基于商空间理论和粗糙集理论的粒计算模型研究
|
摘要
在不同的抽象层次上观察、理解、表示现实世界问题连同其解,并进行分析、综合、推理,是人类问题求解过程的一个明显特征,也是人类问题求解能力的强有力的表现。从一定意义上来说,这就是人类问题求解过程中智能之所在。针对人类问题求解的这种能力和特征,人工智能研究者对其进行了深入的研究,并建立了各种形式化的模型。作为一种正在兴起的人工智能研究领域,粒计算的目的就是建立一种体现人类问题求解特征的一般模型,其基本思想是在不同的粒度层次上进行问题求解。粒是粒计算的最基本的原语,它是一簇点(对象、物体)由于难以区别,或相似、或接近、或某种功能而结合在一起所构成的。从狭义上看,粒计算可以理解为在不同粒度层次上以粒作为运算对象进行计算和推理。从广义上看,作为一种术语,粒计算可以理解为在问题求解过程中使用粒的理论、方法论、技术和工具的统称。粒、由某个粒化准则所得到的粒层、所有粒层构成的层次结构是组成粒计算模型的三个基本组成;粒化、关于粒的计算和推理是粒计算的两个基本问题,而这两者都可以从语义和算法两个方面来进行研究。
本文主要在集合论背景下从理论上研究了两种分明粒计算模型,即基于问题求解的商空间理论和基于粗糙集理论的粒计算模型。具体内容如下:
1、分别从论域的结构和粒化准则这两个角度推广了商空间理论。
从粒计算的角度来看,问题求解的商空间理论用拓扑来描述论域的结构、用等价关系来完成粒化,借助于自然映射实现在不同粒度层次上的转换。Cech意义下的闭包运算是比拓扑更一般的描述论域的结构的数学语言,本文从论域结构的角度对问题求解的商空间理论进行推广,用闭包运算代替拓扑来描述结构而依然用等价关系来完成粒化,结果表明商空间理论的结论可以推广到这种更一般的情形。等价关系的传递性公理在很多实际应用中很难满足,因此可能会限制商空间理论的可应用领域。本文又讨论了在相容关系,即满足自反性、对称性的二元关系,条件下的商空间理论。在这种情形下,拓扑描述结构,相容关系完成粒化而集值映射充当自然映射的角色,本文的研究表明在这种情形下商空间理论的大部分结论依然成立。这两种推广在一定程度上从理论上丰富了问题求解的商空间理论的内容,并进一步扩大了其可应用的范围。
Perception, understanding and representation, as well as analysis, synthesis and reasoning, of real world problem together with its solution at different levels of granularity, is an obvious feature in the process of human problem solving, and it also embodies the outstanding ability of human problem solving. In a sense, it is precisely the intelligence in the process of human problem solving. Considering such ability of human, researchers of artificial intelligence have made some further investigation and presented many formal models. As an emerging research sub-field of artificial intelligence, granular computing, whose philosophy is to implement the problem solving at different levels of granularity, aims to establish much more general model reflecting the process of human problem solving. Granule, a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality, is the primitive notion of granular computing. In a narrow sense, granular computing can be understood as the computing and reasoning with granules at different levels of granularity. In a broader sense, granular computing is a unified notion of theories, methodologies, techniques and tools that utilize granules in the process of problem solving. Three basic components of a granular computing model are granule, granular level derived by a given granulation criteria and hierarchy composed of all granular levels. Granulation and computing or reasoning with granules, either of them can be studied from both the semantic and algorithmic perspectives, are two related issues of granular computing.The thesis mainly studies two crisp models of granular computing in the framework of set theory, namely, quotient space theory of problem solving and rough set theory. More specifically, the research includes:1、 Quotient space theory of problem solving is generalized from two directions, namely the structure of the universe of discourse and granulation criteria. In the light of granular computing, quotient space theory of problem solving exploits topology to describe the structure of the universe of discourse, and utilizes equivalence relation to implement granulation, and relies on the natural mapping to realize the translations among different levels of granularity. Closure operation in the sense of Cech is more general mathematical language that can be used to describe the structure of the universe of discourse. When generalizing quotient space theory of problem sohing. the thesis exploits closure operation of Cech sense to describe structure. It turns out that
conclusions of classical quotient space theory keep being true in such a general framework. The axiom of transitivity is a tough condition for many applications, which may confine the applicable fields of quotient space theory of problem solving, so a weakened version of quotient space theory with topology describing the structure, compatibility relation that is reflexive and symmetric binary relation implementing granulation, set-valued function playing the role of natural mapping, are presented. The discussion implies that there are corresponding conclusions of classical quotient space theory. To some degree, these two generalizations enrich the intension of quotient space theory of problem solving, and extend its applicable fields.2 > The thesis presents the topological interpretation of the upper and lower approximate operators of generalized rough set model based on reflexive binary relation, as well as the order-theoretic interpretation of the upper and lower approximate operators of generalized rough set model based on covering. Classical rough set theory lays its foundation on the indiscemibility relation, which is represented as equivalence relation in mathematics, between objects. Generalized rough set models based on binary relation and covering respectively are two important studies. The thesis proves that the upper approximate operator of generalized based rough set models based on binary relation is precisely quasi-discrete closure operation, in the sense of Cech. derived by the given binary relation. Dually, the lower approximate operator is preciseh- the quasi-discrete interior operation. Then the topological understandings of generalized approximate rough operators are proposed. As far as covering based rough approximate operators are concerned, it is proved, by constructing Galois connection between the power set of object set and the power set of the covering, that the upper approximate operator is preciseh' the closure operator, which satisfies special axioms, in the sense of order theory. Also, the dual result about the lower approximate operator is given. The studies of the thesis provide, to some extent, the foundations for the further application of generalized rough set theory.3-, A generalized multi-valued information system and its corresponding formal language, a -decision logic language, are proposed. Simply, information system, also called information table, is a 2-dimension table used as the carrier of information, which includes the finite set of objects, die finite set of attribute (name), the finite domain for each attribute and the information function for each attribute that maps objects to the domain of this attribute. It is the information function that distinguishes different ft-pe information systems. Based on the analysis of single-valued
information system, multi-valued one, relational attribute one and fuzzified single-valued one, a fuzzified multi-valued information system, which generalizes all of the mentioned ones, is presented in which the information function for each attribute maps a object to a fuzzy set of corresponding domain. Consequently, a -decision logic language that generalizes decision logic language of single information system is proposed. By the selection of parameter a one can describes, analyzes and solves of problem at different level of granularity.Undoubtedly, considering the philosophy of granular computing, the thesis discusses the problem of granular computing at rather coarse granularity. Namely, the granular computing is studied at a very abstract level. Therefore, how to combine discussed theories and methods of granular computing with concrete problems is an important issue of further research.
