Indistinguishability Operators

The notion of equality is essential in any formal theory since it allows us to classify the objects it deals with.

Classifying is one of the most important processes in knowledge, representation and inference since it permits us to relate, construct, generalize, find general laws, etc. It is inconceivable that a scientific knowledge should be without an equality that allows us to classify the objects it studies.

One of the most interesting issues related to indistinguishability operators is their generation, which depends on the way in which the data are given and the use we want to make of them. The four most common ways are:

In many situations, data come packed as a reflexive and symmetric fuzzy matrix or relation R, also known as a proximity or tolerance relation. When, for coherence, transitivity is also required, the relation R must be replaced by a new relation R′ that satisfies the transitivity property. The transitive closure of R is the smallest of such relations among those greater than or equal to R. It is the most popular approximation of R and there are several algorithms for calculating it. In Section 2.1, the sup− T product is introduced to generate it. If a lower approximation is required, transitive openings are a possibility 2.4.

The presence of an indistinguishability operator on a universe determines its granules.

According to Zadeh, granularity is one of the basic concepts that underlie human cognition [146] and the elements within a granule ‘have to be dealt with as a whole rather than individually’ [145].

min -indistinguishability operators are widely used in Taxonomy because they are closely related to hierarchical trees. Indeed, given a min -indistinguishability operator on a set X and α ∈ [0,1], the α-cuts of E are partitions of X and if α ≥ β, then the α-cut is a refinement of the partition corresponding to the β-cut. Therefore, E generates an indexed hierarchical tree. Reciprocally, from an indexed hierarchical tree a min -indistinguishability operator can be generated. These results follow from the fact that 1–E is a pseudo ultrametric. Pseudo ultrametrics are pseudodistances where, in the triangular inequality, the addition is replaced by the more restrictive max operation. The topologies generated by ultrametrics are very peculiar, since if two balls are non disjoint, then one of them is included in the other one.

As explained in Chapter 4, T -indistinguishability operators have a very important metric component. One consequence of this fact is that, if the t-norm is continuous Archimedean, the operators generate metric betweenness relations and their structure can be studied in terms of the different types of betweenness relations they produce. This chapter will revisit the three main methods for building T -indistinguishability operators -with the sup-T product, using the Representation Theorem and by constructing a decomposable relation from a fuzzy subset- in relation to betweenness relations.

The Representation Theorem 2.54 states that every T -indistinguishability operator on a universe X can be generated by a family of fuzzy subsets of X. Nevertheless, there is no uniqueness in the selection of the family. Different families, even having different cardinalities, can generate the same operator. This point lends great interest to the theorem, since if we interpret the elements of the family as degrees of matching between the elements of the universe X and a set of prototypes, we can choose different features in order to establish this matching, thereby giving different semantic interpretations to the same T -indistinguishability operator.

In many situations, there can be more than one indistinguishability operator or, more generally, a T-transitive relation defined on a universe. Let us suppose, for example that we have a set of instances defined by some features. We can generate an indistinguishability operator or a fuzzy preorder from each feature. Also, we can have some prototypes, and again we can define a relation from each of them in our universe. In these cases we may need to aggregate the relations obtained. This is usually done by calculating their minimum (or infimum). Although this has a very clear interpretation in fuzzy logic since the infimum is used to model the universal fuzzy quantifier ∀, it often leads to undesirable results in applications because the minimum has a drastic effect. If, for example, two objects of our universe are very similar or indistinguishable for all but one indistinguishability operator but are very different for this particular operator, then the application of the minimum will give this last measure all others will be forgotten. This can be reasonable and useful if we need a perfect matching with respect to all of our relations, but this is not the case in many situations. When we need to take all relations into account in a less dramatic way, we need other ways of aggregating them. Since if R and S are T-transitive fuzzy relations with respect to a t-norm T then T(R,S) is also a T-transitive fuzzy relation, it seems at first glance that this could be a good way to aggregate them. Nevertheless, if we aggregate in this way, we obtain relations with very low values. In the case of non-strict Archimedean t-norms, it is even worse, since in many cases almost all of the values of the obtained relation are equal to zero. Therefore, other ways need to be found.

A proximity matrix or relation on a finite universe X is a reflexive and symmetric fuzzy relation R on X. In many applications, for coherence-imposition or knowledge-learning reasons, transitivity of R with respect to a t-norm T is required. T-transitive approximation methods for proximities are especially useful in many artificial intelligence areas such as fuzzy clustering [98], non-monotonic reasoning [24], fuzzy database modelling [86] [128], decision-making and approximate reasoning [27] applications. In these cases, R must be replaced by a new relation E that also satisfies transitivity, i.e. T-indistinguishability operators. Of course, it is desirable that E be as close as possible to R. This chapter presents three reasonable -i.e. easy and rapid- ways to find close transitive relations to R when the t-norm is continuous Archimedean,as well as a fourth method for the minimum t-norm.

Fuzzy functions fuzzify the concept of a function between two universes. They have been used in various fields, including vague algebras [37], fuzzy numbers [71], vague lattices and quantum mechanics, and have proven useful to the understanding of approximate reasoning [38], the analysis of input/output systems, fuzzy interpolation [38], [14] and reasoning based on fuzzy rules [80].

In approximate reasoning, imprecise conclusions are inferred from imprecise premises. The typical way to do this is by using IF-THEN rules of the form

If x is A, then y is B.

A and B are modelled by fuzzy subsets μ _A and ν _B , respectively, and the conditions x is A and y is B are measured by μ _A (x) and ν _B (y).

This chapter presents two approaches to approximate reasoning based on indistinguishability operators.

In the crisp case, if (G, ∘ ) is a set with an operation ∘ : G ×G →G and ~ is an equivalence relation on G, then ∘ is compatible with ~ if and only if

In this case, an operation \(\tilde{\circ}\) can be defined on \(\overline{G}=G/\sim\) by where \(\overline{a}\) and \(\overline{b}\) are the equivalence classes of a and b with respect to ~.

Demirci generalized this idea to the fuzzy framework by introducing the concept of vague algebra, which basically consists of fuzzy operations compatible with given indistinguishability operators [37].

The literature contains examples of indistinguishability operators valued in more general structures than the unit interval endorsed by a t-norm. In [60] [58], for example, indistinguishability operators are studied under category theory. In [23], the unit interval was generalized to GL-monoids. These generalizations are very useful, because they simplify the study of some specific cases. For example, many of the results found in this chapter about finitely valued indistinguishability operators can be proved in exactly the same way as in the unit interval case because both are GL-monoids.