# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2022016
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## Concept and attribute reduction based on rectangle theory of formal concept

 Department of Computer Science, Anhui University of Technology, Ma'anshan 243002, China

*Corresponding author: Jianqin Zhou

Received  September 2021 Revised  November 2021 Early access June 2022

Fund Project: The authors are supported by NSF grant of Anhui Province(No.1808085MF178), China

Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative necessary concepts and unnecessary concepts. First, we present the new judgment theorems for relative necessary concepts and unnecessary concepts. Second, we derive the bounds for both the maximum number of relative necessary concepts and the maximum number of unnecessary concepts. Third, based on rectangle theory of formal concept, a fast algorithm for reducing attributes while preserving the extensions for a set of formal concepts is proposed using the extension bit-array technique, which allows multiple context cells to be processed by a single 32-bit or 64-bit operator. Technically, the new algorithm could store both formal context and extension of a concept as bit-arrays, and we can use bit-operations to process set operations "or" as well as "and". One more merit is that the new algorithm does not need to consider other concepts in the concept lattice, thus the algorithm is explicit to understand and fast. Experiments demonstrate that the new algorithm is effective in the computation of attribute reductions.

Citation: Jianqin Zhou, Sichun Yang, Xifeng Wang. Concept and attribute reduction based on rectangle theory of formal concept. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022016
##### References:
 [1] S. Andrews, A 'Best-of-Breed' approach for designing a fast algorithm for computing fixpoints of Galois Connections, Information Sciences, 295 (2015), 633-649.  doi: 10.1016/j.ins.2014.10.011. [2] S. Andrews, In-Close4 Program, 2017, https://sourceforge.net/projects/inclose/files/In-Close/. [3] L. Cao, L. Wei and J. J. Qi, Concept reduction preserving binary relations, Pattern Recogn Artif Intell, 31 (2018), 516-524. [4] C. Carpineto and G. Romano, A lattice conceptual clustering system and its application to browsing retrieval, Machine Learning, 24 (1996), 95-122.  doi: 10.1007/BF00058654. [5] A. Frank and A. Asuncion, UCI Machine Learning Repository, 2010, http://archive.ics.uci.edu/ml. [6] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-59830-2. [7] S. O. Kuznetsov, On computing the size of a lattice and related decision problems, Order, 18 (2001), 313-321.  doi: 10.1023/A:1013970520933. [8] J. Li, C. Mei, L. Wang and J. Wang, On inference rules in decision formal contexts, International Journal of Computational Intelligence Systems, 8 (2015), 175-186. [9] L. Li, J. Mia and B. Xie, Attribute reduction based on maximal rules in decision formal context, International Journal of Computational Intelligence Systems, 7 (2014), 1044-1053.  doi: 10.1080/18756891.2014.963972. [10] P. H. P. Nguyen and D. Corbett, A basic mathematical framework for conceptual graphs, IEEE Transactions on Knowledge and Data Engineering, 18 (2006), 261-271. [11] Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356.  doi: 10.1007/BF01001956. [12] R. Ren and L. Wei, The attribute reductions of three-way concept lattices, Knowl.-Based Syst., 99 (2016), 92-102. [13] M. Shao, H. Yang and W. Wu, Knowledge reduction in formal fuzzy contexts, Knowl.-Based Syst., 73 (2015), 265-275. [14] X. D. Tu, Y. L. Wang and M. L. Zhang, Using formal concept analysis to identify negative correlations in gene expression data, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 13 (2006), 380-391. [15] Q. Wan and L. Wei, Approximate concepts acquisition based on formal contexts, Knowl.-Based Syst., 75 (2015), 78-86. [16] L. Wei, L. Cao, J. J. Qi and W. Zhang, Concept reduction and concept characteristics in formal concept analysis (in Chinese), Sci. Sin. Inform., 50 (2020), 1817-1833.  doi: 10.1360/N112018-00272. [17] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [18] L. A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems, 90 (1997), 111-127.  doi: 10.1016/S0165-0114(97)00077-8. [19] W. X. Zhang, L. Wei and J. J. Qi, Attribute reduction in concept lattice based on discernibility matrix, Lecture Notes in Computer Science, 3642 (2005), 157-165. [20] W. X. Zhang, L. Wei and J. J. Qi, Attribute reduction theory and approach to concept lattice, Sci. China Ser. F, 48 (2005), 713-726.  doi: 10.1360/122004-104. [21] H. L. Zhi, J. J. Qi, T. Qian and L. Wei, Three-way dual concept analysis, International Journal of Approximate Reasoning, 114 (2019), 151-165.  doi: 10.1016/j.ijar.2019.08.010. [22] C. F. Zou, D. Q. Zhang and J. F. Wan, Using concept lattice for personalized recommendation system design, IEEE Systems Journal, 11 (2017), 305-314.

