Reasoning with Rough Sets and Paraconsistent Rough Sets

Sammanfattning: This thesis presents an approach to knowledge representation combining rough sets and para-consistent logic programming.The rough sets framework proposes a method to handle a specific type of uncertainty originating from the fact that an agent may perceive different objects of the universe as being similar, although they may have di®erent properties. A rough set is then defined by approximations taking into account the similarity between objects. The number of applications and the clear mathematical foundation of rough sets techniques demonstrate their importance.Most of the research in the rough sets field overlooks three important aspects. Firstly, there are no established techniques for defining rough concepts (sets) in terms of other rough concepts and for reasoning about them. Secondly, there are no systematic methods for integration of domain and expert knowledge into the definition of rough concepts. Thirdly, some additional forms of uncertainty are not considered: it is assumed that knowledge about similarities between objects is precise, while in reality it may be incomplete and contradictory; and, for some objects there may be no evidence about whether they belong to a certain concept.The thesis addresses these problems using the ideas of paraconsistent logic programming, a recognized technique which makes it possible to represent inconsistent knowledge and to reason about it. This work consists of two parts, each of which proposes a di®erent language. Both languages cater for the definition of rough sets by combining lower and upper approximations and boundaries of other rough sets. Both frameworks take into account that membership of an object into a concept may be unknown.The fundamental difference between the languages is in the treatment of similarity relations. The first language assumes that similarities between objects are represented by equivalence relations induced from objects with similar descriptions in terms of a given number of attributes. The second language allows the user to define similarity relations suitable for the application in mind and takes into account that similarity between objects may be imprecise. Thus, four-valued similarity relations are used to model indiscernibility between objects, which give rise to rough sets with four-valued approximations, called paraconsistent rough sets. The semantics of both languages borrows ideas and techniques used in paraconsistent logic programming. Therefore, a distinctive feature of our work is that it brings together two major fields, rough sets and paraconsistent logic programming.