Considering the increasing interest in fuzzy theory and possible applications, the concept of fuzzy metric space concept has been introduced by several authors from different perspectives. This paper interprets the theory in terms of metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology. We study interrelationships between this theory and other fuzzy theories such as intuit...

We generalize the classical notion of VC-dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of...

We show that the notion of inductive bias in concept learning can be quantified in a way that directl_v relates to learning performance in the framework recently introduced by Valiant. Our measure of bias is based on the growth function introduced by Vapnik and Chervonenkis, and on the Vapnik-Chervonenkis dimension. We measure some common language biases, including restriction to conjunctive co...

We consider inductive limits A of sequences A\ —> Ai —» of finite direct sums of C*-algebras of continuous functions from compact Hausdorff spaces into full matrix algebras. We prove that A has topological stable rank (tsr) one provided that A is simple and the sequence of the dimensions of the spectra of Aι is bounded. For unital A, tsr (̂ 4) = 1 means that the set of invertible elements is den...

In this paper, following the ideas of (continuous) t-norm and interval numbers, a concept of (continuous) interval-valued t-norm is proposed. Based on the interval-valued fuzzy set and the continuous interval-valued t-norm, we propose a notion of interval-valued fuzzy metric space, which is a generalization of fuzzy metric space in the sense of George and Veeramani [Fuzzy Sets and Systems 64 (1...

Inductive queries are queries that generate pattern sets. This paper studies properties of boolean inductive queries, i.e. queries that are boolean expressions over monotonic and anti-monotonic constraints. More specifically, we introduce and study algebraic operations on the answer sets of such queries and show how these can be used for constructing and optimizing query plans. Special attentio...

Our paper was motivated by (Chaudhuri and Rosenfeld, 1996), a paper in which several metrics related to the Hausdorff metric are given for fuzzy sets defined on the same nonempty support set S. The first metric given in (Chaudhuri and Rosenfeld, 1996), denoted F 0 in our paper, requires all the fuzzy sets considered by F 0 to have the same maximum value. Chaudhuri and Rosenfeld (1996) then modi...