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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