Parameterized Metrics and Their Applications in Word Combinatorics

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

“…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 particular, fuzzy metrics (see, e.g., [2]) have been used for pattern recognition (see, e.g., [3]), and partial metrics (see, e.g., [4]) have been used for some theoretical computer-science-related applications (see e.g., [5]). Fuzzy fragmentary metrics [6] and parametrized metrics [7,8] have been introduced in order to measure the distance between two infinite words in a more appropriate way than ordinary metrics. In this paper, we introduce yet another generalization of a fuzzy metric, called a fuzzy approximating metric (Section 3) and, closely related with them, the approximating version of parametrized metrics (Section 4).…”

“…As the first step, we see the practical use of fuzzy approximating metrics in the problems of combinatorics on words, in particular, to study the advantages/disadvantages of approximating parametrized metrics if compared with other metric type structures, particularly those which were used in [7], by analyzing specific numerical examples.…”