Answering Fuzzy Queries over Fuzzy DL-Lite Ontologies

A prominent problem in knowledge representation is how to answer queries taking into account also the implicit consequences of an ontology representing domain knowledge. While this problem has been widely studied within the realm of description logic ontologies, it has been surprisingly neglected within the context of vague or imprecise knowledge, particularly from the point of view of mathematical fuzzy logic. In this paper, we study the problem of answering conjunctive queries and threshold queries w.r.t. ontologies in fuzzy DL-Lite. Specifically, we show through a rewriting approach that threshold query answering w.r.t. consistent ontologies remains in ${AC}^{0}$ in data complexity, but that conjunctive query answering is highly dependent on the selected triangular norm, which has an impact on the underlying semantics. For the idempotent Gödel t-norm, we provide an effective method based on a reduction to the classical case.

Revision of Pseudo-Ultrametric Spaces Based on m-Polar T-Equivalences and Its Application in Decision Making

In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy environment. Dealing with multi-polarity in practical examples with fuzzy data leadsus to introduce a new concept called m-polar T-equivalence relations based on a finitely multivalued t-norm T, and to study the metric behavior of such relations. First, we study the new operators including the m-polar triangular norm T and conorm S as well as m-polar implication I and m-polar negation N, acting on the Cartesian product of [0,1]m-times.Then, using the m-polar negations N, we provide a method to construct a new type of metric spaces, called m-polar S-pseudo-ultrametric, from the m-polar T-equivalences, and reciprocally for constructing m-polar T-equivalences based on the m-polar S-pseudo-ultrametrics. Finally, the link between fuzzy graphs and m-polar S-pseudo-ultrametrics is considered. An algorithm is designed to plot a fuzzy graph based on the m-polar SL-pseudo-ultrametric, where SL is the m-polar Lukasiewicz t-conorm, and is illustrated by a numerical example which verifies our method.

Inequalities in Triangular Norm-Based ∗-fuzzy ( L + ) p Spaces

In this article, we introduce the ∗-fuzzy (L+)p spaces for 1≤p<∞ on triangular norm-based ∗-fuzzy measure spaces and show that they are complete ∗-fuzzy normed space and investigate some properties in these space. Next, we prove Chebyshev’s inequality and Hölder’s inequality in ∗-fuzzy (L+)p spaces.

The Auto-Diagnosis of Granulation of Information Retrieval on the Web

In this paper, a postulation on the relationship between the memory structure of the brain’s neural network and the representation of information granules in the semantic web is presented. In order to show this connection, abstract operations of inducing information granules are proposed to be used for the proposed logical operations systems, hereinafter referred to as: analysis, reduction, deduction and synthesis. Firstly, the searched information is compared with the information represented by the thesaurus, which is equivalent to the auto-diagnosis of this system. Secondly, triangular norm systems (information perception systems) are built for fuzzy or vague information. These are fuzzy sets. The introduced logical operations and their logical values, denoted as problematic, hypothetical, validity and decidability, are interpreted in these fuzzy sets. In this way, the granularity of the information retrieval on the Web is determined according to the type of reasoning.

Solvability of a Bounded Parametric System in Max-Łukasiewicz Algebra

The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system.

On the continuous action of enriched lattice-valued convergence groups: Some examples

Starting with a category SL-CONVGRP, of stratified enriched cl-premonoid-valued convergence groups as introduced earlier, we present a category SL-CONVTGRP, of stratified enriched cl-premonoid-valued convergence transformation groups, the idea behind this category is crept in the notion of convergence transformation group - a generalization of topological transformation group. In this respect, we are able to provide natural examples in support to our endeavor; these examples, however, stem from the action of convergence approach groups on convergence approach spaces, and the action of probabilistic convergence groups under triangular norm on probabilistic convergence spaces. Based on the category of enriched lattice-valued convergence spaces, a Cartesian closed category that enjoys lattice-valued convergence structure on function space, we look into among others, the lattice-valued convergence structures on the group of homeomorphisms of enriched lattice-valued convergence spaces, generalizing a concept of convergence transformation groups on convergence spaces, obtaining a characterization.