Foundations of Computational Intelligence Volume 2

This chapter considers how one might utilize fuzzy sets, near sets, and rough sets, taken separately or taken together in hybridizations as part of a computational intelligence toolbox. These technologies offer set theoretic approaches to solving many types of problems where the discovery of similar perceptual granules and clusters of perceptual objects is important. Perceptual information systems (or, more concisely, perceptual systems) provide stepping stones leading to nearness relations and properties of near sets. This work has been motivated by an interest in finding a solution to the problem of discovering perceptual granules that are, in some sense, near each other. Fuzzy sets result from the introduction of a membership function that generalizes the traditional characteristic function. Near set theory provides a formal basis for observation, comparison and classification of perceptual granules. Near sets result from the introduction of a description-based approach to perceptual objects and a generalization of the traditional rough set approach to granulation that is independent of the notion of the boundary of a set approximation. Near set theory has strength by virtue of the strength it gains from rough set theory, starting with extensions of the traditional indiscernibility relation. This chapter has been written to establish a context for three forms of sets that are now part of the computational intelligence umbrella. By way of introduction to near sets, this chapter considers various nearness relations that define partitions of sets of perceptual objects that are near each other. Every perceptual granule is represented by a set of perceptual objects that have their origin in the physical world. Objects that have the same appearance are considered perceptually near each other, i.e., objects with matching descriptions. Pixels, pixel windows, and segmentations of digital images are given by way of illustration of sample near sets. This chapter also briefly considers fuzzy near sets and near fuzzy sets as well as rough sets that are near sets.The main contribution of this chapter is the introduction of a formal foundation for near sets considered in the context of fuzzy sets and rough sets.

Fuzzy techniques have been originally invented as a methodology that transforms the knowledge of experts formulated in terms of natural language into a precise computer-implementable form. There are many successful applications of this methodology to situations in which expert knowledge exist, the most well known is an application to fuzzy control.

In some cases, fuzzy methodology is applied even when no expert knowledge exists: instead of trying to approximate the unknown control function by splines, polynomials, or by any other traditional approximation technique, researchers try to approximate it by guessing and tuning the expert rules. Surprisingly, this approximation often works fine, especially in such application areas as control and multi-criteria decision making.

In this chapter, we give a mathematical explanation for this phenomenon.

Most traditional examples of fuzziness come from the analysis of commonsense reasoning. When we reason, we use words from natural language like “young”, “well”. In many practical situations, these words do not have a precise true-or-false meaning, they are fuzzy. One may therefore be left with an impression that fuzziness is a subjective characteristic, it is caused by the specific way our brains work. However, the fact that that we are the result of billions of years of successful adjusting-to-the-environment evolution makes us conclude that everything about us humans is not accidental. In particular, the way we reason is not accidental, this way must reflect some real-life phenomena – otherwise, this feature of our reasoning would have been useless and would not have been abandoned long ago. In other words, the fuzziness in our reasoning must have an objective explanation – in fuzziness of the real world. In this chapter, we first give examples of objective real-world fuzziness. After these example, we provide an explanation of this fuzziness – in terms of cognizability of the world.

We have already proposed a paraconsistent annotated logic program called EVALPSN. In EVALPSN, an annotation called an extended vector annotation is attached to each literal. In order to deal with before-after relation between two time intervals, we introduce a new interpretation for extended vector annotations in EVALPSN, which is named Before-after(bf) EVALPSN.

In this chapter, we introduce the bf-EVALPSN and its application to real-time process order control and its safety verification with simple examples. First, the background and overview of EVALPSN are introduced, and paraconsistent annotated logic as the formal background of EVALPSN and EVALPSN itself are recapitulated with simple examples. Then, after bf-EVALPSN is formally defined, how to implement and apply bf-EVALPSN to real-time intelligent process order control and its safety verification with simple practical examples. Last, unique and useful features of bf-EVALPSN are introduced, and conclusions and remarks are provided.