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北京师范大学模糊系统与人工智能方向简介（讨论稿）
北京师范大学模糊数学与人工智能方向是国内最早从事模糊数学及其应用研究的单位之一，可以说是国内模糊数学研究的重要基地。早在1979年北师大数学科学学院开始就开始招收模糊数学研究方向的硕士研究生，是我国最早从事模糊数学研究的硕士学科点。1986年，汪培庄先生牵头，以模糊数学为主申请下来应用数学博士点，这也是我国最早从事模糊数学研究的博士学科点。迄今为止，北师大数学科学学院已培养几十名硕士和博士研究生，并且在各种工作岗位已成为骨干力量。
北京师范大学模糊系统与模糊信息研究中心暨复杂系统实时智能控制实验室创建于2000年。现任中心主任为国家级有突出贡献中青年专家李洪兴教授。目前，实验室拥有博导教授2人，副教授3人，博士后2人，在读博士生15人（其中具有教授职称者2人，副教授4人），硕士研究生19人。该研究中心现有一个应用数学的博士学位授权点，应用数学和控制理论与控制工程两个硕士学位授权点。
1982年至今，北京师范大学模糊数学与人工智能研究群体先后提出并研究了因素空间、真值流推理、随机集落影、模糊计算机、模糊摄动理论、幂结构提升理论、基于变权综合的智能信息处理、模糊系统的插值表示、变论域智能计算、复杂系统建模以及知识表示的数学理论模糊计算机等一些先进的理论方法。近期的主要研究成果包括：
1）给出因素空间理论，建立知识表示的数学框架，并系统研究概念的内涵与外延表示问题，为专家经验、领域知识在软件系统中的表示与计算提供了理论基础；
2）揭示了模糊逻辑系统的数学本质，给出常用模糊逻辑系统地插值表示，并系统研究了模糊逻辑系统的构造、分析以及泛逼近性等理论问题；
3）提出变论域自适应智能信息处理理论，设计了基于变论域思想的一类高精度模糊控制器，在世界上第一个实现了四级倒立摆控制实物系统，经教育部组织专家鉴定，确认这是一项原创性的具有国际领先水平的重大科研成果；
4）引入变权的概念，并给出基于自适应变权理论的智能信息处理方法；
5）提出模糊计算机的概念，并研究了模糊计算机设计的若干理论问题；
6）给出数学神经网络理论，从数学上揭示了模糊逻辑系统与人工神经网络之间的关系，首次定义了“输出返回”的模糊逻辑系统并证明了它与反馈式神经
网络等价；
7）提出一种基于数据集成、规则提取和模糊推理的复杂系统的建模方法，即基于模糊推理的建模方法，由此可突破障碍模糊控制理论发展的一些瓶颈问题，诸如稳定性、能控性、能观测性等的判据问题。
8）揭示了Fuzzy系统的概率论意义，指出Fuzzy系统中常用的清晰化方法，即重心法是合理的且在平均平方意义下是最优的方法. 基于不同的Fuzzy蕴涵算子，给出几种典型的概率分布，如Zadeh分布、Mamdani分布、Lukasiewicz 分布等，它们充当Fuzzy系统的“系统内核”作用.
9）利用随机过程理论刻画了变论域自适应模糊控制的自适应机理，并且为经典自适应控制的自适应机理的源头给出了数学描述。
模糊信息处理与人工智能目前已是计算机科学的重要分支之一，在国际范围里已有相当的影响。据了解，在北美、欧洲和日本的许多大学的计算机系，都有专家在从事模糊信息处理与人工智能研究与教学；他们所取得的成果已显示出模糊信息处理与人工智能在计算机科学研究与发展中的巨大作用。北京师范大学是国内模糊系统及其应用研究的发源地之一，李洪兴及其带领的学术团队在这一领域做出了一系列原创性、具有世界领先或先进水平的研究成果，在国内外具有重要的影响。由于模糊信息处理是新兴学科，我国在这一领域的研究水平与国外差距不大，因此，发展模糊信息处理与智能计算等相关领域的研究对于我国计算机科学研究赶超世界先进水平是很有意义的。
附录该方向近期主要研究成果
一. 获奖情况
[1] 1987年获国家自然科学四等奖。
[2] 2003年获教育部自然科学一等奖。
[3] 2005年获国家科技进步二等奖。
二. 已获得的专利
[1] 发明专利：预焙电解铝生产的变论域自适应模糊控制方法及装置
发明人：李洪兴；王加银；谷云东；冯艳宾
专利号：ZL 02 1 58123.1
授权公告日：2004年12月1日
[2] 实用新型专利：四级倒立摆实验仪
设计人：李洪兴
专利号：ZL 02 2 57287.2
授权公告日：2003年9月17日
三．已出版的主要著作
[1] Hong-Xing Li, P.C. Philip Chen and Han-Pang Huang, Fuzzy Neural Intelligent
Systems, CRC Press, FL, USA, 2001.
