Anna Maria Radzikowska

A comparative study of fuzzy rough sets

The notion of a rough set was originally proposed by Pawlak (1982). Later on, Dubois and Prade (1990) introduced fuzzy rough sets as a fuzzy generalization of rough sets. In this paper, we present a more general approach to the fuzzification of rough sets. Specifically, we define a broad family of fuzzy rough sets, each one of which, called an (I, J)-fuzzy rough set, is determined by an implicator I and a triangular norm J. Basic properties of fuzzy rough sets are investigated. In particular, we define three classes of fuzzy rough sets, relatively to three main classes of implicators well known in the literature, and analyse their properties in the context of basic rough equalities. Finally, we refer to an operator-oriented characterization of rough sets as proposed by Lin and Liu (1994) and show soundness of this axiomatization for the Lukasiewicz fuzzy rough sets.

Fuzzy rough sets based on residuated lattices

Rough sets were developed by Pawlak as a formal tool for representing and processing information in data tables. Fuzzy generalizations of rough sets were introduced by Dubois and Prade. In this paper, we consider L–fuzzy rough sets as a further generalization of the notion of rough sets. Specifically, we take a residuated lattice L as a basic structure. L–fuzzy rough sets are defined using the product operator and its residuum provided by the residuated lattice L. Depending on classes of binary fuzzy relations, we define several classes of L–fuzzy rough sets and investigate properties of these classes.

Characterisation of main classes of fuzzy relations using fuzzy modal operators

Fuzzy modal operators express interactions between binary fuzzy relations and fuzzy sets. Each of these operators is determined by a binary fuzzy relation (on a given universe) and transforms one fuzzy set (in this universe) to another one. In this paper, we define several fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators. Interpretation of these characterisations is presented in the context of fuzzy modal logics. We show that these characterisations constitute the basis for determining characteristic axioms of particular classes of fuzzy modal logics.

Rough approximation operations based on IF sets

Intuitionistic fuzzy sets, originally introduced by Atanassov. allow for representation both degrees of membership and degrees of non-membership of an element to a set. In this paper we present a generalisation of Pawlaks rough approximation operations taking Atanassovs structures as a basis. A special class of residuated lattices is taken as a basic algebraic structure. In the signature of these algebras we have counterparts of two main classes of fuzzy implications. We show that basing on these lattices we can express degrees of weak and strong certainties and possibilities of membership and non-membership of an element to a set.

On L-Fuzzy Rough Sets

In this paper we introduce a new class of algebras, called extended residuated lattices. Basing on this structure we present an algebraic generalization of approximation operators and rough sets determined by abstract counterparts of fuzzy logical operations. We show formal properties of these structures taking into account several classes of fuzzy relations.

Epistemic Approach to Actions with Typical Effects

We study a problem of actions with typical, but not certain effects. We show how this kind of actions can be incorporated in a dynamic/epistemic multi-agents system in which the knowledge, abilities and opportunities of agents are formalized as well as the results of actions they perform. To cope with complexity of a rational agent behaviour, we consider scenarios composed of traditionally viewed basic actions and atomic actions with typical effects. We focus on a specific type of scenarios reflecting a “typical” pattern of an agents behaviour. Adopting a model-theoretic approach we formalize a nonmonotonic preferential strategy for these scenarios in order to reason about the final results of their realizations.

Double Residuated Lattices and Their Applications

In this paper we introduce a new class of double residuated lattices. Basic properties of these algebras are given. Taking double residuated lattices as a basis, we propose a fuzzy generalisation of information relations. We also define several fuzzy information operators and show that some classes of information relations can be characterised by means of these operators.

Lattice-Based relation algebras II

We present classes of algebras which may be viewed as weak relation algebras, where a Boolean part is replaced by a not necessarily distributive lattice. For each of the classes considered in the paper we prove a relational representation theorem.

A fuzzy generalisation of information relations

In this paper we consider a fuzzy generalisation of some information relations. Basic properties of these relations are provided. We give characterisations of these relations formalised by means of fuzzy information operators. For particular classes of fuzzy information relations the corresponding classes of fuzzy information logics are defined and briefly discussed.

On some classes of fuzzy information relations

Recently, E. Orlowska (1999) has proposed a many-valued generalization of modal information logics which are logical systems for reasoning from incomplete and many-valued information. Following her ideas, we consider a fuzzy generalization of some information relations. We provide basic properties of these relations and their characterizations formalized by means of fuzzy relational operators. For particular classes of fuzzy information relations the corresponding classes of fuzzy information logics are defined and briefly discussed.