Skill Level Reduction and Necessary and Sufficient Conditions of Forward-Graded(Backward-Graded) Knowledge Structure in Fuzzy Formal Context
FENG Danlu1, LI Jinjin1,2, LI Zhaowen3, ZHOU Yinfeng4, YANG Taoli1
1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000; 2. Fujian Key Laboratory of Granular Computing and Application, Minnan Normal University, Zhangzhou 363000; 3. Guangxi College and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000; 4. School of Mathematics and Statistics, Shaanxi Normal University, Xian 710119
Abstract Fuzzy skill mapping is a pathway to construct knowledge structure. However, applying the basic local independence model to the forward-graded(backward-graded) knowledge structure results in unrecognizable problem. Therefore, under the premise of fuzzy skill mapping, two problems are solved in this paper: excessive time consumption in skill reduction and searching for the necessary and sufficient conditions of the forward-graded(backward-graded) knowledge structure. Firstly, based on the fuzzy skill context, a pair of operators is constructed, and the simple closure space is acquired directly through the fuzzy skill concept lattice determined by the pair of operators. At the same time, the minimum skill proficiency corresponding to each knowledge state is obtained. Secondly, the concept of skill level reduction is proposed. Redundant skill level is reduced by label skill reduction, and the algorithm of skill level reduction is provided. In addition, the necessary and sufficient conditions for inducing forward-graded(backward-graded) simple closure space from fuzzy skill mapping are presented, along with an algorithm for obtaining the forward-graded problem set and the backward-graded problem set. Finally, comparative experiments on five UCI datasets verify the feasibility and effectiveness of the proposed algorithm, and the forward-graded problem set and the backward-graded problem set are obtained.
Fund:National Natural Science Foundation of China(No.11871259), Natural Science Foundation of Fujian Province(No.2019J01748,2020J02043), Young and Middle-Aged Foundation of Fujian Province(No.JAT210255)
Corresponding Authors:
LI Jinjin, Ph.D., professor. His research interests include information technology, mathematical theories and methods of uncertainty.
About author:: FENG Danlu, master student. Her re-search interests include knowledge space theo-ry. LI Zhaowen, Ph.D., professor. His research interests include topology and its applications, rough set, fuzzy set and information systems.ZHOU Yinfeng, Ph.D. candidate. Her research interests include knowledge space theory.YANG Taoli, master student. Her research interests include knowledge space theory.
[1] DOIGNON J P, FALMAGNE J C. Spaces for the Assessment of Knowledge. International Journal of Man-Machine Studies, 1985, 23(2): 175-196. [2] FALMAGNE J C, DOIGNON J P. Learning Spaces: Interdisciplinary Applied Mathematics. Berlin, Germany: Springer, 2011. [3] DOBLE C, MATAYOSHI J, COSYN E, et al. A Data-Based Simulation Study of Reliability for an Adaptive Assessment Based on Knowledge Space Theory. International Journal of Artificial Intelligence in Education, 2019, 29: 258-282. [4] KOPPEN M, DOIGNON J P. How to Build a Knowledge Space by Querying an Expert. Journal of Mathematical Psychology, 1990, 34(3): 311-331. [5] KOPPEN M. Extracting Human Expertise for Constructing Know-ledge Spaces: An Algorithm. Journal of Mathematical Psychology, 1993, 37(1): 1-20. [6] COSYN E, THIÉRY N. A Practical Procedure to Build a Knowledge Structure. Journal of Mathematical Psychology, 2000, 44(3): 383-407. [7] DOIGNON J P. Knowledge Spaces and Skill Assignments // FIS-CHER G H, LAMING D, eds. Contributions to Mathematical Psychology, Psychometrics, and Methodology. Berlin, Germany: Springer, 1994: 111-121. [8] WILLE R. Restructuring Lattice Theory: An Approach Based on Hie-rarchies of Concepts // RIVAL I. Ordered Sets. Berlin, Germany: Springer, 1982: 445-470. [9] 李金海,魏玲,张卓,等. 概念格理论与方法及其研究展望. 模式识别与人工智能, 2020, 33(7): 619-642. (LI J H, WEI L, ZHANG Z, et al. Concept Lattice Theory and Method and Their Research Prospect. Pattern Recognition and Artificial Intelligence, 2020, 33(7): 619-642.) [10] RUSCH A, WILLE R. Knowledge Spaces and Formal Concept Analysis // BOCK H H, POLASEK W, eds. Data Analysis and Information Systems. Berlin, Germany: Springer, 1995: 427-436. [11] SPOTO A, STEFANUTTI L, VIDOTTO G. Knowledge Space Theo-ry, Formal Concept Analysis, and Computerized Psychological Assessment. Behavior Research Methods, 2010, 42(1): 342-350. [12] 李进金,孙文.知识空间,形式背景和知识基. 