Mohua Banerjee

Roughness of a fuzzy set

An integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set. Several properties of this new measure are established. Some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned.

Rough fuzzy MLP: knowledge encoding and classification

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).

Evolutionary Rough Feature Selection in Gene Expression Data

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.

Rough sets through algebraic logic

While studying rough equality within the framework of the modal system S 5, 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 rough algebra is more structured than a tqBa. Thus, enriching the tqBa with additional axioms, two more structures, viz. pre-rough algebra and rough algebra, are denned. Representation theorems of these algebras are also obtained. Further, the corresponding logical systems L 1 L 2 are proposed and eventually, L 2 is proved to be sound and complete with respect to a rough set semantics.

Algebras from Rough Sets

Rough set theory has seen nearly two decades of research on both foundations and on diverse applications. A substantial part of the work done on the theory has been devoted to the study of its algebraic aspects. ‘Rough algebras’ now abound, and have been shown to be instances of various algebraic structures, both well-established and relatively new, e.g., quasi-Boolean, Stone, double Stone, Nelson, Lukasiewicz algebras, on the one hand, and topological quasi-Boolean, prerough and rough algebras, on the other. More interestingly and importantly, some of these latter algebras find a new dimension (interpretation) through representations as rough structures. An attempt is made here to present the various relationships and to discuss the representation results.

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 Simple Modal Logic for Reasoning about Revealed Beliefs

Even though in Artificial Intelligence, a set of classical logical formulae is often called a belief base, reasoning about beliefs requires more than the language of classical logic. This paper proposes a simple logic whose atoms are beliefs and formulae are conjunctions, disjunctions and negations of beliefs. It enables an agent to reason about some beliefs of another agent as revealed by the latter. This logic, called MEL , borrows its axioms from the modal logic KD , but it is an encapsulation of propositional logic rather than an extension thereof. Its semantics is given in terms of subsets of interpretations, and the models of a formula in MEL is a family of such non-empty subsets. It captures the idea that while the consistent epistemic state of an agent about the world is represented by a non-empty subset of possible worlds, the meta-epistemic state of another agent about the formers epistemic state is a family of such subsets. We prove that any family of non-empty subsets of interpretations can be expressed as a single formula in MEL . This formula is a symbolic counterpart of the Mobius transform in the theory of belief functions.

Rough sets and 3-valued Lukasiewicz logic

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.

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.

Rough Consequence and Rough Algebra

A notion of rough consequence is investigated in detail. Two algebraic structures emerge from the modal system S5. Properties of these structures, called rough algebras, have been studied. A link between rough consequence and one of the rough algebras is established.