We present a language for defining paraconsistent rough sets and reasoning about them. Our framework relates and brings together two major fields: rough sets [23] and paraconsistent logic programming [9]. To model inconsistent and incomplete information we use a four-valued logic.The language discussed in this paper is based on ideas of our previous work [21, 32, 22] developing a four-valued framework for rough sets. In this approach membership function, set containment and set operations are four-valued, where logical values are t (true), f (false), i (inconsistent) and u (unknown).We investigate properties of paraconsistent rough sets as well as develop a paraconsistent rule language, providing basic computational machinery for our approach.

Abstract. This paper presents a language for defining four-valued rough sets and to reason about them. Our framework brings together two major fields: rough sets and paraconsistent logic programming. On the one hand it provides a paraconsistent approach, based on four-valued rough sets, for integrating knowledge from different sources and reasoning in the presence of inconsistencies. On the other hand, it also caters for a specific type of uncertainty that originates from the fact that an agent may perceive different objects of the universe as being indiscernible. This paper extends the ideas presented in [9]. Our language allows the user to define similarity relations and use the approximations induced by them in the definition of other four-valued sets. A positive aspect is that it allows users to tune the level of uncertainty or the source of uncertainty that best suits applications.

Abstract. Rough set approximations of Pawlak [15] are sometimes generalized by using similarities between objects rather than elementary sets. In practical applications, both knowledge about properties of objects and knowledge of similarity between objects can be incomplete and inconsistent. The aim of this paper is to define set approximations when all sets, and their approximations, as well as similarity relations are four-valued. A set is four-valued in the sense that its membership function can have one of the four logical values: unknown (u), false (f), inconsistent (i), or true (t). To this end, a new implication operator and set-theoretical operations on four-valued sets, such as set containment, are introduced. Several properties of lower and upper approximations of four-valued sets are also presented.

Rough sets framework has two appealing aspects. First, it is a mathematical approach to deal with vague concepts. Second, rough set techniques can be used in data analysis to find patterns hidden in the data. The number of applications of rough sets to practical problems in different fields demonstrates the increasing interest in this framework and its applicability.Most of the current rough sets techniques and software systems based on them only consider rough sets defined explicitly by concrete examples given in tabular form. The previous research mostly disregards the following two problems. The first problem is related with how to define rough sets in terms of other rough sets. The second problem is related with how to incorporate domain or expert knowledge.This thesis 1 proposes a language that caters for implicit definitions of rough sets obtained by combining different regions of other rough sets. In this way, concept approximations can be derived by taking into account domain knowledge. A declarative semantics for the language is also discussed. It is then shown that programs in the proposed language can be compiled to extended logic programs under the paraconsistent stable model semantics. The equivalence between the declarative semantics of the language and the declarative semantics of the compiled programs is proved. This transformation provides the computational basis for implementing our ideas. A query language for retrieving information about the concepts represented through the defined rough sets is also defined. Several motivating applications are described. Finally, an extension of the proposed language with numerical measures is discussed. This extension is motivated by the fact that numerical measures are an important aspect in data mining applications.

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