×

Notions from rough set theory in a generalized dependency relation context. (English) Zbl 1446.03087

Summary: In this paper, we introduce a notion of generalized dependency relation between subsets of an arbitrary (not necessarily finite) set \(\Omega\) starting with the classical Armstrong’s rule. More specifically, we fix a given set system \(\mathcal{F}\) on \(\Omega\) and call any transitive binary relation \(\leftarrow\) on the power set \(\mathcal{P}(\Omega)\) such that = 0.5 cm
\(\bullet\)
\(B \subseteq A \in \mathcal{F}\) implies \(B \leftarrow A\);
\(\bullet\)
\(B \leftarrow A\) if and only if \(b \leftarrow A \; \forall b \in B\);
a \([\mathcal{F}]\)-dependency relation on \(\Omega\). We use the generality of such a notion to investigate some common analogies between rough set theory on information tables, formal context analysis, Scott’s information systems and possibility theory. More specifically, taking as inspirational models some classical notions derived by rough set theory on attribute set of an information table, we first generalize and study such notions for any \([\mathcal{F}]\)-dependency relation. Next, we interpret such general results relatively to natural dependency relations derived by rough set theory on objects of an information table, formal context analysis, Scott’s information systems and possibility theory. Finally, we study the generation of \([\mathcal{F}]\)-dependency relations by starting from a fixed set \(\mathcal{D}\) of subset ordered pairs of \(\Omega\).

MSC:

03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence
68T30 Knowledge representation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abramsky, S.; Jung, A., Domain theory, (Abramsky, S.; Gabbay, D. M.; Maibaum, T. S.E., Handbook of Logic in Computer Science, vol. III, (1994), Oxford University Press)
[2] 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)
[3] Aledo, J. A.; Diaz, L. G.; Martínez, S.; Valverde, J. C., On the periods of parallel dynamical systems, Complexity, 2017, (2017) · Zbl 1367.94485
[4] Armstrong, W. W., Dependency structures of data base relationships, (Information Processing, vol. 74, (1974), North-Holland Amsterdam), 580-583 · Zbl 0296.68038
[5] Birkhoff, G., Lattice theory, (1967), American Mathematical Society Providence, RI · Zbl 0126.03801
[6] Bisi, C.; Chiaselotti, G.; Gentile, T.; Oliverio, P. A., Dominance order on signed partitions, Adv. Geom., 17, 1, 5-29, (2017) · Zbl 1383.05020
[7] 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)
[8] Cattaneo, G., Generalized rough sets (preclusivity fuzzy-intuitionistic (BZ) lattices), Stud. Log., 58, 47-77, (1997) · Zbl 0864.03040
[9] Cattaneo, G., Abstract approximation spaces for rough theories, (Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery 1: Methodology and Applications, Stud. Fuzziness Soft Comput., (1998), Physica Heidelberg), 59-98 · Zbl 0927.68087
[10] Cattaneo, G.; Ciucci, D., Algebraic structures for rough sets, (Peter, J. F.; Skowron, A., Transactions on Rough Sets II, Lect. Notes Comput. Sci., vol. 3135, (2004), Springer-Verlag), 208-252, (Special Issue on Foundations of Rough Sets) · Zbl 1109.68115
[11] Cattaneo, G.; Ciucci, D., Lattices with interior and closure operators and abstract approximation spaces, (Peters, J. F.; Skowron, A., Transactions on Rough Sets X, Lect. Notes Comput. Sci., vol. 5656, (2009), Springer-Verlag), 67-116, (Special Issue on Foundations of Rough Sets) · Zbl 1248.06005
[12] Cattaneo, G., An investigation about rough set theory: some foundational and mathematical aspects, Fundam. Inform., 108, 197-221, (2011) · Zbl 1241.03060