本文主要在集合论背景下从理论上研究了两种分明粒计算模型,即基于问题求解的商空间理论和基于粗糙集理论的粒计算模型。具体内容如下:
1、分别从论域的结构和粒化准则这两个角度推广了商空间理论。
从粒计算的角度来看,问题求解的商空间理论用拓扑来描述论域的结构、用等价关系来完成粒化,借助于自然映射实现在不同粒度层次上的转换。Cech意义下的闭包运算是比拓扑更一般的描述论域的结构的数学语言,本文从论域结构的角度对问题求解的商空间理论进行推广,用闭包运算代替拓扑来描述结构而依然用等价关系来完成粒化,结果表明商空间理论的结论可以推广到这种更一般的情形。等价关系的传递性公理在很多实际应用中很难满足,因此可能会限制商空间理论的可应用领域。本文又讨论了在相容关系,即满足自反性、对称性的二元关系,条件下的商空间理论。在这种情形下,拓扑描述结构,相容关系完成粒化而集值映射充当自然映射的角色,本文的研究表明在这种情形下商空间理论的大部分结论依然成立。这两种推广在一定程度上从理论上丰富了问题求解的商空间理论的内容,并进一步扩大了其可应用的范围。
Perception, understanding and representation, as well as analysis, synthesis and reasoning, of real world problem together with its solution at different levels of granularity, is an obvious feature in the process of human problem solving, and it also embodies the outstanding ability of human problem solving. In a sense, it is precisely the intelligence in the process of human problem solving. Considering such ability of human, researchers of artificial intelligence have made some further investigation and presented many formal models. As an emerging research sub-field of artificial intelligence, granular computing, whose philosophy is to implement the problem solving at different levels of granularity, aims to establish much more general model reflecting the process of human problem solving. Granule, a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality, is the primitive notion of granular computing. In a narrow sense, granular computing can be understood as the computing and reasoning with granules at different levels of granularity. In a broader sense, granular computing is a unified notion of theories, methodologies, techniques and tools that utilize granules in the process of problem solving. Three basic components of a granular computing model are granule, granular level derived by a given granulation criteria and hierarchy composed of all granular levels. Granulation and computing or reasoning with granules, either of them can be studied from both the semantic and algorithmic perspectives, are two related issues of granular computing.The thesis mainly studies two crisp models of granular computing in the framework of set theory, namely, quotient space theory of problem solving and rough set theory. More specifically, the research includes:1、 Quotient space theory of problem solving is generalized from two directions, namely the structure of the universe of discourse and granulation criteria. In the light of granular computing, quotient space theory of problem solving exploits topology to describe the structure of the universe of discourse, and utilizes equivalence relation to implement granulation, and relies on the natural mapping to realize the translations among different levels of granularity. Closure operation in the sense of Cech is more general mathematical language that can be used to describe the structure of the universe of discourse. When generalizing quotient space theory of problem sohing. the thesis exploits closure operation of Cech sense to describe structure. It turns out that
conclusions of classical quotient space theory keep being true in such a general framework. The axiom of transitivity is a tough condition for many applications, which may confine the applicable fields of quotient space theory of problem solving, so a weakened version of quotient space theory with topology describing the structure, compatibility relation that is reflexive and symmetric binary relation implementing granulation, set-valued function playing the role of natural mapping, are presented. The discussion implies that there are corresponding conclusions of classical quotient space theory. To some degree, these two generalizations enrich the intension of quotient space theory of problem solving, and extend its applicable fields.2 > The thesis presents the topological interpretation of the upper and lower approximate operators of generalized rough set model based on reflexive binary relation, as well as the order-theoretic interpretation of the upper and lower approximate operators of generalized rough set model based on covering. Classical rough set theory lays its foundation on the indiscemibility relation, which is represented as equivalence relation in mathematics, between objects. Generalized rough set models based on binary relation and covering respectively are two important studies. The thesis proves that the upper approximate operator of generalized based rough set models based on binary relation is precisely quasi-discrete closure operation, in the sense of Cech. derived by the given binary relation. Dually, the lower approximate operator is preciseh- the quasi-discrete interior operation. Then the topological understandings of generalized approximate rough operators are proposed. As far as covering based rough approximate operators are concerned, it is proved, by constructing Galois connection between the power set of object set and the power set of the covering, that the upper approximate operator is preciseh' the closure operator, which satisfies special axioms, in the sense of order theory. Also, the dual result about the lower approximate operator is given. The studies of the thesis provide, to some extent, the foundations for the further application of generalized rough set theory.3-, A generalized multi-valued information system and its corresponding formal language, a -decision logic language, are proposed. Simply, information system, also called information table, is a 2-dimension table used as the carrier of information, which includes the finite set of objects, die finite set of attribute (name), the finite domain for each attribute and the information function for each attribute that maps objects to the domain of this attribute. It is the information function that distinguishes different ft-pe information systems. Based on the analysis of single-valued
information system, multi-valued one, relational attribute one and fuzzified single-valued one, a fuzzified multi-valued information system, which generalizes all of the mentioned ones, is presented in which the information function for each attribute maps a object to a fuzzy set of corresponding domain. Consequently, a -decision logic language that generalizes decision logic language of single information system is proposed. By the selection of parameter a one can describes, analyzes and solves of problem at different level of granularity.Undoubtedly, considering the philosophy of granular computing, the thesis discusses the problem of granular computing at rather coarse granularity. Namely, the granular computing is studied at a very abstract level. Therefore, how to combine discussed theories and methods of granular computing with concrete problems is an important issue of further research.
引文
[Are99] Arenas, F. G. Alexandroff spaces [J]. Acta Math. Univ. Comenianae, Vol LⅩⅤⅢ, 1999, 1: 17-25.
[BBJ02] Boolos, GA., Burges, J. P., Jeffrey, R. Computability and logic, 4th edition [M]. Cambridge University Press, Cambridge, 2002.
[B&M02] Bettini, C., Montanari, A. Research issues and trends in spatial and temporal granularities [J]. Annals of Mathematics and Artificial Intelligence, 2002, 36: 1-4.
[Bir67] Birkhoff, G. Lattice theory, 3rd edtion [M]. Coll. Publ. ⅩⅩⅤ, American Mathematical Society, 1967.
[Bri93] Brink, C. Power structures [J]. Algebra Universalis, 1993, 30: 177-216.
[Bry89] Bryniarski, E. A calculus of rough sets of the first order [J]. Bulletin of the Polish Academy of Sciences, Mathematics, 1989, 37: 71-77.
[B&B98] Bonikowski, Z., Bryniarski, E. Extensions and intensions in the rough set theory [J]. Information Science, 1998, 107: 149-167.
[B&S81] Burris, S., Sankappanavar, H. A course in universal algebra [M]. Springer-verlag, 1981.
[Cech66] Cech. E. Topological spaces (revised by J. Frolik, Z. Katetov, and M. Novak) [M]. Wiley, New York, 1966.
[C&C04] Cheng, J. X., Chen. W. L. Quasi-discrete closure space and generalized rough approximate space based on binary relation [C]. Proceeding of the 3rd International Conference on Machine Learning and Cybernetics, Shanghai, 2004: 26-29.
[CCZ04] 陈万里,程家兴,张持健.基于相容关系推广粗糙集理论[J].计算机工程与应用,2004,40(4):26-28.
[C&C05] 陈万里,程家兴.粒计算的α_决策逻辑语言[J].控制与决策(已录用).
[C&X96] 蔡自兴,徐光祐.人工智能及应用(第二版)[M].北京:清华大学出版社,1996
[D&O98] Demri, S, Orlowska, E. Logical analysis of indiscernibility [C]. In: Incomplete Information: Rough Set Analysis, Orlowska, E. (Ed.). Physica-Verlag, Heidelberg, 1998, 347-380.
