Corrigendum to “An Easy-to-Understand Method to Construct Desired Distance-Like Measures”

Abstract

Metrics and their weaker forms are used to measure difference between two data (or other things). There are many metrics that are available but not desired to a practitioner. This paper recommends in a plausible reasoning manner an easy-to-understand method to construct desired distance-like measures: to fuse easy-to-obtain (or easy to coined by practitioners) pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators (easy to coined by practitioners et al). The simple reason to do this is that data for a real-world problem are sometimes full of multiagents. A distance-like notion, called weak interval-valued pseudometric (briefly, WIVP metric), is defined by using known notions of pseudo-semimetric, pseudometric, and metric; this notion is topological good and shows precision, flexibility, and compatibility than single pseudo-semimetric, pseudometric, or metric. Propositions and detailed examples are given to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems, and WIVP metric and its special cases are characterized by using axioms. Moreover, some WIVP-metrics pertinent to quantitative logic theory or interval-valued fuzzy graphs are constructed, and fixed point theorems and common fixed point theorems in weak interval-valued metric spaces are also presented. Topics and strategies for further study are also put forward concretely and clearly.

In many cases, the measure values of true data are not unique (but two or more) for uncertainty or complexity. For example, there are several agents in China that value and order all periodicals published in China. Peking University Library and Nanjing University Library are generally thought to be the best two and incomparable to each other. For a journal J, assume the orders given by Peking University Library and the Nanjing University Library are m-th and n-th, respectively; then, m and n may be not the same in general. There are also many other examples. In 2012, breakthrough of the selected by the famous journal Science is different from those selected by the famous journal Nature; Gini coefficients in China in 2012 from two different agents are 0.481 and 0.61, respectively; the Chebyshev distance (resp., the Euclidean distance, the Manhattan distance or the city block distance, and the river distance) between two points (0,1) and (1,2) in the Euclidean plane R² is 1 (resp., $$, 2, 3). Please see Proposition 4 for definitions of these metrics; the effective distances used in cluster analysis are many and varied; a given asymptomatic infected people to corona virus disease (COVID-19 for short) is thought to be highly contagious (which can be represented by a fuzzy number A) by experts in one country but lowly contagious (which can also be represented by a fuzzy number B that is much different from A) by experts in another country.

In practice, most people choose just one of the measure values (or choose the arithmetic mean of these measure values) as the true data, such a kind of dispose can be accepted only in rare cases (e.g., the information loss cannot be avoided or make almost no difference). To make an improvement of disposal of these uncertain or complex data, at least two better theories (one is theoretically inspirational, and another is application-motivated; both are based mainly on the idea of fuzzy set) have been proposed which are mostly about measuring values of difference between two abstract “points” (precisely, two elements of a set) whose information or data can be provided by at least two different agents (but cannot be provided satisfactorily by one agent, see the following Example 1).

In this section, we will define the notion of weak interval-valued pseudometric (shortly, WIVP metric) and exemplify in detail how to construct distance-like measures (including WIVP metrics) desired in practice by fusing easy-to-obtain or easy-to-coin pseudo-semimetrics, pseudometrics, or metrics based on operators ∧, ∨, and simple aggregation operators. We also characterize WIVP metric and its special forms intuitively so that practitioners can understand them easily.

In this section, we will demonstrate how to construct by using some of these logic implication operators and some WIVP metrics which may be used in quantitative logic (cf. [23]) and quantitative reasoning (cf. [24]).

Metric graph theory abounds in applications (e.g., it is applicable in such different areas as location theory, theoretical biology and chemistry, combinatorial optimization, and computational geometry, see [42], [p.99–121] for details). In this section, we extend the notion of metric on a graph to interval-valued metric on an interval-valued fuzzy graph (particularly, on a fuzzy graph) and give some related examples.

An J-graph (where J is a completely distributive complete lattice with the least element 0) is a triple G = (V, σ, μ) consisting of a nonempty finite set V and a pair of mappings σ : V⟶J and μ : V × V⟶J which satisfies supp. σ = V and μ(x, y) = μ(y, x) ≤ σ(x)∧σ(y) (∀(x, y) ∈ V × V). The underlying graph of G is defined as [G] = (V, E), where E = {{x, y}⊆V|μ(x, y) > 0}. An J-graph G = (V, σ, μ) is said to be connected if its underlying graph [G] = (V, E) is connected, i.e., for any 2-element subset {x, y}⊆V, there exists an m(x, y)-element subset {z₁, z₂, …, z_{m(x, y)}}⊆V (m_xy ≥ 2) such that x = z₁, z_{m(x, y)} = y, and {z₁, z₂}, {z₂, z₃}, ⋯, {z_{m(x, y)−1}, z_{m(x, y)}} are all in E; the word P = z₁z₂ ⋯ z_{m(x, y)} is called a path from x to y, and the set of all paths from x to y is denoted by $$.

The main results of this section are as follows:

Theorem 18 may be proved based on Lemma 17 and results on contractive type mappings satisfying (1), (4), (5), (7), (11), (12), (14), (18), (43) in [44], and Theorem 19 may be proved based on Lemma 17 and results on contractive type mappings satisfying (176), (179), (180), (182), (186), (187), (193) in [44].

Since data from many real-world problems are not only from multiagents but also becoming more and more big and complex for vagueness and uncertainty, measurement by a single metric do not meet the needs of some practical problems. Motivated by Polya’s plausible reasoning and artificial neural networks, this paper consider a distance-like notion, called weak interval-valued pseudometric (WIVP metric for short), which, as a generalization of the notion of metric, is still topological good. To benefit practitioners, easy-to-understand propositions and much detailed examples are given (in the first half of the paper) to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP metric (even interval-valued metric) in practical problems. To show theoretical applications of WIVP metrics, we exemplify how to construct (by using some logic implication operators, some WIVP metrics which may be useful in quantitative logic [23] and quantitative reasoning [24]) and how to define well matched interval-valued metrics on interval-valued fuzzy graphs. As these WIVP metrics are relatively precision, flexibility and compatibility than single pseudo-semimetric, pseudometric, and metric, more applications should be investigated (even put forward) based on plausible reasoning. Practitioners are also suggested to explore (in the plausible reasoning manner) other complex and more fitted methods to fabricate more desired distance-like measures. For examples, to fuse easy-to-obtain pseudo-semimetrics, pseudometrics, or metrics by making full use of well-known or frequently used t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners or others; strategies also containing making full use of interval numbers and very special triangular fuzzy numbers.

Our future work includes completion of WIVP-metric spaces, interval-valued truth degrees of formulas based on deferent logic implication operators, interval-valued similarity degrees of formulas based on deferent logic implication operators, related approximate reasoning, dynamic systems on interval-valued metric spaces (even on interval-valued pseudometric spaces), and applications of weak interval-valued pseudometrics in medical diagnosis and decision-making problems (see related works [45, 46] for details).

The authors declare that there are no conflicts of interest regarding the publication of this paper.