Fuzzy Quasi-Metric Spaces: Bicompletion, Contractions on Product Spaces, and Applications to Access Predictions

Since L.A. Zadeh introduced the theory of fuzzy sets in 1965, it has been used in a range of areas of mathematics and applied to a great variety of real life scenarios. These scenarios cover complex processes with no simple mathematical model such as industrial control devices, planning and scheduling, pattern recognition, etc. or systems managing inaccurate or highly unpredictable information. Fuzzy topology is one important example of use of L.A. Zadeh’s theory. Through the years, authors of this field have pursued the definition of a fuzzy metric space in order to measure the distance between elements of a set according to degrees of closeness. This work deals with the bicompletion of fuzzy quasi-metric spaces in the sense of Kramosil and Michalek. Sherwood proved that every fuzzy metric space has a completion which is unique up to isometry based on properties of Levy’s metric. Here we prove that each fuzzy quasi-metric space has a bicompletion. Our construction is performed using directly the suprema of subsets of [0, 1] and lower limits of sequences in [0, 1] instead of using Levy’s metric. We take advantage of the bicompleteness and bicompletion of fuzzy quasimetric spaces as well as of the properties of fuzzy and intuitionistic fuzzy metric spaces in order to introduce several applications to computer science problems. Thus, the existence and uniqueness of solution for the recurrence equations associated to certain algorithms with two recursive procedures is studied. To carry out a complexity analysis of algorithms we apply the Banach contraction principle both in a certain product of (non-Archimedean) fuzzy quasi-metrics on the domain of words and in the product quasi-metric of two Schellekens’ complexity quasi-metric spaces. Finally, we study an application of fuzzy metric spaces to information systems based on accesses locality. For that means we use equivalence classes in order to compare elements and we take advantage of the suitability of fuzzy constructions related to problems that evolve during time. This approach allows to define a dynamical framework to decide on an object classification into different classes. As a natural extension of the model we use the notion of an intuitionistic fuzzy metric space to measure both the degree of closeness and remoteness between two elements of a fuzzy set.