[D&P90] Dubois, D., Prade, P. Fuzzy rough sets and rough fuzzy sets [J]. International Journal of General Systems, 1990. 17: 191-209.
[DGO01] Duntsch I, Gediga G, Orlowska E. Relational attribute systems [J]. Int. Journal Human and Computer Studies, 2001, 55: 293-309.
[D&P02] Davey, B. A., Priestley, H. A. Introduction to lattice and order [M]. Cambridge University Press, Cambridge, 1992.
[Euz01] Euzenat, J. Granularity in relational formalisms-with application to time and space representation. Computational Intelligence, 2001, 17: 703-737.
[Gal03] Galton, A. A generalized topological view of motion in discrete space [J]. Theoretical Computer Science, 2003, 305: 111-134.
[G&W92] Ginuchglia, F., Walsh, T. A theory of abstraction [J]. Artificial Intelligence, 1992, 56: 323-390.
[Gra98] Gratzer, G. General lattice theory, 2nd edtion [M]. Birkhauser Verlag, 1998.
[Grz87] Grzymala-Busse, J. W. Rough-set and Dempster-Shafer approaches to knowledge acquisition under uncertainty-a comparison, Manuscript. Department of Computer Science, University of Kansas. 1987.
[Gua96] 关泽群.商空间下的遥感图象分析理论探讨[D].武汉:武汉测绘科技大学,1996
[HCC93] Han, J., Cai, Y., Cercone, N. Data-driven discovery of quantitative rules in databases [J]. IEEE Transactions on Knowledge and Data Engineering, 1993, 5: 29-40.
[H&P96] Hearst, M. A., Pedersen, J. O. Reexamining the cluster hypothesis: Scatter/Gather on retrieval results [C]. Proceedings of SIGIR'96, 1996: 76-84.
[Hob85] Hobbs, J. R. Granularity [C]. Proceedings of the Ninth International Joint Conference on Artificial Intelligence. 1985: 432-435.
[Hor01] Horusby, K. Temporal zooming [C]. Transactions in GIS 5. 2001: 255-272.
[HSL01] Hu, K., Sui, Y., Lu, Y., Wang, J. and Shi, C. Concept approximation in concept lattice, knowledge discovery and data mining [C]. Proceedings of the 5th Pacific-Asia conference. PAKDD 2001. Lecture Notes in Computer Science 2035, 2001: 167-173.
[HSL00] 胡可云,陆玉昌,石纯一.基于概念格的分类和关联规则的集成挖掘方法[J].软件学报.2000,11(11):1478-1484.
[Ime87] Imielinski, T. Domain abstraction and limited reasoning [C]. Proceedings of the 10th International Joint Conference on Artificial Intelligence. 1987: 997-1003.
[IHT03] Inuiguchi, M., Hirano, S., Tsumoto, S. (Eds.). Rough Set Theory and Granular Computing [M]. Springer, Berlin, 2003.
[K&F88] Klir, G. J., Folger, T. A. Fuzzy sets, uncertainty, and information. Prentice Hall, Englewood Cliffs, 1988.
[Kno93] Knoblock, C. A. Generating abstraction hierarchies: an automated approach to reducing search in planning [M]. Kluwer Academic Publishers. Bosto, 1993.
[Kel55] Kely, J. L. General topology [M]. Springer-Verlag, 1955. (北京:世界图书出版社,2001).
[Ken96] Kent, R. E. Rough concept analysis: a synthesis of rough sets and formal concept analysis [J]. Fundamenta Informaticae, 1996, 27: 69-181.
[Kol97] Kolman, B. Discrete mathematics structures(影印版)[M].北京:清华大学出版社,1997.
[L&D89] Lau, K. F., Dill, K. A. A lattice statistical mechanics model of the conformational and sequence space of proteins [J]. Macromolecules. 1989. 22: 3986-3997.
[L&D90] Lau, K. F., Dill. K. A. Theory for protein mutability and biogenesis [C]. Proceedings of the National Academy of Sciences USA, 1990, 87: 638-642.
[Lee87] Lee, T. T. An information-theoretic analysis of relational databases-part Ⅰ: data dependencies and information metric [J]. IEEE Transactions on Software Engineering SE-13, 1987: 1049-1061.
[Ler83] Leron, U. Structuring mathematical proofs [J]. American Mathematical Monthly, 1983, 90: 174-185.
[Lin88] Lin, T. Y. Neighborhood systems and approximation in relational databases and knowledge bases [C]. Proceedings of the 4th International Symposium on Methodologies of Intelligent Systems, 1988.
[Lin92] Lin, T. Y. Topological and fuzzy rough sets. In: Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Slowinski, R. (Ed.). Kluwer Academic Publishers, Boston, 1992: 287-304.
[Lin97a] Lin, T. Y. Neighborhood systems-application to qualitative fuzzy and rough sets [C]. in: Advances in Machine Intelligence and Soft-Computing edited by P. P. Wang, Department of Electrical Engineering, Duke University, Durham, North Carolina, USA, 1997: 132-155.
[Lin97b] Lin, T. Y. Granular computing, announcement of the BISC special interest group on granular computing. 1997. http://www.cs.uregina.ca/～yyao/GrC/.
[Lin98a] Lin, T. Y., Zhong, N., Dong, J., Ohsuga, S. Frameworks for mining binary relations in data [C]. Rough sets and Current Trends in Computing, Proceedings of the 1st International Conference. Lecture Notes in Artificial Intelligence 1424. 1998: 387-393.
[Lin98b] Lin, T. Y. Granular computing on binary relations Ⅰ: data mining and neighborhood systems, Ⅱ: rough set representations and belief functions [C]. In: Rough Sets in Knowledge Discovery 1. Polkowski, L., Skowron, A. (Eds.). Physica-Verlag, Heidelberg, 1998: 107-140.
[Lin01] Lin, T. Y. Generating concept hierarchies/networks: mining additional semantics in relational data [C]. Advances in Knowledge Discovery and Data Mining. Proceedings of the 5th Pacific-Asia Conference. Lecture Notes on Artificial Intelligence 2035, 2001: 174-185.
[Lin02] Lin. T. Y., Yao, Y. Y., Zadeh, L. A. (Eds.) Rough Sets, Granular Computing and Data Mining [M]. Physica-Verlag, Heidelberg, 2002.
[Lin03] Lin. T. Y. Granular computing: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing [C]. Proceedings of the 9th International Conference, Lecture Notes in Artificial Intelligence 2639, 2003: 16-24.
[Lip81] Lipski W. On databases with incomplete information [J]. Journal of the ACM, 1981, 28: 41-70.
[Luce56] Luce, R. D. Semi-orders and a theory of utility discrimination [J]. Econometrica, 1956. 24.
[Li04] 李进金.覆盖广义粗集理论的拓扑学方法[J].模式识别与人工智能,2004,17(1):7-10.
[Liu01] 刘清.Rough集与Rough推理[M].北京:科学出版社,2001.
[Liu04] 刘清,黄兆华.G-逻辑及其归结原理[J].计算机学报,2004,27(7):865-873.
[L&W00] 刘普寅,吴孟达.模糊集理论与应用[M].长沙:国防科技大学出版社,2000.
