2969
[79]Xiaodong Pan, Yang Xu. Lattice implication ordered semigroups, Information Science. 2008,178:403-413
[80]潘小东.关于格蕴涵代数方程的研究.西南交通大学硕士研究生学位论文,2005
[81]Pavelka, J. On fuzzy logic I:Many-valued rules of inference, II:Enriched residuated lattices and semantics of propositional calculi, III:Semantical completeness of some many-valued propositional calculi. Zeitschr. F. Math. Logik und Grundlagend. Math.. 1979,25:45-52,119-134,447-464
[82]Z. Pei, D. Ruan, J. Liu, Y. Xu, Linguistic Values based Intelligent Information Processing: Theory, Methods, and Application, Atlantis Computational Intelligence Systems-Vol.1, Atlantis press & World Scientic,2009.
[83]Perfilieva, I. Correct models of fuzzy IF-THEN rules are continuous. Fuzzy Sets and Systems.2006,157:3188-3197
[84]P. P. Wang (ed.). Computing with words. John Wiley and Sons, Inc,2001:347-366
[85]Qin, K.Y., Xu, Y. Lattice-valued proposition logic (Ⅱ). J. Southwest Jiaotong University. 1994,2(1):22-27
[86]Qiu, D., Wang, H. A probabilistic model of computing with words. Journal of Computer and System Sciences.2005,70:176-200
[87]Rasiowa, H. An Algebraic Approach to Non-Classical Logics. Stud. In Logic. North Holland, Amsterdam,1974
[88]Rescher, N. Many-Valued Logic. McGraw-Hill,1969, New York
[89]R. Horcik, P. Cintula, Product Lukasiewicz logic, Arch. Math. Logic 43:477-503,2004
[90]Ruan, D, Kerre, E.E. (Eds.) Fuzzy Sets Theory and Applications. Kluwer Academic Publishers,2000:81-105
[91]宋士吉,吴澄.模糊推理的反向三I算法.中国科学E.2002,32(2)：230-246
[92]Terano, T., Sugeno, M., Mukaidono, K. Shigematsu (eds.) Fuzzy Engineering toward Human Friendly System. Ohmusha, Japan,1991:60-69
[93]Turksen, I. B. Type 2 representation and reasoning for CWW. Fuzzy Sets and Systems. 2002,127(1):17-36
[94]Turunen, E. A note on Pavelka's fuzzy logic. Zeitschr. f. Math. Logik und Grundlagen d. Math.1991,37:39-40
[95]Turunen, E. Well-defined Fuzzy Sentential Logic. Mathematical Logic Quarterly.1995,41: 236-248
[96]Wang, H., Qiu, D. Computing with words via Turing machines:a formal approach. IEEE Trans. Fuzzy Systems.2003,11(6):742-753
[97]王国俊.模糊逻辑与模糊推理.全国第七届多值逻辑与模糊逻辑学术会议论文集.西安,1996：82-96
[98]王国俊.一类代数上的逻辑学(Ⅰ).陕西师范大学学报.1997,25(1)：1-8
[99]王国俊.一类代数上的逻辑学(Ⅱ).陕西师范大学学报.1997,25(3)：1-8
[100]王国俊.Fuzzy命题演算的一种形式演绎系统.科学通报.1997,42(10)：1041-1045
[101]Wang, G. J. On the logic foundation of fuzzy reasoning. Information Sciences.1999, 117:47-88
[102]王国俊.模糊推理的全蕴涵三Ⅰ算法.中国科学E.1999,29(1)：43-53
[103]王国俊.非经典数理逻辑与近似推理.科学出版社.北京,2000
[104]王国俊,任燕.Lukasiewicz命题集的发散性与相容性.工程数学学报.2003,20(3)：13-18
[105]Wang, G. J. Formalized theory of general fuzzy reasoning. Information Sciences.2004, 160:251-266
[106]王国俊.计量逻辑学(Ⅰ).工程数学学报.2006,23(2)：191-215
[107]Guojun Wang, Hongjun Zhou, Quantitative logic.179 (2009) 226-247
[108]H. Wu, The generalized truth degree of quantitative logic in the logic system Ln* (n-valued NM-logic system), Computers and Mathematics with Applications (2010), doi:10.1016/j.camwa.2010.01.024
[109]Guo-Jun Wang, Xiao-Jing Hui, Jian-She Song. The Ro-type fuzzy logic metric space and an algorithm for solving fuzzy modus ponens. Computers and Mathematics with Applications.2008,55:1974-1987
[110]Wang, P. P. Computing with words. John Wiley and Sons, Inc.,2001
[111]WANG Xue-fang, ZHENG Feng-bin, XU Yang. Closure Operators of Lattice-valued Propositional Logic LP(X). Chin. Quart. J. of Math.2005,20 (3):301-308
[112]吴望名.模糊推理的原理与方法.贵州科技出版社.贵阳,1994
[113]谢祥云.序半群引论.科学出版社,北京.2001
[114]徐扬,乔全喜,陈超平,秦克云.不确定性推理.西南交通大学出版社.成都,1994
[115]徐扬.格蕴涵代数.西南交通大学学报.1993,89(1)：20-27
[116]Xu, Y., Qin, K.Y. Lattice-valued propositional logic (Ⅰ). J. Southwest Jiaotong University.1993,1(2):123-128