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##### References:
 [1] S. Andrews, A 'Best-of-Breed' approach for designing a fast algorithm for computing fixpoints of Galois Connections, Information Sciences, 295 (2015), 633-649.  doi: 10.1016/j.ins.2014.10.011. [2] S. Andrews, In-Close4 Program, 2017, https://sourceforge.net/projects/inclose/files/In-Close/. [3] L. Cao, L. Wei and J. J. Qi, Concept reduction preserving binary relations, Pattern Recogn Artif Intell, 31 (2018), 516-524. [4] C. Carpineto and G. Romano, A lattice conceptual clustering system and its application to browsing retrieval, Machine Learning, 24 (1996), 95-122.  doi: 10.1007/BF00058654. [5] A. Frank and A. Asuncion, UCI Machine Learning Repository, 2010, http://archive.ics.uci.edu/ml. [6] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-59830-2. [7] S. O. Kuznetsov, On computing the size of a lattice and related decision problems, Order, 18 (2001), 313-321.  doi: 10.1023/A:1013970520933. [8] J. Li, C. Mei, L. Wang and J. Wang, On inference rules in decision formal contexts, International Journal of Computational Intelligence Systems, 8 (2015), 175-186. [9] L. Li, J. Mia and B. Xie, Attribute reduction based on maximal rules in decision formal context, International Journal of Computational Intelligence Systems, 7 (2014), 1044-1053.  doi: 10.1080/18756891.2014.963972. [10] P. H. P. Nguyen and D. Corbett, A basic mathematical framework for conceptual graphs, IEEE Transactions on Knowledge and Data Engineering, 18 (2006), 261-271. [11] Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356.  doi: 10.1007/BF01001956. [12] R. Ren and L. Wei, The attribute reductions of three-way concept lattices, Knowl.-Based Syst., 99 (2016), 92-102. [13] M. Shao, H. Yang and W. Wu, Knowledge reduction in formal fuzzy contexts, Knowl.-Based Syst., 73 (2015), 265-275. [14] X. D. Tu, Y. L. Wang and M. L. Zhang, Using formal concept analysis to identify negative correlations in gene expression data, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 13 (2006), 380-391. [15] Q. Wan and L. Wei, Approximate concepts acquisition based on formal contexts, Knowl.-Based Syst., 75 (2015), 78-86. [16] L. Wei, L. Cao, J. J. Qi and W. Zhang, Concept reduction and concept characteristics in formal concept analysis (in Chinese), Sci. Sin. Inform., 50 (2020), 1817-1833.  doi: 10.1360/N112018-00272. [17] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [18] L. A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems, 90 (1997), 111-127.  doi: 10.1016/S0165-0114(97)00077-8. [19] W. X. Zhang, L. Wei and J. J. Qi, Attribute reduction in concept lattice based on discernibility matrix, Lecture Notes in Computer Science, 3642 (2005), 157-165. [20] W. X. Zhang, L. Wei and J. J. Qi, Attribute reduction theory and approach to concept lattice, Sci. China Ser. F, 48 (2005), 713-726.  doi: 10.1360/122004-104. [21] H. L. Zhi, J. J. Qi, T. Qian and L. Wei, Three-way dual concept analysis, International Journal of Approximate Reasoning, 114 (2019), 151-165.  doi: 10.1016/j.ijar.2019.08.010. [22] C. F. Zou, D. Q. Zhang and J. F. Wan, Using concept lattice for personalized recommendation system design, IEEE Systems Journal, 11 (2017), 305-314.
