Md. Aquil Khan

Formal reasoning with rough sets in multiple-source approximation systems

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

A study of multiple-source approximation systems

The article continues an investigation of multiple-source approximation systems (MSASs) [1,2]. These are collections of Pawlak approximation spaces over the same domain, and embody the situation where information arrives from a collection of sources. Notions of strong/weak lower and upper approximations of a subset of the domain were introduced in [1]. These result in a division of the domain into five mutually disjoint sets. Different kinds of definability of a set are then defined. In this paper, we study further properties of all these concepts in a structure called multiple-source approximation system with group knowledge base (MSASG), where we also have equivalence relations representing the combined knowledge base of each group of sources. Some of the properties of combined knowledge base are presented and its relationship with the strong/weak lower and upper approximation is explored. Specifically, ordered structures that arise from these concepts are studied in some detail. In this context, notions of dependency, that reflect how much the information provided by a MSASG depends on an individual source or group of sources, are introduced. Membership functions for MSASs were investigated in [2]. These are also studied afresh here.

Multiple-source approximation systems: membership functions and indiscernibility

The work presents an investigation of multiple-source approximation systems, which are collections of Pawlak approximation spaces over the same domain. We particularly look at notions of definability of sets in such a collection µ. Some possibilities for membership functions in µ are explored. Finally, a relation that reflects the degree to which objects are (in)discernible in µ is also presented.

A Logic for Complete Information Systems

A logic LIS for complete information systems is proposed. The language of LIS contains constants corresponding to attribute and attribute-values. A sound and complete deductive system for the logic is presented. Decidability is also proved.

Logics for information systems and their dynamic extensions

The article proposes logics for information 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 of information and information update are 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 update logic for information systems

Updates in a knowledge base, given as an information system in rough set theory, may need to be made due to changes in (i) the set of attributes, (ii) attribute-values, or (iii) the set of objects (instances). In this article, we propose a logic for information systems which incorporates all these three aspects of updates. The logic can capture the flow of information as well as its effects on the approximations of concepts. A sound and complete deductive system for the logic is presented. The decidability issue is also discussed.

A modal logic for multiple-source tolerance approximation spaces

Notions of lower and upper approximations are proposed for multiple-source tolerance approximation spaces which consist of a number of tolerance relations over the same domain. A modal logic is proposed for reasoning about the defined notions of approximations. A sound and complete deductive system for the logic is presented. Decidability is also proved.

A preference-based multiple-source rough set model

We propose a generalization of Pawlaks rough set model for the multi-agent situation, where information from an agent can be preferred over that of another agent of the system while deciding membership of objects. Notions of lower/upper approximations are given which depend on the knowledge base of the sources as well as on the position of the sources in the hierarchy giving the preference of sources. Some direct consequences of the definitions are presented.

Formal reasoning in preference-based multiple-source rough set model

We propose a generalization of Pawlaks rough set model for the multi-agent situation, where information from an agent can be preferred over that of another agent of the system while deciding membership of objects. Notions of lower/upper approximations are given, which depend on the knowledge base of the sources as well as on the position of the sources in the hierarchy giving the preference of sources. A quantified modal logic is proposed to reason about the properties of the proposed approximations. A sound and complete deductive system for the logic is also presented. Moreover, it is shown how the properties of the proposed approximations can be deduced as theorems of this deductive system.

A probabilistic approach to rough set theory with modal logic perspective

We propose the notion of probabilistic information system (PIS) to capture situations where information regarding attributes of objects are not precise, but given in terms of probability. Notions of indistinguishability relations and corresponding approximation operators based on PISs are proposed and studied. It is shown that the deterministic information systems (DISs), incomplete information systems (IISs) and non-deterministic information systems (NISs) are all special instances of PISs. Moreover, the approximation operators defined on DIS (relative to indiscernibility), IISs and NISs (relative to similarity relations) are all originated from a single approximation operator defined on PISs. Further, a logic LPIS for PISs is proposed that can be used to reason about the proposed approximation operators. A sound and complete deductive system for the logic is given. Decidability of the logic is also proved.