[117]Xu Y.,Qin K.Y., Song, Z.M. Syntax of First Order Lattice Valued Logic System FM. Chinese Science Bulletin.1997,42(16):1337-1340
[118]Xu, Y., Qin, K.Y, Liu, J., Song, Z.M. L-valued propositional logic Lvpl.Inform. Sci.. 1999,114:205-235
[119]Xu, Y, Liu, J., Song, Z.M., Qin, K.Y. On semantics of L-valued first-order logic Lvfl. Internat. J. Gen. Systems.2000,29(1):53-79
[120]Xu, Y., Kerre, E.E., Ruan, D., Song, Z. M. Fuzzy Reasoning Based on the Extension Principle. Int. J. of Intelligent Systems.2001,16(4):469-495
[121]Xu, Y., Song, Z.M., Qin, K.Y., Liu, J. Syntax of L-valued first-order logic Lvfl. Int. J. Multiple-Valued Logic.2001,7:213-257
[122]Xu, Y, Liu, J., Ruan, D., LI, W. J. Fuzzy Reasoning Based on Generalized Fuzzy If-Then Rules. Int. J. of Intelligent Systems.2002,17(10):977-1006
[123]Xu, Y., Ruan, D., Qin, K.Y., Liu, J. Lattice-Valued Logic-An alternative approach to treat fuzziness and incomparability. Springer-Verlag,2003
[124]Xu, Y, Liu, J., Ruan, D., Lee, T. T. On the consistency of rule bases based on lattice-valued first-order logic LF(X). International Journal of Intelligent Systems.2006, 21(4):399-424
[125]Xu Y, Chen Shuwei, Jun Liu. Linguistic Truth-Valued Lattice Implication Algebra and Its Properties. Proc. IMACS Multiconference on "Computational Engineering in Systems Applications" (CESA2006), October 4-6,2006, Beijing, China,1413-1418
[126]Yager, R.R. Using approximate reasoning to represent default knowledge. Artificial Intelligence.1987,31:99-112
[127]Yager, R.R. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Systems, Man and Cybernet.1988,18:183-190
[128]Yager, R.R. On weighted median aggregation, International Journal of Uncertainty. Fuzziness and Knowledge-Based Systems.1994,2:101-113
[129]Yager, R.R. Uncertainty reasoning and conflict resolution. Internat. J. Uncertainty Reason.2000,25:15-42
[130]Yager, R. R. On the retranslation process in Zadeh's paradigm of computing with words. IEEE Trans. Systems, Man, and Cybernetics, Part B:Cybernetics.2004,34(2): 1184-1195
[131]Yeung, D.S, Tsang, E.C.C. Weighted fuzzy production rules. Fuzzy Sets and Systems. 1997,88(3):299-313
[132]Ying, M.S. Fuzzy topology based complete residuated lattice-valued logic (Ⅰ). Fuzzy Sets and Systems.1993,56:337-373
[133]Ying, M.S. Fuzzy topology based on residuated lattice-valued logic. Acta Math. Sinica (English Series).2001,17:89-102
[134]Ying, M. A formal model of computing with words. IEEE Trans. Fuzzy Systems. 2002,10(5):640-652
[135]Zadeh, L. A. The concept of a linguistic variable and its applications to approximate reasoning, Part Ⅰ, Part Ⅱ, Part Ⅲ. Information Sciences.1975,8:199-249,301-357,9: 43-80
[136]Zadeh, L.A. Fuzzy logic and approximate reasoning. Syntheses.1975,30:407-428
[137]Zadeh, L. A. Fuzzy logic=computing with words. IEEE Trans. Fuzzy Syst.,1996,4: 103-111
[138]Zadeh, L. A. Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems.1997,90:111-127
[139]Zadeh, L. A. From computing with numbers to computing with words-from manipulation of measurements to manipulation of perceptions. IEEE Trans. Circuits Systems.1999,45:105-119
[140]Zadeh, L. A. Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. Journal of Statistical Planning and Inference.2002,105(1): 233-264
[141]张文修,梁怡.不确定性推理.西安交通大学出版社.西安,1996
[142]Zhou, C., Ruan, D. Fuzzy control rules extraction from perception-based information using computing with words. Information Sciences.2002,142:275-290
[143]周红军,王国俊.系统L*中极大相容理论的结构刻画和紧致性定理.模糊系统与数学.2008,22(4)：8-14