[13] 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
[14] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Preclusivity and simple graphs, (Proc. RSFDGrC 2015, Lect. Notes Comput. Sci., vol. 9437, (2015), Springer), 127-137
[15] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Preclusivity and simple graphs: the n-cycle and n-path cases, (Proc. RSFDGrC 2015, Lect. Notes Comput. Sci., vol. 9437, (2015), Springer), 138-148
[16] Chiaselotti, G.; Ciucci, D.; Gentile, T., Simple graphs in granular computing, Inf. Sci., 340-341, 279-304, (2016) · Zbl 1395.68260
[17] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., The granular partition lattice of an information table, Inf. Sci., 373, 57-78, (2016)
[18] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Generalizations of rough set tools inspired by graph theory, Fundam. Inform., 148, 207-227, (2016) · Zbl 1388.03045
[19] Chiaselotti, G.; Gentile, T.; Infusino, F.; Oliverio, P. A., Rough sets for n-cycles and n-paths, Appl. Math. Inf. Sci., 10, 1, 117-124, (2016)
[20] Chiaselotti, G.; Gentile, T.; Infusino, F.; Oliverio, P. A., The adjacency matrix of a graph as a data table. A geometric perspective, Ann. Mat. Pura Appl., 196, 3, 1073-1112, (2017) · Zbl 1366.05029
[21] Chiaselotti, G.; Ciucci, D.; Gentile, T.; Infusino, F., Rough set theory and digraphs, Fundam. Inform., 153, 291-325, (2017) · Zbl 1400.05193
[22] Chiaselotti, G.; Gentile, T.; Infusino, F., Knowledge pairing systems in granular computing, Knowl.-Based Syst., 124, 144-163, (2017)
[23] Chiaselotti, G.; Gentile, T.; Infusino, F., Dependency structures for decision tables, Int. J. Approx. Reason., 88, 333-370, (2017) · Zbl 1418.68187
[24] Chiaselotti, G.; Gentile, T.; Infusino, F., Simplicial complexes and closure systems induced by indistinguishability relations, C. R. Acad. Sci. Paris, Ser. I, 355, 991-1021, (2017) · Zbl 1371.05327
[25] Chiaselotti, G.; Gentile, T.; Infusino, F., Pairings and related symmetry notions, Ann. Univ. Ferrara, (Jan. 2018)
[26] Chiaselotti, G.; Gentile, T.; Infusino, F., Granular computing on information tables: families of subsets and operators, Inf. Sci., 442-443, 72-102, (2018)
[27] Ciucci, D., Back to the beginnings: Pawlak’s definitions of the terms information system and rough set, (Thriving Rough Sets, Stud. Comput. Intell., vol. 708, (2017), Springer), 225-235
[28] Codd, E. F., A relation model of data for large shared data banks, Commun. ACM, 13, 6, 377-387, (June 1970)
[29] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press · Zbl 0444.94049
[30] Dubois, D.; Prade, H., Possibility theory, (1988), Plenum New York · Zbl 0645.68108
[31] Ganter, B.; Wille, R., Formal concept analysis. mathematical foundations, (1999), Springer-Verlag · Zbl 0909.06001
[32] Koshevoy, G. A., Choice functions and abstract convex geometries, Math. Soc. Sci., 38, 35-44, (1999) · Zbl 0943.91031
[33] Larsen, K. G.; Winskell, G., Using information systems to solve recursive domain equations, Inf. Comput., 91, 232-258, (1991) · Zbl 0731.68071
[34] Leung, Y.; Ma, J.; Zhang, W.; Li, T., Dependence-space-based attribute reductions in inconsistent decision information systems, Int. J. Approx. Reason., 49, 623-630, (2008) · Zbl 1184.68514
[35] Malishevski, A. V., Path independence in serial-parallel data processing, Math. Soc. Sci., 21, 335-367, (1994) · Zbl 0884.68040
[36] Matsuyama, K., Chernoff’s dual axiom, revealed preference and weak rational choice functions, J. Econ. Theory, 35, 155-165, (1985) · Zbl 0553.90010
[37] Moulin, H., Choice functions over a finite set: a summary, Soc. Choice Welf., 2, 147-160, (1985) · Zbl 0576.90004
