Some Remarks on Strong Fuzzy Metrics and Strong Fuzzy Approximating Metrics with Applications in Word Combinatorics

Abstract

Noticing that ordinary metrics do not present an adequate tool for the study of analytic problems of word combinatorics, as well as in the research of some problems related to theoretical computer science, we propose to use fuzzy metrics in this type of problems. Specifically, the so-called strong fuzzy metric seems to be more appropriate here. In the first part of the paper, we study some special classes of strong fuzzy metrics, topological and lattice properties of certain families of strong fuzzy metrics, and, more generally, strong k-fuzzy metrics. Noticing that one of the standard axioms of a strong fuzzy metric can be easily violated when applied in real situations, in the second part of the paper we introduce more general, approximating fuzzy metrics and illustrate their applicability with some numerical examples.

Fuzzy Approximating Metrics, Approximating Parametrized Metrics and Their Relations with Fuzzy Partial Metrics

“…When comparing x and y at every step t , thus having information up to t on both strings and not knowing yet whether x = y, we obviously have only relation M(x, x, t) ≥ M(x, y, t). We view these observations as justification for the following definitions first introduced in our paper [18]. Definition 4.…”

“…In our previous papers, we presented several approaches and constructions of fuzzy metrics [6,24], parametrized metrics [7] and fuzzy approximating metrics [18], which describe the distance between any infinite words. In these papers, we stress inappropriate numerical results for ordinary metrics on the universe of the infinite words.…”

“…In these papers, we stress inappropriate numerical results for ordinary metrics on the universe of the infinite words. Here, for the evaluation of the distance between infinite words, we use a strong approximating parametrized metric constructed by means of Theorem 1 from a strong fuzzy approximating metric considered in [18].…”

“…It is provable that the mapping M c : X × X × R + → (0, 1] is a strong fuzzy approximating pseudometric in the case of the drastic t-norm T D (for details, see [18]). Now from Theorem 1, we define an approximating parametrized metric P M c (x, y, t) = 1−M c (x,y,t) M c (x,y,t) and define lim t→∞ P M c (x, y, t) = Γ(x, y).…”