[Mar00] Martin T. Lecture notes on biological sequence analysis. 2000.
[Man98] Mani, I. A theory of granularity and its application to problems of polysemy and underspecification of meaning [C]. Principles of Knowledge Representation and Reasoning, Proceedings of the Sixth International Conference, 1998: 245-255.
[MGB92] McCalla, G., Greer, J., Barrie, J., Pospisil, P. Granularity hierarchies [J]. Computers and Mathematics with Applications, 1992, 23: 363-375.
[Mun04] Munkres, J. R. Topology, 2nd edition[M].北京:机械工业出版社,2004.
[Nov03] Novak, V. Intentional theory of granular computing [J]. Soft computing, 2003.
[NSZ99] Nayak, A., Sinclair, A, Zwick. U. Journal Comp. Biol. 1999, 6: 13
[O&P87] Orlowska E, Pawlak Z. Representation of non-deterministic information [J]. Theoretical Computer Science, 1987, 29: 27-39.
[Orl85] Orlowska, E. Logic of indiscernibility relations [J]. Bulletin of the Polish Academy of Sciences. Mathematics, 1985, 33: 475-485.
[Paw82] Pawlak, Z. Rough sets [J]. Imernational Journal of Computer and Information Sciences, 1982, 11: 341-356.
[Paw91] Pawlak, Z. Rough Sets: Theoretical aspects of reasoning about data [M]. Kluwer Academic Publishers, Boston. 1991.
[Paw98] Pawlak, Z. Granularity of knowledge, indiscernibility and rough sets [C]. Proceedings of 1998 IEEE International Conference on Fuzzy Systems, 1998: 106-110.
[Paw02] Pawlak Z. Rough sets, decision algorithm and Bayes(?)s theorems [J]. European Journal of Operational research, 2002, 136: 181-189.
[Ped01] Pedrycz, W. Granular computing: an emerging paradigm [M]. Springer-Verlag, Berlin, 2001.
[P&S98] Polkowski, L., Skowron, A. Towards adaptive calculus of granules [C]. Proceedings of 1998 IEEE International Conference on Fuzzy Systems, 1998: 111-116.
[Pag93] Pagliani, P. From concept lattices to approximation spaces: algebraic structures of some spaces of partial objects [J]. Fundamenta Informaticae, 1993, 18: 1-25.
[Pei04] 裴道武.两类广义粗糙集模型[J].模式识别与人工智能,2004,17(3):281-285.
[PBL02] 卜东波,白硕,李国杰.聚类/分类中的粒度原理[J].计算机学报,2002,25(8):10-16.
[Qua00] Quafafou M. α_RST: a generalization of rough set theory [J]. Information Science, 2000, 124: 301-316.
[R&N95] Russell, S., Norvig, P. Artificial Intelligence: a modern approach [M]. Pearson Education, Inc., Prentice-Hall, 1995.(北京:人民邮电出版社,2002.)
[Rus69] Ruspini, E. H. A new approach to clustering [J]. Information and Control, 1969, 15: 2-32.
[R&H00] 阮达,黄祟福 译,模糊集与模糊信息粒理论[M].北京:北京师范大学出版社,2000.
[S&Z98] Saitta, L., Zucker, J. D. Semantic abstraction for concept representation and learning [C]. Proceedings of the Symposium on Abstraction, Reformulation and Approximation, 1998: 103-120. http://www.cs.vassar.edu/_ellman/sara98/papers/.
[S&M83] Salton, G., McGill, M. Introduction to modern reformation retrieval [M]. McGraw Hill, New York, 1983.
[Sha76] Sharer, G. A mathematical theory of evidence [M]. Princeton University Press, Princeton, 1976.
[Sko01] Skowron, A. Toward intelligent systems: calculi of information granules [J]. Bulletin of International Rough Set Society, 2001, 5: 9-30.
[S&G94] Skowron, A., Grzymala-Busse, J. From rough set theory to evidence theory [C]. In: Advances in the Dempster-Shafer Theory of Evidence, Yager, R. R., Fedrizzi, M., Kacprzyk, J. (Eds.). Wiley, New York, 1994: 193-236.
[S&S98] Skowron, A., Stepaniuk, J. Information granules and approximation spaces [C]. Proceedings of 7th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, 1998: 354-361.
[Sla03] Slapal, J. Closure operations for digital topology [J]. Theoretical Computer Science, 2003, 305: 457-471.
[S&S01] Skowron, A., Stepaniuk, J. Information granules: towards foundations of granular computing [J]. International Journal of Iutelligent Systems 16 (2001): 57-85.
[Smi89] Smith. E. E. Concepts and induction [M]. In: Foundations of Cognitive Science, Posner, M. I. (Ed.). The MIT Press, Cambridge (1989), 501-526.
[Smi95] Smith, M. B. Semi-metrics, closure spaces and digital topology: [J]. Theoretical Computer Science, 1995, 151: 257-276.
[Sow84] Sowa, J. F. Conceptual structures [C]. Information Processing in Mind and Machine. Addison-Wesley, Reading, 1984.
[S&W98] Stell, J. G., Worboys, M. F. Stratified map spaces: a formal basis for multi-resolution spatial databases [C]. Proceedings of the 8th International Symposium on Spatial Data Handling, 1998: 180-189.
[S&V97] Slowinski, R., Vanderpooten, D. Similarity relation as a basis for rough approximations [C]. In P. P. Wang (ed.), Advances in Machine Intelligence and Soft-Computing, Department of Electrical Engineering, Duke University, Durham, NC, USA. 1997: 17-33.
[S&V00] Slowinski, R., Vanderpooten, D. A generalized definition of rough approximation based on similarity [J]. IEEE Transactions on Knowledge and Data Engineering, 2000, 12 (2): 331-336.
[Sun83] 孙以丰译.基础拓扑学[M].北京:北京大学出版社,1983.
[Tar56] Tarski, A. Semantics, metamathematics and logic [M]. Oxford University Press, Oxford, 1956.
[Tve77] Tversky, A. Features of similarity [J]. Psychological review, 1977, 84(4): 327-352.
[Wil92] Wille, R.: Concept lattices and conceptual knowledge systems [J]. Computers Mathematics with Applications, 1992, 23: 493-515.
[Wol93] Wolff, K. E. A first course in formal concept analysis__ how to understand line diagram [C]. In: Faulbaum, F. (ed.) SoftStat'93, Advances in Statistical Software, 1993, 4: 429-438.
[W&S] William, H., Sorin. I. HP Benclunarks. http://www.cs.sandia.gov/tech_reports/compbio/tortilla-hp-benchmarks.html
[Wang00] 王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2000.
[Wang01] 王国胤.Rough集理论与知识获取[M].西安:西安交通大学出版社,2001.
[WMZ96] 王钰.苗夺谦,周育健.关于Rough Set理论与应用的研究[J].模式识别与人工智能,1996,9(4):337-344.
[Xio02] 熊金城.点集拓扑讲义(第2版)[M].北京:高等教育出版社,2002
[Xu04] 徐峰.基于商空间的计算智能及其在金融工程中的应用研究[D].合肥:安徽大学,2004.
[Yao95] Yao, Y. Y., Noroozi, N. A unified framework for set-based computations [C]. Proceedings of the 3rd International Workshop on Rough Sets and Soft Computing. The Society for Computer Simulation, 1995: 252-255.
