Logics for some dynamic spaces-I

We study a collection of logics L(T,I) with models based on ‘dynamic I spaces’, which are finite sequences of Kripke I frames with a common domain, I being any of the normal modal systems K, K4, T, B, S4, KTB, KB4 and S5. The language of L(T,I) has modal connectives for ‘possibility’ and ‘necessity’, as well as temporal connectives. The semantics of L(T,I) can be determined through a kind of fibring over a combination of temporal and Kripke I frames corresponding to the modal system I. This article presents, in a schematic manner, tableau-based proof procedures for this class of logics. Comparisons with closely related systems are made. We briefly look at possible applications of the logics as well. The study, in fact, generalizes the work on the logic temporal rough logic (TRL) by Banerjee and Khan [2] for Pawlak's rough set theory (RST), models of which are based on dynamic S5 spaces. The motivation behind TRL was to capture reasoning with rough sets in the scenario of a knowledge base evolving with time, when the latter is represented by a partition on the domain of discourse. RST has been generalized in many ways over the years, in particular to situations when the knowledge base is not necessarily represented by an equivalence relation, but, for instance, by tolerances or pre-orders. The logics presented here enable one to address reasoning with concepts in the context of such generalized knowledge bases evolving with time.