[2] Hong-Xing Li and Vincent C.Y en, Fuzzy Sets and Fuzzy Decision-Making,
CRC Press, FL, USA, 1995.
[3] Peizhuang Wang and Hong-Xing Li, Fuzzy Information Processing and Fuzzy
Computers, Science Press New Y ork/Beijing, 1997.
[4] Puyin Liu and Hong-Xing Li, Fuzzy Nerual Network Theory and Application,
W orld Scientific Press, Singapore, 2004.
[5]汪培庄，李洪兴, 知识表示的数学理论，天津科技出版社，1994.
[6]汪培庄，李洪兴, 模糊系统理论与模糊计算机，科学出版社，1996.
[7]李洪兴等, 工程模糊数学方法及应用，天津科技出版社，1993.
[8]李洪兴，汪培庄, 模糊数学，国防工业出版社，1994.
四. 已发表的主要论文（按时间顺序）
[1]Hong-Xing Li, Fuzzy perturbation analysis, part 1, Fuzzy Sets and Systems,
17(2), 1985, 189-197. (SCI)
[2]Hong-Xing Li, Fuzzy perturbation analysis, part 2, Fuzzy Sets and Systems,
19(2), 1986, 165-175. (SCI)
[3]Hong-Xing Li, Multifactorial fuzzy sets and multifactorial degree of nearness,
Fuzzy Sets and Systems, 19(3), 1986, 291-297. (SCI)
[4]Hong-Xing Li, Fuzzy clustering methods based on perturbation, Fuzzy Sets and
Systems, 32(3), 1989, 291-302. (SCI)
[5]Hong-Xing Li, Multifactorial functions in fuzzy sets theory, Fuzzy Sets and
Systems, 33(1), 1990, 69-84. (SCI)
[6]Hong-Xing Li, Chengzhong Luo and Peizhuang Wang, The cardinality of fuzzy
sets and the continuum hypothesis, Fuzzy Sets and Systems, 55(1993), 61-77.
(SCI)
[7]Hong-Xing Li etc, The operations of fuzzy cardinalities, Journal of
Mathematical Analysis and Applications, 182(1994),768-778. (SCI)
[8]Hong-Xing Li, Interpolation mechanism of fuzzy control, Science in China (中
国科学, Series E), 41(3), 1998, 312-320. (SCI)
[9]Hong-Xing Li, Factor spaces theory and its applications to fuzzy information
processing (I), Fuzzy Sets and Systems, 95(2), 1998, 147-160. (SCI)
[10]H ong-Xing Li, The relationship between fuzzy controllers and PID controllers,
Science in China (中国科学, Series E), 42(2), 1999, 215-224. (SCI)
[11]H ong-Xing Li, Adaptive fuzzy controllers based on variable universe, Science in
China (中国科学, Series E), 42(1), 1999, 10-20. (SCI)
[12]H ong-Xing Li and L.X. Li, Representing diverse mathematical problems using
neural networks in hybrid intelligent systems, Expert Systems, 16(4), 1999, 262-272. (SCI)
[13]H ong-Xing Li and C.L. Philip Chen, The equivalence between fuzzy logic
systems and feedforward neural networks, IEEE Transactions on Neural Networks, 11(2), 2000, 356-365. (SCI)
[14]H ong-Xing Li, Output-back fuzzy logic systems and equivalence with feedback
neural networks, Chinese Science Bulletin（科学通报）, 45(7), 2000, 592-596.