[Yao96] Yao, Y. Y. Two views of the theory of rough sets in finite universes [J]. International Journal of Approximation Reasoning, 1996, 15: 291-317.
[Y&W97] Yao, Y. Y., Wong, S. K. M., Lin, T. Y. A review of rough set models [C]. In: Rough Sets and Data Mining: Analysis for Imprecise Data, Lin, T. Y., Cercone, N. (Eds.). Kluwer Academic Publishers, Boston, 1997: 47-75.
[Yao98] Yao, Y. Y. Relational interpretations of neighborhood operators and rough set approximation operator [J]. Information Science, 1998, 111: 239-259.
[Yao99] Yao, Y. Y. Granular computing using neighborhood systems [C]. In: Advances in Soft Computing: Engineering Design and Manufacturing, Roy, R., Furuhashi, T., and Chawdhry, P. K. (Eds), Springer-Verlag, London, 1999: 539-553.
[Yao00] Yao, Y. Y. Granular computing: basic issues and possible solutions [C]. Proceedings of the 5th Joint Conference on Information Sciences. 2000: 186-189.
[Yao01a] Yao, Y. Y. Information granulation and rough set approximation [J]. International Journal of Intelligent Systems, 2001, 16: 87-104.
[Yao01b] Yao, Y. Y. Modeling data mining with granular computing [C]. Proceedings of the 25th Annual International Computer Software and Applications Conference (COMPSAC 2001), 2001: 638-643.
[Yao02] Yao, Y. Y., Liau, C. J. A generalized decision logic language for granular computing [C]. Proceedings of FUZZ-IEEE'02 in the 2002 IEEE World Congress on Computational Intelligence, 2002: 1092-1097.
[Yao03] Yao, Y. Y., Liau, C. J., Zhong, N. Granular computing based on rough sets, quotient space theory, and belief functions [C]. Proceedings of ISMIS'03, 2003: 152-159.
[Y&Zh02] Yao, Y. Y., Zhong, N. Granular computing using information tables [C]. In: Data Mining, Rough Sets and Granular Computing, Lin, T. Y., Yao. Y. Y., Zadeh, L. A. (Eds.). Physica-Verlag, Heidelberg, 2002: 102-124.
[Y&C97] Yao, Y. Y., Chen, X. C. Neighborhood based information systems [C]. Proceedings of the 3rd Joint Conference on Information Sciences, Volume 3: Rough Set & Computer Science Research Triangle Park, North Carolina; USA, 1997, March 1-5, 154-157.
[Y&F98] Yager, R. R., Filev, D. Operations for granular computing: mixing words with numbers [C]. Proceedings of 1998 IEEE International Conference on Fuzzy Systems, 1998: 123-128.
[Yao04a] Yao, Y. Y. Concept lattices in rough set theory [C]. Proceedings of 23rd International meeting of the North American Fuzzy Information Processing Society, 2004.
[Yao04b] Yao, Y. Y. A partition model of granular computing. 2004.http://www.cs.uregina.ca/～yyao/GrC/.
[Y&C04] Yao, Y. Y. and Chen, Y. H. Rough set approximations in formal concept analysis [C]. Proceedings of 23rd International Meeting of the North American Fuzzy Information Processing Society, 2004.
[Yao05] Yao, Y. Y. Granular computing 2005. http://www.cs.uregina.ca/～yyao/GrC/.
[You97] 尤承业.基础拓扑学讲义[M].北京:北京大学出版社.1997.
[Zad65] Zadeh, L. A. Fuzzy sets [J]. Information and Control, 1965, 8: 338-353.
[Zad71] Zadeh, L. A. Similarity relations and fuzzy ordering [J]. Information Science, 1971, 3: 171-200.
[Zad79] Zadeh, L. A. Fuzzy sets and information granularity [J]. In: Advances in Fuzzy Set Theory and Applications, Gupta, N., Ragade, R., Yager, R. (Eds.). North-Holland. Amsterdam 1979: 3-18.
[Zad96] Zadeh, L. A Fuzzy logic=computing with words [J]. IEEE Transactions on Fuzzy Systems, 1996, 4: 103-111.
[Zad97a] Zadeh, L. A. Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic [J]. Fuzzy Sets and Systems, 1997, 90: 111-127.
[Zad97b] Zadeh L. A. Annotmcement of GrC. 1997. http://www.cs.uregina.ca/～yyao/GrC/.
[Zad98] Zadeh, L. A. Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information/intelligent systems [J]. Soft Computing, 1998, 2: 23-25.
[Zak83] Zakowski, W. Axiomization in the space (U, Π). Demonstratio Mathematica, 1983, 16: 761-769.
[Z&Z90] 张钹,张铃.问题求解理论及应用[M].北京:清华大学出版社,1990.(英文版: Bo Zhang and Ling Zhang, Theory and Application of Problem Solving, North-Holland. Elsevier Science Publishers B. V. 1992)
[Z&Z03a] Zhang, L., Zhang, B. The quotient space theory of problem solving [C]. Proceedings of International Conference on Rough Sets, Fuzzy Set, Data Mining and Granular Computing, Lecture Notes in Artrifical Intelligence 2639, 2003: 11-15.
[Z&Z03b] 张铃,张钹.模糊商空间理论(模糊粒度计算方法)[J].软件学报,2003,14(4):70-776.
[Z&Z03c] 张铃,张钹.商空间理论与粒度计算[C].CRSSC2003,重庆,2003.
[Z&Z04] Zhang, L. Zhang, B. The quotient space theory of problem solving [J]. Fundamenta Informaticae, IOS press, 2004, 59: 287-298.
[Zhang05] 张铃.浅谈粒度计算,2005(手稿).
[Z&W00] 张文修,吴伟志.粗糙集理论介绍与研究综述[J].模糊系统与数学,2000,14(4):1-12.
[Z&W02] 祝峰,王飞跃.关于覆盖广义粗集的一些基本结果[J].模式识别与人工智能,2002,15(1):6-13.
[Zhu00] 朱雪龙.应用信息论基础[M].北京:清华大学出版社,2000.
[Zu088] 左孝凌.离散数学[M].上海:上海科学技术文献出版社.1988.
[BBJ02] Boolos, GA., Burges, J. P., Jeffrey, R. Computability and logic, 4th edition [M]. Cambridge University Press, Cambridge, 2002.
[B&M02] Bettini, C., Montanari, A. Research issues and trends in spatial and temporal granularities [J]. Annals of Mathematics and Artificial Intelligence, 2002, 36: 1-4.
[Bir67] Birkhoff, G. Lattice theory, 3rd edtion [M]. Coll. Publ. ⅩⅩⅤ, American Mathematical Society, 1967.
[Bri93] Brink, C. Power structures [J]. Algebra Universalis, 1993, 30: 177-216.
[Bry89] Bryniarski, E. A calculus of rough sets of the first order [J]. Bulletin of the Polish Academy of Sciences, Mathematics, 1989, 37: 71-77.
[B&B98] Bonikowski, Z., Bryniarski, E. Extensions and intensions in the rough set theory [J]. Information Science, 1998, 107: 149-167.
