# Fuzzy Multisets and Fuzzy Computing

DOI: 10.4018/978-1-4666-9798-0.ch002
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## Abstract

Multisets and Multirelations are extensions of ordinary sets and relations while fuzzy multisets and fuzzy multirelations are fuzzy extensions of these concept. Although many generalizations tend to be meaningless, these are quite useful as one can model things we see every day. In addition, these structures have found uses in the theory of computation. In particular, fuzzy multisets are used in fuzzy models of computation, that is, models of computation where vagueness is manifested by fuzzy sets play central role. These fuzzy models of computation include fuzzy P systems, the fuzzy chemical abstract machine, and fuzzy Petri nets.
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## Introduction

Vagueness is a linguistic phenomenon as well as a property of physical objects. In the English language the word fuzzy is a synonym of the word vague. Typically, the term vague is used to denote something uncertain, imprecise or ambiguous. Nevertheless, it is widely accepted that a term is vague to the extent that it has borderline cases, that is, cases in which it seems impossible either to apply or not to apply this term. The Sorites Paradox, which was introduced by Eubulides of Miletus, is a typical example of an argument that shows what it is meant by borderline cases. Also, the paradox is one of the so called little-by-little arguments. The term sorites derives from the Greek word for heap. The paradox is about the number of grains of wheat that makes a heap. All agree that a single grain of wheat does not comprise a heap. The same applies for two grains of wheat as they do not comprise a heap, etc. However, there is a point where the number of grains becomes large enough to be called a heap, but there is no general agreement as to where this occurs. Since it is quite possible that some (?) readers may find almost absurd the idea that vagueness exists in the real world, it is necessary to present an example that hopefully sheds light on this idea. The following example, which was presented originally by E.J. Lowe (Lowe, 1994), shows that vagueness exists at the subatomic level:

Suppose (to keep matters simple) that in an ionization chamber a free electron a is captured by a certain atom to form a negative ion which, a short time later, reverts to a neutral state by releasing an electron b. As I understand it, according to currently accepted quantum mechanical principles there may simply be no objective fact of the matter as to whether or not a is identical with b. It should be emphasized that what is being proposed here is not merely that we may well have no way of telling whether or not a and b are identical,which would imply only an epistemic indeterminacy. It is well known that the sort of indeterminacy presupposed by orthodox interpretations of quantum theory is more than merely epistemic—it is ontic. The key feature of the example is that in such an interaction electron a and other electrons in the outer shell of the relevant atom enter an “entangled” or “superposed” state in which the number of electrons present is determinate but the identity of any one of them with a is not, thus rendering likewise indeterminate the identity of a with the released electron b.

The idea behind this example is that “identity statements represented by a = b are ‘ontically’ indeterminate in the quantum mechanical context”(French & Krause, 2003). In different words, in the quantum mechanical context a is equal to b to some degree, which is one of the fundamental ideas behind fuzzy set theory.

Fuzzy set theory is a mathematical model of vagueness. . It is widely accepted that there are (at least) three different expressions of vagueness(Sorensen, 2008):

• Many-valued Logics and Fuzzy Logic Borderline statements are assigned truth-values that are between absolute truth and absolute falsehood .

• Supervaluationism The idea that borderline statements lack a truth value.

• Contextualism The truth value of a proposition depends on its context (i.e., a person may be tall relative to American men but short relative to NBA players).

There is a fourth, more recent, expression of vagueness that is based on the use of paraconsistent logics.

The basic idea behind fuzzy sets is that a set contains elements to some degree. Let us give an example that will illustrate this idea. Consider a cloud and the space it occupies. Certain points are definitely inside the cloud and many more points are definitely outside the cloud, but there are some points where one cannot say for sure whether they are inside or outside the cloud. For some, it might be more likely to be inside the cloud but for some it would be more likely to be outside the cloud. Because of this uncertainty, we say that these points belong to some degree to the cloud.

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