Abstract The study of rough set theory in multi-agent/multi-source contexts, and its extension to attribute-relative approximations, constitutes a significant line of research in the rough set literature. Motivated by these, we propose a unified modal framework based on multiple-source information structures, where accessibility relations are indexed by sources and parameterized by attribute sets. We introduce two semantics, W-semantics and S-semantics, interpreting modal formulas via weak and strong approximation operators from multi-granulation rough set models. These provide a formal system for reasoning about uncertainty that is both attribute-relative and source-sensitive. We establish sound and complete axiomatizations for the corresponding logics over classes of models satisfying structural properties like reflexivity, symmetry and transitivity. Our completeness proofs use a step-by-step construction adapted to the semantics, avoiding reliance on copying techniques. This work deepens the modal foundations of rough set theory and contributes to the logic of complex approximation systems.

Motivated by rough set theory, we introduce a novel semantics for the basic modal language based on the possibility lower approximation operator of subset approximation structures. The study investigates axiomatization, expressiveness, and invariance results related to this semantics. From a rough set perspective, it provides a formal language for reasoning about the possibility lower approximation operator. Additionally, the axiomatization results obtained here provide characterizing properties of the operator.

The article discusses temporal information systems (TISs) that add the dimension of time to complete or incomplete information systems. Through TISs, one can accommodate the possibility of domains or attribute values for objects changing with time or the availability of currently missing information with time. Different patterns of flow of information give different TISs. The corresponding logics with sound and complete axiomatization are presented.

The notions of strong/weak approximations have been studied extensively in recent years. These approximations are based on a structure of the form (W,{Ri}i∈ N), called the multiple-source approximation system, whereRiis an equivalence relation onW, andNis an initial segment of the set N of natural numbers. We propose and explore a simple modal language and semantics that can be used to reason about the strong/weak approximations of concepts. Moreover, our study is not confined to collections of equivalence relations only, but other types of relations are also considered. This study is important, keeping in view the notions of generalized approximation spaces with relations other than equivalence.

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.

The article proposes logics forinformation systems, which provide information about a set of objects regarding a set of attributes. Both “complete” and “incomplete” information systems are dealt with. The language of these logics contains modal operators, and constants corresponding to attributes and attribute values. Sound and complete deductive systems for these logics are presented, and the problem of decidability is addressed. Furthermore, notions ofinformationandinformation updateare defined, and dynamic extensions of the above logics are presented to accommodate these notions. A set of reduction axioms enables us to obtain a complete axiomatization of the dynamic logics.