Dialogue in Hierarchical Learning of a Concept Using Prototypes and Counterexamples

Abstract

While dealing with vague concepts often it puts us in fix to determine whether to a particular situation/case/state a particular concept applies or not. A human perceiver can determine some cases as the positive instances of the concept, and some as the negative instances of the same; but there always remain cases, which might have some similarities with some positive cases, and also have some similarities with some negative cases of the concept. So we propose to learn about the applicability of a concept to a particular situation using a notion of similarity of the situation with the available prototypes (positive instances) and counterexamples (negative instances) of the concept. Perceiving a vague concept, due to the inherent nature of vagueness, is subjective, and thus never can be exhausted by listing down all the positive and negative instances of the concept. Rather we may come to realize about the applicability, or non-applicability, or applicability to some extent, of a concept to a situation in a step-by-step hierarchical manner by initiating dialogue between a perceiver and the situation descriptor. Hence, the main key ingredients of this proposal are (i) prototypes and counterexamples of a concept, (ii) similarity based arguments in favour and against of applicability of a concept at a particular situation, and (iii) hierarchical learning of the concept through dialogues. Similarity based reasoning [3], hierarchical learning of concepts [1], dialogue in the context of approximation space [2] all are separately important directions of research. For our purpose, in this presentation we would concentrate on combining these aspects from a different angle. In [4], a preliminary version of logic of prototypes and counterexamples has been set. To make this paper self-contained, we recapitulate the necessary definitions below. We start with a set S of finitely many situations, member of W may be called a world. We now consider a fuzzy approximation space W, Sim, where Sim is a fuzzy similarity relation between worlds of W. That is, Sim : W × W → [0, 1], and we assume Sim to satisfy the following properties. (i) Sim(ω, ω) = 1 (reflexivity) (ii) Sim(ω, ω ′) = Sim(ω ′ , ω) (symmetry) (iii) Sim(ω, ω ′) * Sim(ω ′ , ω ′′) ≤ Sim(ω, ω ′′) (transitivity). Following [3], the fuzzy approximation space W, Sim is based on the unit