(SCI)
[15]H ong-Xing Li and Li-Da Xu, A neural network representation of linear
programming, European Journal of Operational Research, 124(2), 2000, 224-234. (SCI)
[16]H ong-Xing Li, V.C. Y en and E.S. Lee, Model of neurons based on factor space,
Computers and Mathematics with Applications, 39(2000), 91-100. (SCI) [17]H ong-Xing Li etc, Factor spaces theory and its applications to fuzzy information
processing: Two kinds of factor space canes, Computers and Mathematics with Applications, 40(2000), 835-843. (SCI)
[18]H ong-Xing Li etc, Factor space theory and fuzzy information processing: Fuzzy
decision making based on the concepts of feedback extension, Computers and Mathematics with Applications, 40(2000), 845-864. (SCI)
[19]H ong-Xing Li, V.C. Y en and E.S. Lee, Factor space theory in fuzzy information
processing: Composition of states of factors and multifactorial decision making, Computers and Mathematics with Applications, 39(2000), 245-265. (SCI) [20]P u-Yin Liu and Hong-Xing Li, Approximation of generalized fuzzy systems to
integrable functions, Science in China (中国科学, Series E), 43(6), 2000, 613-624. (SCI)
[21]Q ing He and Hong-Xing Li, Extension principles and fuzzy set categories,
Computers and Mathematics with Applications, 39(2000), 45-53. (SCI) [22]V.C. Y en and Hong-Xing Li, Concept representation, factor space theory and
information systems research, Systems Research and Behavioral Science, 17(2000), 163-172. (SCI)
[23]H ong-Xing Li and Li D Xu, Feature space theory—a mathematical foundation
for data mining, Knowledge-Based Systems, 14(5-6), 2001, 253-257. (SCI) [24]P u-Yin Liu and Hong-Xing Li, Analyses for L p( )-norm approximatio n
capability of generalized Mamdani fuzzy systems, Information Sciences,138 (2001), 195-210. (SCI)
[25]B ing-Xue Yao and Hong-Xing Li, Weak HX-ring on a ring, Italian Journal of
Pure and Applied Mathematics, No.10, 2001, 125-131.
[26]H ong-Xing Li,Zhi-Hong Miao, Jia-Yin Wang, V ariable universe adaptive fuzzy
control on the quadruple inverted pendulum, Science in China (中国科学, Series
E), 2002, 45(2), 213-224. (SCI)
[27]H ong-Xing Li, Zhi-Hong Miao, Jia-Yin Wang, V ariable universe stable adaptive
fuzzy control of nonlinear system, Science in China (中国科学, Series E), 2002, 45(3), 225-240. (SCI)
[28]H ong-Xing Li, Jia-Yin Wang, Zhi-Hong Miao, Modeling on fuzzy control
systems, Science in China (中国科学, Series A), 2002, 45(12), 1506-1517.
(SCI)
[29]X ue-Hai Yuan, Hong-Xing Li and E. Stanley Lee, Categories of fuzzy sets and
weak topos, Fuzzy Sets and Systems, 127(2002), 291-297. (SCI)
[30]Pu-Yin Liu and Hong-Xing Li, Image restoration techniques based on fuzzy
neural networks, Science in China (中国科学, Series F), 2002, 45(4), 274-285.