[B&S81] Burris, S., Sankappanavar, H. A course in universal algebra [M]. Springer-verlag, 1981.
[Cech66] Cech. E. Topological spaces (revised by J. Frolik, Z. Katetov, and M. Novak) [M]. Wiley, New York, 1966.
[C&C04] Cheng, J. X., Chen. W. L. Quasi-discrete closure space and generalized rough approximate space based on binary relation [C]. Proceeding of the 3rd International Conference on Machine Learning and Cybernetics, Shanghai, 2004: 26-29.
[CCZ04] 陈万里,程家兴,张持健.基于相容关系推广粗糙集理论[J].计算机工程与应用,2004,40(4):26-28.
[C&C05] 陈万里,程家兴.粒计算的α_决策逻辑语言[J].控制与决策(已录用).
[C&X96] 蔡自兴,徐光祐.人工智能及应用(第二版)[M].北京:清华大学出版社,1996
[D&O98] Demri, S, Orlowska, E. Logical analysis of indiscernibility [C]. In: Incomplete Information: Rough Set Analysis, Orlowska, E. (Ed.). Physica-Verlag, Heidelberg, 1998, 347-380.
[D&P90] Dubois, D., Prade, P. Fuzzy rough sets and rough fuzzy sets [J]. International Journal of General Systems, 1990. 17: 191-209.
[DGO01] Duntsch I, Gediga G, Orlowska E. Relational attribute systems [J]. Int. Journal Human and Computer Studies, 2001, 55: 293-309.
[D&P02] Davey, B. A., Priestley, H. A. Introduction to lattice and order [M]. Cambridge University Press, Cambridge, 1992.
[Euz01] Euzenat, J. Granularity in relational formalisms-with application to time and space representation. Computational Intelligence, 2001, 17: 703-737.
[Gal03] Galton, A. A generalized topological view of motion in discrete space [J]. Theoretical Computer Science, 2003, 305: 111-134.
[G&W92] Ginuchglia, F., Walsh, T. A theory of abstraction [J]. Artificial Intelligence, 1992, 56: 323-390.
[Gra98] Gratzer, G. General lattice theory, 2nd edtion [M]. Birkhauser Verlag, 1998.
[Grz87] Grzymala-Busse, J. W. Rough-set and Dempster-Shafer approaches to knowledge acquisition under uncertainty-a comparison, Manuscript. Department of Computer Science, University of Kansas. 1987.
[Gua96] 关泽群.商空间下的遥感图象分析理论探讨[D].武汉:武汉测绘科技大学,1996
[HCC93] Han, J., Cai, Y., Cercone, N. Data-driven discovery of quantitative rules in databases [J]. IEEE Transactions on Knowledge and Data Engineering, 1993, 5: 29-40.
[H&P96] Hearst, M. A., Pedersen, J. O. Reexamining the cluster hypothesis: Scatter/Gather on retrieval results [C]. Proceedings of SIGIR'96, 1996: 76-84.
[Hob85] Hobbs, J. R. Granularity [C]. Proceedings of the Ninth International Joint Conference on Artificial Intelligence. 1985: 432-435.
[Hor01] Horusby, K. Temporal zooming [C]. Transactions in GIS 5. 2001: 255-272.
[HSL01] Hu, K., Sui, Y., Lu, Y., Wang, J. and Shi, C. Concept approximation in concept lattice, knowledge discovery and data mining [C]. Proceedings of the 5th Pacific-Asia conference. PAKDD 2001. Lecture Notes in Computer Science 2035, 2001: 167-173.
[HSL00] 胡可云,陆玉昌,石纯一.基于概念格的分类和关联规则的集成挖掘方法[J].软件学报.2000,11(11):1478-1484.
[Ime87] Imielinski, T. Domain abstraction and limited reasoning [C]. Proceedings of the 10th International Joint Conference on Artificial Intelligence. 1987: 997-1003.
[IHT03] Inuiguchi, M., Hirano, S., Tsumoto, S. (Eds.). Rough Set Theory and Granular Computing [M]. Springer, Berlin, 2003.
[K&F88] Klir, G. J., Folger, T. A. Fuzzy sets, uncertainty, and information. Prentice Hall, Englewood Cliffs, 1988.
[Kno93] Knoblock, C. A. Generating abstraction hierarchies: an automated approach to reducing search in planning [M]. Kluwer Academic Publishers. Bosto, 1993.
[Kel55] Kely, J. L. General topology [M]. Springer-Verlag, 1955. (北京:世界图书出版社,2001).
[Ken96] Kent, R. E. Rough concept analysis: a synthesis of rough sets and formal concept analysis [J]. Fundamenta Informaticae, 1996, 27: 69-181.
[Kol97] Kolman, B. Discrete mathematics structures(影印版)[M].北京:清华大学出版社,1997.
[L&D89] Lau, K. F., Dill, K. A. A lattice statistical mechanics model of the conformational and sequence space of proteins [J]. Macromolecules. 1989. 22: 3986-3997.
[L&D90] Lau, K. F., Dill. K. A. Theory for protein mutability and biogenesis [C]. Proceedings of the National Academy of Sciences USA, 1990, 87: 638-642.
[Lee87] Lee, T. T. An information-theoretic analysis of relational databases-part Ⅰ: data dependencies and information metric [J]. IEEE Transactions on Software Engineering SE-13, 1987: 1049-1061.
[Ler83] Leron, U. Structuring mathematical proofs [J]. American Mathematical Monthly, 1983, 90: 174-185.
[Lin88] Lin, T. Y. Neighborhood systems and approximation in relational databases and knowledge bases [C]. Proceedings of the 4th International Symposium on Methodologies of Intelligent Systems, 1988.
[Lin92] Lin, T. Y. Topological and fuzzy rough sets. In: Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Slowinski, R. (Ed.). Kluwer Academic Publishers, Boston, 1992: 287-304.
[Lin97a] Lin, T. Y. Neighborhood systems-application to qualitative fuzzy and rough sets [C]. in: Advances in Machine Intelligence and Soft-Computing edited by P. P. Wang, Department of Electrical Engineering, Duke University, Durham, North Carolina, USA, 1997: 132-155.
[Lin97b] Lin, T. Y. Granular computing, announcement of the BISC special interest group on granular computing. 1997. http://www.cs.uregina.ca/～yyao/GrC/.
[Lin98a] Lin, T. Y., Zhong, N., Dong, J., Ohsuga, S. Frameworks for mining binary relations in data [C]. Rough sets and Current Trends in Computing, Proceedings of the 1st International Conference. Lecture Notes in Artificial Intelligence 1424. 1998: 387-393.
[Lin98b] Lin, T. Y. Granular computing on binary relations Ⅰ: data mining and neighborhood systems, Ⅱ: rough set representations and belief functions [C]. In: Rough Sets in Knowledge Discovery 1. Polkowski, L., Skowron, A. (Eds.). Physica-Verlag, Heidelberg, 1998: 107-140.
[Lin01] Lin, T. Y. Generating concept hierarchies/networks: mining additional semantics in relational data [C]. Advances in Knowledge Discovery and Data Mining. Proceedings of the 5th Pacific-Asia Conference. Lecture Notes on Artificial Intelligence 2035, 2001: 174-185.