[38] Pawlak, Z., Rough sets. theoretical aspects of reasoning about data, (1991), Kluwer Academic Publisher · Zbl 0758.68054
[39] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Inf. Sci., 177, 3-27, (2007) · Zbl 1142.68549
[40] Pawlak, Z.; Skowron, A., Rough sets: some extensions, Inf. Sci., 177, 28-40, (2007) · Zbl 1142.68550
[41] Pawlak, Z.; Skowron, A., Rough sets and Boolean reasoning, Inf. Sci., 177, 41-73, (2007) · Zbl 1142.68551
[42] Pedrycz, W., Granular computing: an emerging paradigm, (2001), Springer-Verlag Berlin · Zbl 0966.00017
[43] Polkowski, L., Rough sets: mathematical foundations, (2002), Springer-Verlag · Zbl 1040.68114
[44] Polkowski, L., On fractal dimension in information systems. toward exact sets in infinite information systems, Fundam. Inform., 50, 3-4, 305-314, (2002) · Zbl 1012.68218
[45] Polkowski, L.; Polkowska, M. S., Granular rough mereological logics with applications to dependencies in information and decision systems, Trans. Rough Sets, 12, 1-20, (2010) · Zbl 1288.68211
[46] Polkowski, L., Approximate reasoning by parts. an introduction to rough mereology, (2011), Springer
[47] Sanahuja, S. M., New rough approximations for n-cycles and n-paths, Appl. Math. Comput., 276, 96-108, (2016)
[48] Scott, D. S., Domains for denotational semantics, (Automata, Languages and Programming, Lect. Notes Comput. Sci., vol. 140, (1982)), 577-613
[49] Simovici, D. A.; Djeraba, C., Mathematical tools for data mining, (2014), Springer-Verlag London · Zbl 1303.68006
[50] Skowron, A.; Rauszer, C., The discernibility matrices and functions in information systems, (Intelligent Decision Support, Theory Decis. Libr. Ser., vol. 11, (1992), Springer Netherlands), 331-362
[51] Ślezak, D., Rough sets and functional dependencies in data: foundations of association reducts, (Transactions on Computational Science V, Lect. Notes Comput. Sci., vol. 5440, (2009)), 182-205 · Zbl 1246.68221
[52] Tanga, J.; Shea, K.; Min, F.; Zhu, W., A matroidal approach to rough set theory, Theor. Comput. Sci., 471, 1-11, (2013) · Zbl 1258.05022
[53] Wang, S.; Zhu, W.; Zhu, Q.; Min, F., Four matroidal structures of covering and their relationships with rough sets, Int. J. Approx. Reason., 54, 9, 1361-1372, (2013) · Zbl 1316.05025
[54] Yamaguchi, D., Attribute dependency functions considering data efficiency, Int. J. Approx. Reason., 51, 89-98, (2009)
[55] Yao, Y., Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15, 291-317, (1996) · Zbl 0935.03063
[56] Yao, Y., Constructive and algebraic methods of the theory of rough sets, Inf. Sci., 109, 21-47, (1998) · Zbl 0934.03071
[57] Yao, Y. Y.; Zhong, N., Granular computing using information tables, (Data Mining, Rough Sets and Granular Computing, (2002), Physica-Verlag), 102-124 · Zbl 1017.68053
[58] Yao, Y., A partition model of granular computing, (Transactions on Rough Sets I, Lect. Notes Comput. Sci., vol. 3100, (2004), Springer-Verlag), 232-253 · Zbl 1104.68776
[59] Yao, Y.; Zhao, Y., Discernibility matrix simplification for constructing attribute reducts, Inf. Sci., 179, 867-882, (2009) · Zbl 1162.68704
[60] Yao, Y., Notes on rough set approximations and associated measures, J. Zhejiang Ocean Univ. (Nat. Sci.), 29, 5, 399-410, (2010)
[61] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst., 100, Suppl., 9-34, (1999)
[62] Ziarko, W., Probabilistic approach to rough sets, Int. J. Approx. Reason., 49, 272-284, (2008) · Zbl 1191.68705
[63] Zhu, W.; Wang, S., Rough matroids based on relations, Inf. Sci., 232, 241-252, (2013) · Zbl 1293.05036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.