Abstract The strictly convex function is introduced into the research of knowledge granularity for the first time. Based on the strictly convex function, a theory framework for constructing knowledge granularity is proposed. A series of knowledge granularity measuring functions is derived under this framework. It is proved that the existing knowledge granularity measuring functions are the special cases or variations of knowledge granularity measures which are derived by strictly convex functions. The definition of the relative knowledge granularity based on strictly convex function is given. Its monotonicity is proved for some special strictly convex functions and the corresponding equality conditions are provided, although it does not hold for general strictly convex functions. It is proved that the existing two conditional information entropies are the special forms of the proposed relative knowledge granularity. Their knowledge granularity essence is revealed. For a consistent decision table, it is proved that the relative knowledge granularity is equivalent to positive region for each other. Therefore, the attribute reduction judgment method of algebraic reduction is presented by the relative granularity in consistent decision table. The correctness of the proposed conclusions is showed by a numerical example.
[1] Xu Jiucheng, Shi Jinling, Sun Lin. Attribute Reduction Algorithm Based on Relative Granularity in Decision Tables. Computer Science, 2009, 36(3): 205-207 (in Chinese) (徐久成,史进玲,孙 林.一种基于相对粒度的决策表约简算法.计算机科学, 2009, 36(3): 205-207) [2] Chen Yuming, Wu Keshou, Xie Rongsheng. Reduction for Decision Table Based on Relative Knowledge Granularity. Journal of Shandong University: Engineering Science, 2012, 42(6): 8-12(in Chinese) (陈玉明,吴克寿,谢荣生.基于相对知识粒度的决策表约简.山东大学学报:工学版, 2012, 42(6): 8-12) [3] Wang Guoyin, Zhang Qinghua, Ma Xiao, et al. Granular Computing Models for Knowledge Uncertainty. Journal of Software, 2011, 22(4): 676-694 (in Chinese) (王国胤,张清华,马希骜,等.知识不确定性问题的粒计算模型.软件学报, 2011, 22(4): 676-694) [4] Liang Jiye, Wang Junhong, Qian Yuhua. A New Measure of Uncertainty Based on Knowledge Granulation for Rough Sets. Information Sciences, 2009, 179(4): 458-470 [5] Miao Duoqian, Fan Shidong. The Calculation of Knowledge Granulation and Its Application. Systems Engineering-Theory & Practice, 2012, 22(1): 48-56 (in Chinese) (苗夺谦,范世栋.知识的粒度计算及其应用.系统工程理论与实践, 2002, 22(1): 48-56) [6] Liang Jiye, Shi Zhongzhi. The Information Entropy, Rough Entropy and Knowledge Granulation in Rough Set Theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2004, 12 (1): 37-46 [7] Liang Jiye, Li Deyu. The Uncertainty and Knowledge Acquisition in Information System. Beijing, China: Science Press, 2005 (in Chinese) (梁吉业,李德玉.信息系统中的不确定性与知识获取.北京:科学出版社, 2005) [8] Liang Jiye, Shi Zhongzhi, Wierman M J. Information Entropy, Rough Entropy and Knowledge Granulation in Incomplete Information Systems. International Journal of General Systems, 2006, 35(6): 641-654 [9] Liang Jiye, Qian Yuhua. Information Granularity and Entropy Theory in Information System. Science in China: Series E, 2008, 38(12): 2048-2065 (in Chinese) (梁吉业,钱宇华.信息系统中的信息粒与熵理论.