[Lin02] Lin. T. Y., Yao, Y. Y., Zadeh, L. A. (Eds.) Rough Sets, Granular Computing and Data Mining [M]. Physica-Verlag, Heidelberg, 2002.
[Lin03] Lin. T. Y. Granular computing: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing [C]. Proceedings of the 9th International Conference, Lecture Notes in Artificial Intelligence 2639, 2003: 16-24.
[Lip81] Lipski W. On databases with incomplete information [J]. Journal of the ACM, 1981, 28: 41-70.
[Luce56] Luce, R. D. Semi-orders and a theory of utility discrimination [J]. Econometrica, 1956. 24.
[Li04] 李进金.覆盖广义粗集理论的拓扑学方法[J].模式识别与人工智能,2004,17(1):7-10.
[Liu01] 刘清.Rough集与Rough推理[M].北京:科学出版社,2001.
[Liu04] 刘清,黄兆华.G-逻辑及其归结原理[J].计算机学报,2004,27(7):865-873.
[L&W00] 刘普寅,吴孟达.模糊集理论与应用[M].长沙:国防科技大学出版社,2000.
[Mar00] Martin T. Lecture notes on biological sequence analysis. 2000.
[Man98] Mani, I. A theory of granularity and its application to problems of polysemy and underspecification of meaning [C]. Principles of Knowledge Representation and Reasoning, Proceedings of the Sixth International Conference, 1998: 245-255.
[MGB92] McCalla, G., Greer, J., Barrie, J., Pospisil, P. Granularity hierarchies [J]. Computers and Mathematics with Applications, 1992, 23: 363-375.
[Mun04] Munkres, J. R. Topology, 2nd edition[M].北京:机械工业出版社,2004.
[Nov03] Novak, V. Intentional theory of granular computing [J]. Soft computing, 2003.
[NSZ99] Nayak, A., Sinclair, A, Zwick. U. Journal Comp. Biol. 1999, 6: 13
[O&P87] Orlowska E, Pawlak Z. Representation of non-deterministic information [J]. Theoretical Computer Science, 1987, 29: 27-39.
[Orl85] Orlowska, E. Logic of indiscernibility relations [J]. Bulletin of the Polish Academy of Sciences. Mathematics, 1985, 33: 475-485.
[Paw82] Pawlak, Z. Rough sets [J]. Imernational Journal of Computer and Information Sciences, 1982, 11: 341-356.
[Paw91] Pawlak, Z. Rough Sets: Theoretical aspects of reasoning about data [M]. Kluwer Academic Publishers, Boston. 1991.
[Paw98] Pawlak, Z. Granularity of knowledge, indiscernibility and rough sets [C]. Proceedings of 1998 IEEE International Conference on Fuzzy Systems, 1998: 106-110.
[Paw02] Pawlak Z. Rough sets, decision algorithm and Bayes(?)s theorems [J]. European Journal of Operational research, 2002, 136: 181-189.
[Ped01] Pedrycz, W. Granular computing: an emerging paradigm [M]. Springer-Verlag, Berlin, 2001.
[P&S98] Polkowski, L., Skowron, A. Towards adaptive calculus of granules [C]. Proceedings of 1998 IEEE International Conference on Fuzzy Systems, 1998: 111-116.
[Pag93] Pagliani, P. From concept lattices to approximation spaces: algebraic structures of some spaces of partial objects [J]. Fundamenta Informaticae, 1993, 18: 1-25.
[Pei04] 裴道武.两类广义粗糙集模型[J].模式识别与人工智能,2004,17(3):281-285.
[PBL02] 卜东波,白硕,李国杰.聚类/分类中的粒度原理[J].计算机学报,2002,25(8):10-16.
[Qua00] Quafafou M. α_RST: a generalization of rough set theory [J]. Information Science, 2000, 124: 301-316.
[R&N95] Russell, S., Norvig, P. Artificial Intelligence: a modern approach [M]. Pearson Education, Inc., Prentice-Hall, 1995.(北京:人民邮电出版社,2002.)
[Rus69] Ruspini, E. H. A new approach to clustering [J]. Information and Control, 1969, 15: 2-32.
[R&H00] 阮达,黄祟福 译,模糊集与模糊信息粒理论[M].北京:北京师范大学出版社,2000.
[S&Z98] Saitta, L., Zucker, J. D. Semantic abstraction for concept representation and learning [C]. Proceedings of the Symposium on Abstraction, Reformulation and Approximation, 1998: 103-120. http://www.cs.vassar.edu/_ellman/sara98/papers/.
[S&M83] Salton, G., McGill, M. Introduction to modern reformation retrieval [M]. McGraw Hill, New York, 1983.
[Sha76] Sharer, G. A mathematical theory of evidence [M]. Princeton University Press, Princeton, 1976.
[Sko01] Skowron, A. Toward intelligent systems: calculi of information granules [J]. Bulletin of International Rough Set Society, 2001, 5: 9-30.
[S&G94] Skowron, A., Grzymala-Busse, J. From rough set theory to evidence theory [C]. In: Advances in the Dempster-Shafer Theory of Evidence, Yager, R. R., Fedrizzi, M., Kacprzyk, J. (Eds.). Wiley, New York, 1994: 193-236.
[S&S98] Skowron, A., Stepaniuk, J. Information granules and approximation spaces [C]. Proceedings of 7th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, 1998: 354-361.
[Sla03] Slapal, J. Closure operations for digital topology [J]. Theoretical Computer Science, 2003, 305: 457-471.
[S&S01] Skowron, A., Stepaniuk, J. Information granules: towards foundations of granular computing [J]. International Journal of Iutelligent Systems 16 (2001): 57-85.
[Smi89] Smith. E. E. Concepts and induction [M]. In: Foundations of Cognitive Science, Posner, M. I. (Ed.). The MIT Press, Cambridge (1989), 501-526.
[Smi95] Smith, M. B. Semi-metrics, closure spaces and digital topology: [J]. Theoretical Computer Science, 1995, 151: 257-276.
[Sow84] Sowa, J. F. Conceptual structures [C]. Information Processing in Mind and Machine. Addison-Wesley, Reading, 1984.
[S&W98] Stell, J. G., Worboys, M. F. Stratified map spaces: a formal basis for multi-resolution spatial databases [C]. Proceedings of the 8th International Symposium on Spatial Data Handling, 1998: 180-189.
[S&V97] Slowinski, R., Vanderpooten, D. Similarity relation as a basis for rough approximations [C]. In P. P. Wang (ed.), Advances in Machine Intelligence and Soft-Computing, Department of Electrical Engineering, Duke University, Durham, NC, USA. 1997: 17-33.
[S&V00] Slowinski, R., Vanderpooten, D. A generalized definition of rough approximation based on similarity [J]. IEEE Transactions on Knowledge and Data Engineering, 2000, 12 (2): 331-336.
[Sun83] 孙以丰译.基础拓扑学[M].北京:北京大学出版社,1983.
[Tar56] Tarski, A. Semantics, metamathematics and logic [M]. Oxford University Press, Oxford, 1956.
[Tve77] Tversky, A. Features of similarity [J]. Psychological review, 1977, 84(4): 327-352.
[Wil92] Wille, R.: Concept lattices and conceptual knowledge systems [J]. Computers Mathematics with Applications, 1992, 23: 493-515.
