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Formal reasoning with rough sets in multiple-source approximation systems. (English) Zbl 1191.68684
Summary: We focus on families of Pawlak approximation spaces, called multiple-source approximation systems (MSASs). These reflect the situation where information arrives from multiple sources. The behaviour of rough sets in MSASs is investigated – different notions of lower and upper approximations, and definability of a set in a MSAS are introduced. In this context, a generalized version of an information system, viz. multiple-source knowledge representation (KR)-system, is discussed. Apart from the indiscernibility relation which can be defined on a multiple-source KR-system, two other relations, viz. similarity and inclusion are considered. To facilitate formal reasoning with rough sets in MSASs, a quantified propositional modal logic $LMSAS$ is proposed. Interpretations for sets of well-formed formulae (wffs) of $LMSAS$ are defined on MSASs, and the various properties of rough sets in MSASs translate into logically valid wffs of the system. $LMSAS$ is shown to be sound and complete with respect to this semantics. Some decidable problems are addressed. In particular, it is shown that for any $LMSAS$-wff $\alpha$, it is possible to check whether $\alpha$ is satisfiable in a certain class of interpretations with MSASs of a given finite cardinality. Moreover, it is also decidable whether any wff $\alpha$ is satisfiable in the class of all interpretations with MSASs having domain of a given finite cardinality.
##### MSC:
 68T37 Reasoning under uncertainty
##### References:
 [1] Banerjee, M.; Khan, M. A.: Propositional logics from rough set theory, Transactions on rough sets 6, 1-25 (2007) · Zbl 1186.68446 · doi:10.1007/978-3-540-71200-8_1 [2] Bull, R. A.: On modal logic with propositional quantifiers, Journal of symbolic logic 34, No. 2, 257-263 (1969) · Zbl 0184.28101 · doi:10.2307/2271102 [3] Fagin, R.; Halpern, J. Y.; Moses, Y.; Vardi, M. Y.: Reasoning about knowledge, (1995) · Zbl 0839.68095 [4] Del Cerro, L. Farinas; Orłowska, E.: DAL – a logic for data analysis, Theoretical computer science 36, 251-264 (1997) · Zbl 0565.68032 · doi:10.1016/0304-3975(85)90046-5 [5] S. Greco, B. Matarazzo, R. Słowiński, Parametrized rough set model using rough membership and Bayesian confirmation measures, International Journal of Approximate Reasoning, in press, doi:10.1016/j.ijar.2007.05.018. [6] Hughes, G. E.; Cresswell, M. J.: A new introduction to modal logic, (1996) [7] Katzberg, J.; Ziarko, W.: Variable precision rough sets with asymmetric bounds, Proceedings of the international workshop on rough sets and knowledge discovery (RSKD’93), Banff, Canada, October 1993, 163-191 (1993) [8] Kripke, S. A.: A completeness theorem in modal logic, Journal of symbolic logic 24, No. 1, 1-14 (1959) · Zbl 0091.00902 · doi:10.2307/2964568 [9] Liau, C. J.: An overview of rough set semantics for modal and quantifier logics, International journal of uncertainty fuzziness and knowledge based systems 8, No. 1, 93-118 (2000) · Zbl 1113.03309 · doi:10.1142/S0218488500000071 [10] Orłowska, E.; Pawlak, Z.: Representation of nondeterministic information, Theoretical computer science 29, 27-39 (1984) · Zbl 0537.68098 · doi:10.1016/0304-3975(84)90010-0 [11] Orłowska, E.: Kripke semantics for knowledge representation logics, Studia logica 49, 255-272 (1990) · Zbl 0726.03023 · doi:10.1007/BF00935602 [12] Pagliani, P.; Chakraborty, M. K.: Information quanta and approximation spaces I: Non-classical approximation operators, , 605-610 (2005) [13] Pagliani, P.: Pretopologies and dynamic spaces, Fundamenta informaticae 59, No. 2 – 3, 221-239 (2004) · Zbl 1098.68131 [14] Pawlak, Z.; Skowron, A.: Rough sets: some extensions, Information sciences 177, 28-40 (2007) · Zbl 1142.68550 · doi:10.1016/j.ins.2006.06.006 [15] Pawlak, Z.: Rough sets, International journal of computer & information sciences 11, No. 5, 341-356 (1982) [16] Pawlak, Z.: Rough sets. Theoretical aspects of reasoning about data, (1991) · Zbl 0758.68054 [17] Rasiowa, R.: Mechanical proof systems for logic of reaching consensus by groups of intelligent agents, International journal of approximate reasoning 5, No. 4, 415-432 (1991) · Zbl 0738.68073 · doi:10.1016/0888-613X(91)90020-M [18] Rauszer, C.: Knowledge representation systems for groups of agents, Philosophical logic in Poland, 217-238 (1994) [19] Skowron, A.; Stepaniuk, J.: Tolerance approximation spaces, Fundamenta informaticae 27, 245-253 (1996) · Zbl 0868.68103 [20] A. Skowron, Approximate reasoning in MAS: rough set approach, in: Proceedings 2006 IEEE/WIC/ACM Conference on Intelligent Agent Technology, IEEE Computer Society, Washington, 2006, pp. 12 – 18. [21] Śle¸, D.; Zak; Ziarko, W.: The investigation of the Bayesian rough set model, International journal of approximate reasoning 40, 81-91 (2005) [22] Vakarelov, D.: Abstract characterization of some knowledge representation systems and the logic NIL of nondeterministic information, Artificial intelligence II, 255-260 (1987) [23] Wong, S. K. M.: A rough set model for reasoning about knowledge, Rough sets in knowledge discovery 1: methodology and applications, 276-285 (1998) · Zbl 0946.68136 [24] Yao, Y. Y.; Wong, S. K. M.; Lin, T. Y.: A review of rough set models, Rough sets and data mining: analysis for imprecise data, 47-75 (1997) · Zbl 0861.68101 [25] Y.Y. Yao, Probabilistic rough set approximations, International Journal of Approximate Reasoning, in press, doi:10.1016/j.ijar.2007.05.019. [26] W. Ziarko, Probabilistic approach to rough sets, International Journal of Approximate Reasoning, in press, doi:10.1016/j.ijar.2007.06.014.