Formal reasoning with rough sets in multiple-source approximation systems

AbstractWe 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 α, it is possible to check whether α is satisfiable in a certain class of interpretations with MSASs of a given finite cardinality. Moreover, it is also decidable whether any wff α is satisfiable in the class of all interpretations with MSASs having domain of a given finite cardinality

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page