中国科学:E辑, 2008, 38(12): 2048-2065) [10] Wang Junhong, Liang Jiye, Qian Yuhua. Uncertainty Measure of Rough Sets Based on a Knowledge Granulation for Incomplete Information Systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2008, 16(2): 233-244 [11] Qian Yuhua, Liang Jiye, Wang Feng. A New Method for Measuring the Uncertainty in Incomplete Information Systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2009, 17(6): 855-880 [12] Qian Yuhua, Liang Jiye. Combination Entropy and Combination Granulation in Rough Set Theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2008, 16(2): 179-193 [13] Liang Jiye, Qian Yuhua. Axiomatic Approach of Knowledge Granulation in Information System // Proc of the 19th Australian Joint Conference on Artificial Intelligence. Hobart, Australia, 2006: 1074-1078 [14] Beaubouef T, Petry F E, Arora G. Information-Theoretic Measures of Uncertainty for Rough Sets and Rough Relational Databases. Information Sciences, 1998, 109(1/2/3/4): 185-195 [15] Beaubouef T, Petry F E. Fuzzy Rough Set Techniques for Uncertainty Processing in a Relational Database. International Journal of Intelligent Systems, 2000, 15(5): 389-424 [16] Zhao Mingqing, Yang Qiang. Axiomatic Definition and Measure Method of Knowledge Granularity in Incomplete Information System. Pattern Recognition and Artificial Intelligence, 2008, 21(6): 728-731 (in Chinese) (赵明清,杨 强.不完备信息系统中知识粒度的公理化定义与度量方法.模式识别与人工智能, 2008, 21(6): 728-731) [17] Zhao Mingqing, Yang Qiang, Gao Dezhi. Axiomatic Definition of Knowledge Granularity and Its Constructive Method // Proc of the 3rd International Conference on Rough Sets and Knowledge Technology. Chengdu, China, 2008: 348-354 [18] Wang Guoyin, Yu Hong, Yang Dachun. Decision Table Reduction Based on Conditional Information Entropy. Chinese Journal of Computers, 2002, 25(7): 759-766 (in Chinese) (王国胤,于 洪,杨大春.基于条件信息熵的决策表约简.计算机学报, 2002, 25(7): 759-766) [19] Liang Jiye, Chin K S, Dang Chuangyin, et al. A New Method for Measuring Uncertainty and Fuzziness in Rough Set Theory. International Journal of General Systems, 2002, 31(4): 331-342 [20] Zhu Ping. An Improved Axiomatic Definition of Information Granulation. Fundamenta Informaticae, 2012, 120(1): 93-109 [21] Zeng Fanzhi, Huang Guoshun, Wen Han. Some Equivalent Representations of HUs Attribute Reduction Based on Discernibility Matrix. Computer Engineering, 2011, 37(16): 65-67 (in Chinese) (曾凡智,黄国顺,文 翰.差别矩阵HU属性约简的几种等价表示.计算机工程, 2011, 37(16): 65-67) [22] Wang Guoyin, Zhao Jun, An Jiujiang, et al. A Comparative Study of Algebra Viewpoint and Information Viewpoint in Attribute Reduction. Fundamenta Informaticae, 2005, 68(3): 289-301 [23] Hu Xiaohua, Cerene N. Learning in Relational Databases: A Rough Set Approach. Computation Intelligence, 1995, 11(2): 323-338 [24] Xu Zhangyan,Yang Bingru, Song Wei, et al. Illustrating the Attribute Reduction Based on Hus Discernibility Matrix with Information View. Computer Science, 2007, 34(9): 191-193 (in Chinese) (徐章艳,杨炳儒,宋 威,等.差别矩阵属性约简的信息观解释.计算机科学, 2007, 34(9): 191-193) [25] Huang Guoshun, Zeng Fanzhi, Chen Guangyi, et al. Efficient Algorithm of the Attribute Reduction Using Hus Discernibility Matrix. Journal of Huazhong University of Science and Technology: Natural Science, 2012, 40(4): 8-12 (in Chinese) (黄国顺,曾凡智,陈广义,等.一种HU差别矩阵属性约简的高效算法.华中科技大学学报:自然科学版, 2012, 40(4): 8-12) [26] Yao Yiyu, Zhao Liquan. A Measurement Theory View on the Granularity of Partitions[EB/OL].[2012-04-01].http://dx.doi.org/10.1016/j.ins.2012.05.021 [27] Yan Xuehai, Li Hongxing, Sun Kaibiao. Granular Computing Theory Based on Hypergroups. Fuzzy Systems and Mathematics, 2011, 25(3): 133-142 (in Chinese) (袁学海,李洪兴,孙凯彪.基于超群的粒计算理论.模糊系统与数学, 2011, 25(3): 133-142) [28] Miao Duoqian, Xu Feifei, Yao Yiyu, et al. Set-Theoretic Formulation of Granular Computing. Chinese Journal of Computers, 2012, 35(2): 351-363 (in Chinese) (苗夺谦,徐菲菲,姚一豫,等.粒计算的集合论描述.计算机学报, 2012, 35(2): 351-363)
2013, Vol. 26 
