Granular computing on information tables: families of subsets and operators. (English) Zbl 1440.68288

Summary: In this work we use the granular computing paradigm to study specific types of families of subsets, operators and families of ordered pairs of sets of attributes which are naturally induced by information tables. In an unifying perspective, by means of some representation results, we connect the study of finite closure systems, matroids and finite lattice theory in the scope of the more general notion of attribute dependency based on information tables. For a fixed finite set \(\Omega\) and for a corresponding information table \(\mathcal{J}\) having attribute set \(\Omega \), the fundamental tool we use to proceed in our investigation is the equivalence relation \(\approx_{\mathcal{J}}\) on the power set \(\mathcal{P}(\Omega)\) which identifies two any sets of attributes inducing the same indiscernibility relation on the object set of \(\mathcal{J} \). We interpret the attribute dependency as a preorder \(\geq_{\mathcal{J}}\) on \(\mathcal{P}(\Omega)\) whose induced equivalence relation coincides with \(\approx_{\mathcal{J}} \). Then we investigate in detail the links between the preorder \(\geq_{\mathcal{J}}\), a closure system and an abstract simplicial complex on \(\Omega\) naturally induced by \(\approx_{\mathcal{J}}\) and specific families of ordered pairs of sets of attributes on \(\Omega \).


68T37 Reasoning under uncertainty in the context of artificial intelligence
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[1] Aledo, J. A.; Diaz, L. G.; Martínez, S.; Valverde, J. C., On periods and equilibria of computational sequential systems, Inf. Sci., 409, 27-34 (2017) · Zbl 1429.68167
[2] Armstrong, W. W., Dependency structures of data base relationships, Inf. Process., 74, 580-583 (1974) · Zbl 0296.68038
[3] Birkhoff, G.; Frink, O., Representations of lattices by sets, Trans. Am. Math. Soc., 64, 299-316 (1948) · Zbl 0032.00504
[4] Birkhoff, G., Lattice Theory (1967), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0126.03801
[5] Bisi, C.; Chiaselotti, G.; Gentile, T.; Oliverio, P. A., Dominance order on signed partitions, Adv. Geometry, 17, 1, 5-29 (2017) · Zbl 1383.05020
[6] Bisi, C.; Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Micro and macro models of granular computing induced by the indiscernibility relation, Inf. Sci., 388-389, 247-273 (2017)
[7] Cattaneo, G., Generalized rough sets (preclusivity fuzzy-intuitionistic (BZ) lattices), Stud. Log., 58, 47-77 (1997) · Zbl 0864.03040
[8] Cattaneo, G.; Chiaselotti, G.; Oliverio, P. A.; Stumbo, F., A new discrete dynamical system of signed integer partitions, Eur. J. Comb., 55, 119-143 (2016) · Zbl 1333.05026
[9] Cattaneo, G.; Chiaselotti, G.; Ciucci, D.; Gentile, T., On the connection of hypergraph theory with formal concept analysis and rough set theory, Inf. Sci., 330, 342-357 (2016) · Zbl 1390.68618
[10] Chiaselotti, G.; Ciucci, D.; Gentile, T., Simple graphs in granular computing, Inf. Sci., 340-341, 1, 279-304 (2016) · Zbl 1395.68260
[11] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., The granular partition lattice of an information table, Inf. Sci., 373, 57-78 (2016) · Zbl 1429.68270
[12] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Generalizations of rough set tools inspired by graph theory, Fundam. Inf., 148, 207-227 (2016) · Zbl 1388.03045
[13] Chiaselotti, G.; Gentile, T.; Infusino, F.; Oliverio, P. A., The adjacency matrix of a graph as a data table. a geometric perspective, Annali di Matematica Pura e Applicata, 196, 3, 1073-1112 (2017) · Zbl 1366.05029
[14] Chiaselotti, G.; Gentile, T.; Infusino, F., Knowledge pairing systems in granular computing, Knowl. Based Syst., 124, 144-163 (2017)
[15] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Rough set theory and digraphs, Fundam. Inf., 153, 291-325 (2017) · Zbl 1400.05193
[16] Chiaselotti, G.; Gentile, T.; Infusino, F., Dependency structures for decision tables, Int. J. Approx. Reason., 88, 333-370 (2017) · Zbl 1418.68187
[17] Ciucci, D. E., Temporal dynamics in information tables, Fundam. Inf., 115, 1, 57-74 (2012) · Zbl 1237.68212
[18] Ciucci, D., Back to the beginnings: Pawlak’ s definitions of the terms information system and rough set, in thriving rough sets, Stud. Comput. Intell., 708, 225-235 (2017)
[19] Duntsch, I.; Gediga, G., Algebraic aspects of attribute dependencies in information systems, Fundam. Inf., 29, 1,2, 119-133 (1997) · Zbl 0868.68052