Formal context $(G, M, I)$
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 0 1 1 0 0 2 1 1 0 0 0 3 1 0 0 0 0 4 0 0 0 0 1 5 0 0 0 1 1 6 0 0 1 1 1 7 1 1 1 0 0
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 0 1 1 0 0 2 1 1 0 0 0 3 1 0 0 0 0 4 0 0 0 0 1 5 0 0 0 1 1 6 0 0 1 1 1 7 1 1 1 0 0
Formal concepts in Table 1
 $C_0=(\{1, 2, 3, 4, 5, 6, 7\}, \emptyset)$ $C_4=(\{2, 3, 7\}, \{a_1\})$ $C_3=(\{1, 2, 7\}, \{a_2\})$ $C_2=(\{1, 6, 7\}, \{a_3\})$ $C_1=(\{4, 5, 6\}, \{a_5\})$ $C_5=(\{2, 7\}, \{a_1, a_2\})$ $C_6=(\{1, 7\}, \{a_2, a_3\})$ $C_7=(\{5, 6\}, \{a_4, a_5\})$ $C_9=(\{7\}, \{a_1, a_2, a_3\})$ $C_8=(\{6\}, \{a_3, a_4, a_5\})$ $C_{10}=(\emptyset, \{a_1, a_2, a_3 , a_4, a_5\})$
 $C_0=(\{1, 2, 3, 4, 5, 6, 7\}, \emptyset)$ $C_4=(\{2, 3, 7\}, \{a_1\})$ $C_3=(\{1, 2, 7\}, \{a_2\})$ $C_2=(\{1, 6, 7\}, \{a_3\})$ $C_1=(\{4, 5, 6\}, \{a_5\})$ $C_5=(\{2, 7\}, \{a_1, a_2\})$ $C_6=(\{1, 7\}, \{a_2, a_3\})$ $C_7=(\{5, 6\}, \{a_4, a_5\})$ $C_9=(\{7\}, \{a_1, a_2, a_3\})$ $C_8=(\{6\}, \{a_3, a_4, a_5\})$ $C_{10}=(\emptyset, \{a_1, a_2, a_3 , a_4, a_5\})$
Formal context $(G, M, I)$
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 1 1 1 0 0 2 1 1 1 0 0 3 1 1 1 0 0 4 0 1 1 1 1 5 0 1 1 1 1 6 0 1 1 1 1
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 1 1 1 0 0 2 1 1 1 0 0 3 1 1 1 0 0 4 0 1 1 1 1 5 0 1 1 1 1 6 0 1 1 1 1
Formal context $(G, M, I)$
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 1 1 1 0 0 2 1 1 1 0 0 3 1 1 1 0 0 4 0 1 1 1 1 5 0 1 1 1 1 6 0 1 1 1 1 7 1 1 0 0 0 8 1 0 1 0 0
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 1 1 1 0 0 2 1 1 1 0 0 3 1 1 1 0 0 4 0 1 1 1 1 5 0 1 1 1 1 6 0 1 1 1 1 7 1 1 0 0 0 8 1 0 1 0 0
Formal context $(G, M, I)$
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 1 1 0 0 0 2 1 0 1 0 0 3 1 0 0 1 0 4 1 0 0 0 1 5 0 1 1 0 0 6 0 1 0 1 0 7 0 1 0 0 1 8 0 0 1 1 0 9 0 0 1 0 1 10 0 0 0 1 1
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ 1 1 1 0 0 0 2 1 0 1 0 0 3 1 0 0 1 0 4 1 0 0 0 1 5 0 1 1 0 0 6 0 1 0 1 0 7 0 1 0 0 1 8 0 0 1 1 0 9 0 0 1 0 1 10 0 0 0 1 1
Formal context $(G, M, I )$
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ 1 1 1 0 0 2 1 0 1 0 3 1 0 0 1 4 0 1 1 0 5 0 1 0 1 6 0 0 1 1 7 1 1 1 0 8 1 1 0 1 9 1 0 1 1 10 0 1 1 1
 $G$ $a_1$ $a_2$ $a_3$ $a_4$ 1 1 1 0 0 2 1 0 1 0 3 1 0 0 1 4 0 1 1 0 5 0 1 0 1 6 0 0 1 1 7 1 1 1 0 8 1 1 0 1 9 1 0 1 1 10 0 1 1 1
Computation of attribute reductions with the new algorithm
 Mushroom Nursery $|G|\times |M|$ $8,124\times 115$ $12,960\times 30$ Number of concepts 512 512 Time in seconds 4.218 0.582 Number of reduced columns 19 2
 Mushroom Nursery $|G|\times |M|$ $8,124\times 115$ $12,960\times 30$ Number of concepts 512 512 Time in seconds 4.218 0.582 Number of reduced columns 19 2
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