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.

Rough sets are investigated from the viewpoint of topos theory. Two categories RSC and ROUGH of rough sets and a subcategory ξ-RSC are focussed upon. It is shown that RSC and ROUGH are equivalent. Generalizations RSC(𝒞) and ξ-RSC(𝒞) are proposed over an arbitrary topos 𝒞. RSC(𝒞) is shown to be a qu asitopos, while ξ-RSC(𝒞) forms a topos in the special case when 𝒞 is Boolean. An example of RSC(𝒞) is given, through which one is able to define monoid actions on rough sets. Next, the algebra of strong subobjects of an object in RSC is studied using the notion of relative rough complementation. A class of contrapositionally complemented ‘c. ∨ c.’ lattices is obtained as a result, from the object class of RSC. Moreover, it is shown that such a class can also be obtained if the construction is generalized over an arbitrary Boolean algebra.

A pair of approximation operators, based on the notion of granules in generalized approximation spaces, was studied in an earlier work by the authors. In this article, we investigate algebraic structures formed by the definable sets and also by the rough sets determined by this pair of approximation operators. The definable sets are open sets of an Alexandrov topological space, and form a completely distributive lattice in which the set of completely join irreducible elements is join dense. The collection of rough sets also forms a similar structure. Representation results for such classes of completely distributive lattices as well as Heyting algebras in terms of definable and rough sets are obtained. Further, two unary operators on rough sets are considered, making the latter constitute a structure that is named a ‘rough lattice’. Representation results for rough lattices are proved.

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 analyses prevalent definitions of rough sets from the foundational and mathematical perspectives. In particular, the issue of language dependency in the definitions, and implications of the definitions on the issue of vagueness are discussed in detail.

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.

An evolutionary rough feature selection algorithm is used for classifying microarray gene expression patterns. Since the data typically consist of a large number of redundant features, an initial redundancy reduction of the attributes is done to enable faster convergence. Rough set theory is employed to generate reducts, which represent the minimal sets of nonredundant features capable of discerning between all objects, in a multiobjective framework. The effectiveness of the algorithm is demonstrated on three cancer datasets.

A new scheme of knowledge encoding in a fuzzy multilayer perceptron (MLP) using rough set-theoretic concepts is described. Crude domain knowledge is extracted from the data set in the form of rules. The syntax of these rules automatically determines the appropriate number of hidden nodes while the dependency factors are used in the initial weight encoding. The network is then refined during training. Results on classification of speech and synthetic data demonstrate the superiority of the system over the fuzzy and conventional versions of the MLP (involving no initial knowledge).

A propositional logic for rough sets was proposed in [2]. The present work establishes a relationship between the finitary fragment of this logic and 3-valued Lukasiewicz logic ℒ3. It is also observed that there is an embedding from ℒ3 into the modal system S 5 .

While studying rough equality within the framework of the modal system S5 , an algebraic structure called rough algebra [1], came up. Its features were abstracted to yield a topological quasi-Boolean algebra (tqBa). In this paper, it is observed that

In this paper, 'identity' is eventually considered to be determined by a specified collection of 'properties' represented extensionally by subsets. Generally, the collection may contain fuzzy subsets representing vague properties, thus rendering vagueness to the concept of identity. Saturatedness condition, which in essence is a portion of Leibniz's principle of identity, has been dealt with in some detail. A hierarchy of identities is obtained. This hierarchy also indicates some relationship between the concepts 'identity' and 'property'. Some applications of the results to the modelling of indistinguishability in information systems have been indicated.