[Wol93] Wolff, K. E. A first course in formal concept analysis__ how to understand line diagram [C]. In: Faulbaum, F. (ed.) SoftStat'93, Advances in Statistical Software, 1993, 4: 429-438.
[W&S] William, H., Sorin. I. HP Benclunarks. http://www.cs.sandia.gov/tech_reports/compbio/tortilla-hp-benchmarks.html
[Wang00] 王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2000.
[Wang01] 王国胤.Rough集理论与知识获取[M].西安:西安交通大学出版社,2001.
[WMZ96] 王钰.苗夺谦,周育健.关于Rough Set理论与应用的研究[J].模式识别与人工智能,1996,9(4):337-344.
[Xio02] 熊金城.点集拓扑讲义(第2版)[M].北京:高等教育出版社,2002
[Xu04] 徐峰.基于商空间的计算智能及其在金融工程中的应用研究[D].合肥:安徽大学,2004.
[Yao95] Yao, Y. Y., Noroozi, N. A unified framework for set-based computations [C]. Proceedings of the 3rd International Workshop on Rough Sets and Soft Computing. The Society for Computer Simulation, 1995: 252-255.
[Yao96] Yao, Y. Y. Two views of the theory of rough sets in finite universes [J]. International Journal of Approximation Reasoning, 1996, 15: 291-317.
[Y&W97] Yao, Y. Y., Wong, S. K. M., Lin, T. Y. A review of rough set models [C]. In: Rough Sets and Data Mining: Analysis for Imprecise Data, Lin, T. Y., Cercone, N. (Eds.). Kluwer Academic Publishers, Boston, 1997: 47-75.
[Yao98] Yao, Y. Y. Relational interpretations of neighborhood operators and rough set approximation operator [J]. Information Science, 1998, 111: 239-259.
[Yao99] Yao, Y. Y. Granular computing using neighborhood systems [C]. In: Advances in Soft Computing: Engineering Design and Manufacturing, Roy, R., Furuhashi, T., and Chawdhry, P. K. (Eds), Springer-Verlag, London, 1999: 539-553.
[Yao00] Yao, Y. Y. Granular computing: basic issues and possible solutions [C]. Proceedings of the 5th Joint Conference on Information Sciences. 2000: 186-189.
[Yao01a] Yao, Y. Y. Information granulation and rough set approximation [J]. International Journal of Intelligent Systems, 2001, 16: 87-104.
[Yao01b] Yao, Y. Y. Modeling data mining with granular computing [C]. Proceedings of the 25th Annual International Computer Software and Applications Conference (COMPSAC 2001), 2001: 638-643.
[Yao02] Yao, Y. Y., Liau, C. J. A generalized decision logic language for granular computing [C]. Proceedings of FUZZ-IEEE'02 in the 2002 IEEE World Congress on Computational Intelligence, 2002: 1092-1097.
[Yao03] Yao, Y. Y., Liau, C. J., Zhong, N. Granular computing based on rough sets, quotient space theory, and belief functions [C]. Proceedings of ISMIS'03, 2003: 152-159.
[Y&Zh02] Yao, Y. Y., Zhong, N. Granular computing using information tables [C]. In: Data Mining, Rough Sets and Granular Computing, Lin, T. Y., Yao. Y. Y., Zadeh, L. A. (Eds.). Physica-Verlag, Heidelberg, 2002: 102-124.
[Y&C97] Yao, Y. Y., Chen, X. C. Neighborhood based information systems [C]. Proceedings of the 3rd Joint Conference on Information Sciences, Volume 3: Rough Set & Computer Science Research Triangle Park, North Carolina; USA, 1997, March 1-5, 154-157.
[Y&F98] Yager, R. R., Filev, D. Operations for granular computing: mixing words with numbers [C]. Proceedings of 1998 IEEE International Conference on Fuzzy Systems, 1998: 123-128.
[Yao04a] Yao, Y. Y. Concept lattices in rough set theory [C]. Proceedings of 23rd International meeting of the North American Fuzzy Information Processing Society, 2004.
[Yao04b] Yao, Y. Y. A partition model of granular computing. 2004.http://www.cs.uregina.ca/～yyao/GrC/.
[Y&C04] Yao, Y. Y. and Chen, Y. H. Rough set approximations in formal concept analysis [C]. Proceedings of 23rd International Meeting of the North American Fuzzy Information Processing Society, 2004.
[Yao05] Yao, Y. Y. Granular computing 2005. http://www.cs.uregina.ca/～yyao/GrC/.
[You97] 尤承业.基础拓扑学讲义[M].北京:北京大学出版社.1997.
[Zad65] Zadeh, L. A. Fuzzy sets [J]. Information and Control, 1965, 8: 338-353.
[Zad71] Zadeh, L. A. Similarity relations and fuzzy ordering [J]. Information Science, 1971, 3: 171-200.
[Zad79] Zadeh, L. A. Fuzzy sets and information granularity [J]. In: Advances in Fuzzy Set Theory and Applications, Gupta, N., Ragade, R., Yager, R. (Eds.). North-Holland. Amsterdam 1979: 3-18.
[Zad96] Zadeh, L. A Fuzzy logic=computing with words [J]. IEEE Transactions on Fuzzy Systems, 1996, 4: 103-111.
[Zad97a] Zadeh, L. A. Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic [J]. Fuzzy Sets and Systems, 1997, 90: 111-127.
[Zad97b] Zadeh L. A. Annotmcement of GrC. 1997. http://www.cs.uregina.ca/～yyao/GrC/.
[Zad98] Zadeh, L. A. Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information/intelligent systems [J]. Soft Computing, 1998, 2: 23-25.
[Zak83] Zakowski, W. Axiomization in the space (U, Π). Demonstratio Mathematica, 1983, 16: 761-769.
[Z&Z90] 张钹,张铃.问题求解理论及应用[M].北京:清华大学出版社,1990.(英文版: Bo Zhang and Ling Zhang, Theory and Application of Problem Solving, North-Holland. Elsevier Science Publishers B. V. 1992)
[Z&Z03a] Zhang, L., Zhang, B. The quotient space theory of problem solving [C]. Proceedings of International Conference on Rough Sets, Fuzzy Set, Data Mining and Granular Computing, Lecture Notes in Artrifical Intelligence 2639, 2003: 11-15.
[Z&Z03b] 张铃,张钹.模糊商空间理论(模糊粒度计算方法)[J].软件学报,2003,14(4):70-776.
[Z&Z03c] 张铃,张钹.商空间理论与粒度计算[C].CRSSC2003,重庆,2003.
[Z&Z04] Zhang, L. Zhang, B. The quotient space theory of problem solving [J]. Fundamenta Informaticae, IOS press, 2004, 59: 287-298.
[Zhang05] 张铃.浅谈粒度计算,2005(手稿).
[Z&W00] 张文修,吴伟志.粗糙集理论介绍与研究综述[J].模糊系统与数学,2000,14(4):1-12.
[Z&W02] 祝峰,王飞跃.关于覆盖广义粗集的一些基本结果[J].模式识别与人工智能,2002,15(1):6-13.
[Zhu00] 朱雪龙.应用信息论基础[M].北京:清华大学出版社,2000.
[Zu088] 左孝凌.离散数学[M].上海:上海科学技术文献出版社.1988.