Four-Valued Extension of Rough Sets

Abstract

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.

Paraconsistent Logic Programs with Four-Valued Rough Sets

“…Table 1 provides the semantics for implication in both logics, L t and L k . Observe that the implication → t , introduced in [9], is a four-valued extension of the usual logical implication, suitable for determining set containment and approximations in the case of four-valued sets. …”

“…The rule language makes it possible to define four-valued relations. A defined relation can then be used to specify approximations of another four-valued relation, as discussed in [9] and in Section 2. Such an approximation is itself a four-valued relation.…”

“…The language discussed in this paper is based on ideas of our previous work [9] that presents 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).…”

“…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.…”

Modeling and Reasoning with Paraconsistent Rough Sets

“…Namely, an object o belongs to the lower approximation of a given set A whenever all objects indiscernible from o belong to A and o belongs to its upper approximation, when there are objects indiscernible from o belonging to A. Indiscernibility is modeled by similarity relations reflecting limited perceptual capabilities as well as incomplete and imprecise knowledge. Such approximations naturally lead to three-and four-valued logics (see, e.g., [6,10,14,21,32,22]). …”