Rough-Neural Computing Techniques for Computing with Words
Soft computing comprises various paradigms dedicated to approximately solving real-world problems, e.g., in decision making, classification or learning; among these paradigms are fuzzy sets, rough sets, neural networks, and genetic algorithms. It is well
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[email protected] Summary. In this chapter, we propose a brief survey of rough mereology, i.e., approximate calculus of parts, an approach to reasoning under uncertainty based on the notion of an approximate part (part to a degree). Rough mereology is mentioned or applied in chapters by Gomolinska; Pal, Peters et al.; Skowron and Stepaniuk; Skowron and Swinarski; Peters, Ramanna et al.; and in some chapters in this book, ideas of approximations by parts are presented implicitly in the formal apparatus chosen there. Therefore, it seems desirable to give an account of this approach.
We exploit this approach as the basis for a model of rough-neural computations whose architecture is modeled on neural network architecture but whose computations are performed according to the ideas of rough set theory. In the sections that follow, we review in a nutshell rough set theoretical notions (Sect. I), ontology and mereology (Sect. 2), rough mereology (Sect. 3), and we present a rough-neural computation model based on rough mereological notions and inspired by a neural computing paradigm. As rough mereology may justly be regarded as a common generalization of rough and fuzzy set theories, this model of computations is a particular instance of a rough-fuzzyneural computing paradigm (see chapter by Pal, Peters et al.).
1 Rough Sets: First Notions Motivation for the approach called rough mereology comes from rough set theory as well as from fuzzy set theory. The two approaches to uncertainty and vagueness stem from a common inspiration: to describe inexact concepts by elaborating a language with which it would be possible to express properties of boundary objects, which every inexact concept must possess [14,26]. Rough set theory approaches this problem with descriptors in an attribute-valuebased language, viz., given a finite set U of objects and a finite set A of attributes, we call a descriptor [14] any pair of the form (a, v) where a E A, v E Va, and Va is the value set of the attribute a : U ---> Va formally understood as a function.
S. K. Pal et al. (eds.), Rough-Neural Computing © Springer-Verlag Berlin Heidelberg 2004
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L. Polkowski
Descriptors allow for a logical language: we call descriptors elementary formulas in descriptor logic (DL), and by formulas of (DL), we mean inscriptions formed from elementary formulas by means of propositional functors of disjunction V, conjunction /\, and negation '. Thus, any formula of logic DL may be written in the disjunctive normal form (DNF) as (1)
where we observe that due to finiteness of U and A, we may dispense with negation. We may add a semantic ingredient to logic DL by interpreting the formulas in set U: for a formula a, the meaning [a] of a is defined by
[aJ={xEU:xFa}, where the semantic satisfaction
(2)
F is defined by recurrence as follows
x F (a, v) iff a(x) = v, xFyvaiffxFyorxFa, x F y/\a iff x F yand x F a.
(3)
A concept X ~ U is exact when there exists a formula a of form (I) with the property that X = [a]. Otherwise, the concept X is inexact; in this c
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