[20] Hońko, P., Description and classification of complex structured objects by applying similarity measures, Int. J. Approx. Reason., 49, 3, 539-554 (2008) · Zbl 1184.68483
[21] Hońko, P., Compound approximation spaces for relational data, Int. J. Approx. Reason., 71, 89-111 (2016) · Zbl 1352.68244
[23] Jarvinen, J., Representations of Information Systems and Dependences Spaces, and Some Basic Algorithms (1997), University of Turku, Department of Mathematics: University of Turku, Department of Mathematics FIN-20014 TURKU
[24] Li, X. N.; Liu, S. Y., Matroidal approaches to rough sets via closure operators, Int. J. Approx. Reason., 53, 513-527 (2012) · Zbl 1246.68233
[25] Li, X. N.; Yi, H. J.; Liu, S. Y., Rough sets and matroids from a lattice-theoretic view-point, Inf. Sci., 342, 37-52 (2016) · Zbl 1403.06018
[26] Lin, T. Y., Data mining and machine oriented modeling: a granular approach, Appl. Intell., 13, 113-124 (2000)
[27] Mao, H., Characterization and reduct of concept lattices through matroid theory, Inf. Sci., 281, 338-354 (2014) · Zbl 1355.68248
[28] Novotny, M., Dependence spaces of information systems, and applications of dependence spaces, Chapters 7 and 8 of E. Orlowska (ed.), Incomplete Information: Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol 13 (1998), Physica: Physica Heidelberg
[29] Pagliani, P., From concept lattices to approximation spaces: algebraic structures of some spaces of partial objects, Fundam. Inf., 18, 1, 1-25 (1993) · Zbl 0776.06005
[30] Pagliani, P., A pure logic-algebraic analysis of rough top and rough bottom equalities, W.P. Ziarko (eds), Rough Sets, Fuzzy Sets and Knowledge Discovery, Workshops in Computing (1994), Springer: Springer London · Zbl 0819.04010
[31] Pagliani, P., Rough set theory and logic-algebraic structures, Incomplete Information: E. Orlowska (ed.), Rough Set Analysis. Studies in Fuzziness and Soft Computing, vol 13 (1998), Physica: Physica Heidelberg
[32] Pagliani, P.; Chakraborty, M., A geometry of approximation, Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns (2008), Springer
[33] Pawlak, Z., Rough sets. Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publisher · Zbl 0758.68054
[34] Pedrycz, W., Granular Computing: An Emerging Paradigm (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 0966.00017
[35] Pedrycz, A.; Hirota, K.; Pedrycz, W.; Dong, F., Granular representation and granular computing with fuzzy sets, Fuzzy Sets Syst., 203, 17-32 (2012)
[36] Polkowski, L.; Skowron, A., A new paradigm for approximate reasoning, Int. J. Approx. Reason., 15, 333-365 (1996) · Zbl 0938.68860
[37] Polkowski, L., On fractal dimension in information systems. Toward exact sets in infinite information systems, Fundam. Inf., 50, 3-4, 305-314 (2002) · Zbl 1012.68218
[38] Polkowski, L., Approximate reasoning by parts, An Introduction to Rough Mereology (2011), Springer
[39] Rauszer, C. M., Algebraic properties of functional dependencies, Bull. Polish Acad. Sci. Math., 33, 561-569 (1985) · Zbl 0582.68066
[40] Ślezak, D., Approximate entropy reducts, Fundam. Inf., 53, 365-390 (2002) · Zbl 1092.68676
[41] Stell, J. G., Relational granularity for hypergraphs, RSCTC, Lect. Notes Comput. Sci., 6086, 267-276 (2010)
[42] Tanga, J.; Shea, K.; Min, F.; Zhu, W., A matroidal approach to rough set theory, Theor. Comput. Sci., 471, 3, 1-11 (2013) · Zbl 1258.05022
[43] Welsh, D. J.A., Matroid Theory (1976), Academic Press · Zbl 0343.05002
[44] Yao, Y. Y.; Lin, T. Y., Generalization of rough sets using modal logics, Intell. Autom. Soft Comput., 2, 2, 103-120 (1996)
[45] Yao, Y., Constructive and algebraic methods of the theory of rough sets, Inf. Sci., 109, 21-47 (1998) · Zbl 0934.03071
[46] Yao, Y., A partition model of granular computing, Transactions on Rough Sets I, Lecture Notes in Computer Science, volume 3100, 232-253 (2004), Springer-Verlag · Zbl 1104.68776
[47] Yao, Y.; Zhang, N.; Miao, D.; Xu, F., Set-theoretic approaches to granular computing, Fundam. Inf., 21, 1001-1018 (2010)
[48] Zadeh, L. A., Fuzzy sets and information granularity, (Gupta, N.; Ragade, R.; Yager, R., Advances in Fuzzy Set Theory and Applications (1979), North-Holland: North-Holland Amsterdam), 3-18
[49] Zadeh, L. A., Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets Syst., 19, 111-127 (1997) · Zbl 0988.03040
[50] Zhu, W.; Wang, S., Rough matroids based on relations, Inf. Sci., 232, 241-252 (2013) · Zbl 1293.05036
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