(SCI)
[31]Hong-Xing Li,Li D Xu, Jia-Yin Wang and Zhi-Wen Mo, Feature space theory
in data mining: transformations between extensions and intensions in knowledge representation, Expert Systems, 20(2), 2003, 60-71. (SCI)
[32]Hong-Xing Li,Jia-Yin Wang and Zhi-Hong Miao, Marginal linearization
method in modeling on fuzzy control systems, Progress in Nature Science, 2003, 13(7), 489-496. (SCI)
[33]H ong-Xing Li,Ling-Xia Li and Jia-Yin Wang, Interpolation representation of
feedforward neural networks, Mathematical and Computer Modeling, 37, 7-8, (2003), 829-847. (SCI)
[34]Hong-Xing Li and E. S. Lee, Interpolation representations of fuzzy logic
systems, Computers & Mathematics with Applications, V olume 45, Issues 10-11, Pages 1683-1693 (May - June 2003). (SCI)
[35]Hong-Xing Li , Yan-Da Li, Zhi-Hong Miao and E. S. Lee, Control functions of
fuzzy controllers, Computers & Mathematics with Applications, V olume 46, Issues 5-6, Pages 875-890 (September2003). (SCI)
[36]Hong-Xing Li and E. S. Lee, Interpolation functions of feedforward neural
networks, Computers & Mathematics with Applications, V olume 46, Issue 12, Pages 1861-1874 (December 2003). (SCI)
[37]S hang-Ming Zhou, Hong-Xing Li and Li-Da Xu, A variational approach to
intensity approximation for remote sensing images using dynamic neural, Expert Systems, 20(4), 2003, 163-170. (SCI)
[38]Qing He, Hong-Xing Li, Zhong-Zhi Shi and E.S. Lee, Fuzzy clustering method
based on perturbation, Computers & Mathematics with Applications, V olume 46, Pages 929-946 ( 2003). (SCI)
[39]H ong-Xing Li, Li LX, W ang Jiayin, Mo Zhiwen, Li Y anda, Fuzzy decision
making based on variable weights, Mathematical and Computer Modeling, 39(2004), 163-179. (SCI)
[40]H ong-Xing Li, Y ou Fei and Peng Jiayin, Fuzzy controllers based on some fuzzy
implication operators and their response functions, Progress in Nature Science, 2004, 14(1), 15-20. (SCI)
[41]Pu-Yin Liu and Hong-Xing Li, Fuzzy techniques in restoration research—A
survey (invited paper), International Journal of Computational Cognition, 2(2), 2004, 131-149.
[42]Song-Chol Han and Hong-Xing Li, Indices and periods of incline matrices,
Linear Algebra and Its Applications, 387(2004), 143-165. (SCI)
[43]Pu-Yin Liu and Hong-Xing Li, Efficient learning algorithms for three-layer
regular feedforward fuzzy neural networks, IEEE Transactions on Neural Networks, 15(3), 2004, 545-558. (SCI)
[44]Xiao-Jun Tong, Mianyun Chen, Hong-Xing L i, Pan-operators, strunture with
non-idempotent pan-addition, Fuzzy Sets and Systems, 145, 2004, 463-470.
(SCI)
[45]Song-Chol Han and Hong-Xing Li, Invertible incline matrices and Cramer’s
rule over inclines, Linear Algebra and Its Applications, 389(2004), 121-138.
(SCI)
[46]Song-Chol Han, Hong-Xing Li and Yun-Dong Gu, Standard eigenvectors of
incline matrices, Linear Algebra and Its Applications, 389(2004), 235-248.
(SCI)
[47]Song-Chol Han, Hong-Xing Li, The semigroup of incline Hall matrices, Linear
Algebra and Its Applications, 390(2004), 183-196. (SCI)
[48]Hong-Xing Li,Jia-Yin Wang, Y an-Bin Feng and Yun-Dong Gu, Hardware
implementation of the quadruple inverted pendulum with single motor, Progress in Nature Science, 2004, 14(9), 822-827. (SCI)
[49]Zhihong Miao, Hongxing Li, Jiayin Wang. Optimal adaptive fuzzy control
systems with H tracking performance. Advances in Systems Science and Applications, 2004, 4(3): 367-374.
[50]Pu-Yin Liu and Hong-Xing Li, Approximation analysis of feedforward regular
fuzzy netural network with two hidden layers, Fuzzy Sets and Systems, 150, 2005, 373-396. (SCI)
[51]Pu-Yin Liu and Hong-Xing L i, Hierarchical TS fuzzy system and its universal
approximation, Information Sciences, 169(3-4), 2005, 279-303. (SCI) [52]Song-Chol Han, Hong-Xing Li, Notes on “Pseudo-t-norms and implication
operators on a complete Brouwerian lattice” and “Pseu do-t-norms and implication operators: direct products and direct product decompositions”, Fuzzy Sets and Systems, 153(2005), 289-294. (SCI)
五. 承担科研基金情况（只列主持的项目，参加的项目略去不列）
注：国家自然科学基金项目“基于论域的自适应模糊控制器(批准号：69974006)”结题后被国家自然科学基金委评为特优项目。