西北大学学报(自然科学版), 2019, 49(4): 517-526. (LI J J, SUN W. Knowledge Space, Formal Context and Know-ledge Base. Journal of Northwest University(Natural Science Edition),2019, 49(4): 517-526.) [13] 周银凤,李进金. 形式背景下的技能约简与评估. 计算机科学与探索, 2022, 16(3): 692-702. (ZHOU Y F, LI J J. Skill Reduction and Assessment in Formal Context. Journal of Frontiers of Computer Science and Technology, 2022, 16(3): 692-702.) [14] ZHOU Y F, LI J J, YANG H L, et al. Knowledge Structure Construction and Skill Reduction Methods Based on Multi-scale Context. Journal of Experimental and Theoretical Artificial Intelligence, 2023. DOI: 10.1080/0952813X.2023.2183266. [15] 周银凤,李进金,冯丹露,等. 形式背景下的学习路径与技能评估. 模式识别与人工智能, 2021, 34(12): 1069-1084. (ZHOU Y F, LI J J, FENG D L, et al. Learning Paths and Skills Assessment in Formal Context. Pattern Recognition and Artificial Intelligence, 2021, 34(12): 1069-1084.) [16] XU F F, MIAO D Q, YAO Y Y, et al. Analyzing Skill Sets with Or-Relation Tables in Knowledge Spaces // Proc of the 8th IEEE International Conference on Cognitive Informatics. Washington, USA: IEEE, 2009: 174-180. [17] DÜNTSCH I, GEDIGA G. Skills and Knowledge Structures. British Journal of Mathematical and Statistical Psychology, 1995, 48(1): 9-27. [18] FALMAGNE J C, DOIGNON J P. A Class of Stochastic Procedures for the Assessment of Knowledge. British Journal of Mathematical and Statistical Psychology, 1988, 41(1): 1-23. [19] FALMAGNE J C, KOPPEN M, VILLANO M, et al. Introduction to Knowledge Spaces: How to Build, Test, and Search Them. Psychological Review, 1990, 97(2): 201-224. [20] STEFANUTTI L. A Logistic Approach to Knowledge Structures. Journal of Mathematical Psychology, 2006, 50(6): 545-561. [21] SPOTO A, STEFANUTTI L, VIDOTTO G. On the Unidentifiability of a Certain Class of Skill Multi Map Based Probabilistic Know-ledge Structures. Journal of Mathematical Psychology, 2012, 56(4): 248-255. [22] SPOTO A, STEFANUTTI L, VIDOTTO G. Considerations about the Identification of Forward-and Backward-Graded Knowledge Structures. Journal of Mathematical Psychology, 2013, 57(5): 249-254. [23] STEFANUTTI L, SPOTO A, VIDOTTO G. Detecting and Explaining BLIM's Unidentifiability: Forward and Backward Parameter Transformation Groups. Journal of Mathematical Psychology, 2018, 82: 38-51. [24] STEFANUTTI L, SPOTO A. BLIM's Identifiability and Parameter Invariance under Backward and Forward Transformations. Journal of Mathematical Psychology, 2020, 95. DOI: 10.1016/j.jmp.2019.102314. [25] SPOTO A, STEFANUTTI L. On the Necessary and Sufficient Conditions for Delineating Forward-and Backward-Graded Knowledge Structures from Skill Maps. Journal of Mathematical Psychology, 2020, 99. DOI: 10.1016/j.jmp.2020.102451. [26] SUN W, LI J J, GE X, et al. Knowledge Structures Delineated by Fuzzy Skill Maps. Fuzzy Sets and Systems, 2021, 407: 50-66. [27] KRAJCI S. Cluster Based Efficient Generation of Fuzzy Concepts. Neural Network World, 2003, 13(5): 521-530. [28] YAHIA S B, JAOUA A. Discovering Knowledge from Fuzzy Concept Lattice // KANDEL A, LAST M, BUNKE H, eds. Data Mining and Computational Intelligence. Berlin, Germany: Sprin-ger, 2001: 167-190. [29] NIU J J, CHEN D G. Incremental Calculation Approaches for Granular Reduction in Formal Context with Attribute Updating. International Journal of Machine Learning and Cybernetics, 2022, 13(9): 2763-2784. [30] NIU J J, CHEN D G, LI J H, et al. Fuzzy Rule-Based Classification Method for Incremental Rule Learning. IEEE Transactions on Fuzzy Systems, 2022, 30(9): 3748-3761. [31] LIN Y D, LI J J, TAN A H, et al. Granular Matrix-Based Know-ledge Reductions of Formal Fuzzy Contexts. International Journal of Machine Learning and Cybernetics, 2020, 11(3): 643-656. [32] 林艺东,李进金,张呈玲. 基于矩阵的模糊-经典概念格属性约简. 模式识别与人工智能, 2020, 33(1): 21-31. (LIN Y D, LI J J, ZHANG C L. Attribute Reductions of Fuzzy-Crisp Concept Lattices Based on Matrix. Pattern Recognition and Artificial Intelligence, 2020, 33(1): 21-31.) [33] SHI L L, YANG H L. Object Granular Reduction of Fuzzy Formal Contexts. Journal of Intelligent and Fuzzy Systems, 2018, 34(1): 633-644. [34] GANTER B, WILLE R. Formal Concept Analysis: Mathematical Foundations. Berlin, Germany: Springer, 1999. [35] ORE O. Galois Connexions. Transactions of the American Mathematical Society, 1944, 55(3): 493-513. [36] BIRKHOFF G. Lattice Theory. Providence, USA: American Ma-thematical Society, 1940. [37] GOGUEN J A. L-Fuzzy Sets. Journal of Mathematical Analysis and Applications, 1967, 18(1): 145-174. [38] GEORGESCU G, POPESCU A. Non-Dual Fuzzy Connections. Archive for Mathematical Logic, 2004, 43: 1009-1039.
2023, Vol. 36 
