An Easy-to-Understand Method to Construct Desired Distance-Like Measures

Fu, Wen Qing; Li, Sheng Gang; Garg, Harish; Liu, Heng; Khalil, Ahmed Mostafa; Zhao, Jingjing

doi:https://doi.org/10.1155/2021/5571546

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Abstract Introduction and Preliminaries Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Cognitive Computing Solutions for Complexity Problems in Computational Social Systems

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Research Article | Open Access

Volume 2021 | Article ID 5571546 | https://doi.org/10.1155/2021/5571546

An Easy-to-Understand Method to Construct Desired Distance-Like Measures

Wen Qing Fu,^1,2Sheng Gang Li ,¹Harish Garg,³Heng Liu,⁴Ahmed Mostafa Khalil,⁵and Jingjing Zhao⁶

Academic Editor: Wei Wang

Received05 Feb 2021

Accepted10 Jun 2021

Published08 Jul 2021

Abstract

Metrics and their weaker forms are used to measure the difference between two data (or other things). There are many metrics that are available but not desired by 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 be coined by practitioners) pseudo-semi-metrics, pseudo-metrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators (easy to be coined by practitioners). The simple reason to do this is that data for a real world problem are sometimes from multiagents. A distance-like notion, called weak interval-valued pseudo-metrics (briefly, WIVP-metrics), is defined by using known notions of pseudo-semi-metrics, pseudo-metrics, and metrics; this notion is topologically good and shows precision, flexibility, and compatibility than single pseudo-semi-metrics, pseudo-metrics, or metrics. 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.

1. Introduction and Preliminaries

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 who 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 that the orders given by Peking University Library and the Nanjing University Library are -th and -th, respectively; then, and may be not the same in general. There are also many other examples. In 2012, breakthrough selected by the famous journal Science are 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 and in the Euclidean plane is 1 (resp., , 2, 3). Please see Proposition 1 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) are thought to be highly contagious (which can be represented by a fuzzy number ) by experts in one country but lowly contagious (which can also be represented by a fuzzy number that is much different from ) 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 in the disposal of these uncertain or complex data, at least two better theories (One is theoretically inspirational, 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 Example 1).

Example 1. (1) The distance between two fuzzy sets and cannot be expressed accurately (or satisfactorily) in a real number but can be expressed satisfactorily in a 4-element set , or an interval number . can be determined by two elements (even if it is an infinite set). Therefore, all sets involved here can be thought to be finite. (2) The distance between two bodies (i.e., the closed ball in the 3-dimensional Euclidean space with center and radius ) and (i.e., the ball in with center and radius ) cannot be expressed accurately (or satisfactorily) in a real number but can be expressed satisfactorily in several numbers (where ). For instance, it can be expressed in a 3-element set (in this way information loss can be almost avoided) or be expressed by any 2-element subset of (in this way much more information loss can be avoided), where ( is the ordinary Euclidean metric on ), is the Euclidean distance between the centroid of and the centroid of , and is the Hausdorff distance between and (notice that ). (3) During the time of COVID-19 (especially, the first three months), a lot of things (particularly, traveling and meeting) had to stop in China. The expected College back-to-University time can be forecasted rather than accurately based on several (even the first two of them) time series (the time series of suspected cases, the time series of imported cases, the time series of close contact cases, the time series of no symptoms cases, the time series of cluster infection cases, the time series of confirmed cases, etc.) but cannot be forecasted satisfactorily by one of them.The first way to make an improvement is using a fuzzy metric. Several authors have introduced the concepts of fuzzy metric and fuzzy metric space from different points of view [1–4]. Erceg [1] defined a fuzzy pseudo-metric on a set consisting of fuzzy sets (e.g., fuzzifications of data or information to be disposed) by generalizing the Hausdorff distance between usual sets. The motivation behind the fuzzy metric of Kramosil and Michalek [3] was a statistical metric; recently, these kinds of fuzzy metric spaces have stimulated a lot of interest (see [5] and references herein for details). Kaleva and Seikkala’s metric [2] defined the distance between two points as a fuzzy number for the reason that sometimes uncertainty is due to fuzziness rather than randomness. As is known, Kaleva and Seikkala’s fuzzy metric spaces possess rich structures with suitable choices of binary operations. Much theoretical work related to Kaleva and Seikkala’s fuzzy metric spaces has been done in recent years (see [2, 5–9] and references herein). By presenting intuitive level forms for the triangle inequalities in the definition of Kaleva and Seikkala’s fuzzy metric, Huang and Wu [6] offered a new and convenient tool for the description and analysis of fuzzy metric spaces (cf. their subsequent work [7] on the existence and uniqueness of completion of special fuzzy metric spaces). Under some intuitive and mild assumptions, Fang [5] (as an improvement on Huang and Wu [7]) also proved the existence and the uniqueness of completion of fuzzy metric spaces. Xiao et al. [10] took a closer step toward possible applications of fuzzy metrics (in integral equations, differential equations, qualitative behavior, dynamic systems, and other nonlinear problems), in which the authors studied the existence and uniqueness of fixed points (under a weaker assumption) for nonlinear contractions in fuzzy metrics spaces in the sense of Kaleva and Seikkala [2]. By virtue of a level-cut method, they established relationships between a fuzzy metric and a family of quasi-metrics (so that the utilizing of results and skills in metric spaces becomes more and more possible).
The second way is using a dissimilarity measure or a distance measure, which is defined on a set consisting of special fuzzy sets (e.g., fuzzifications of data or information to be disposed); it takes many steps toward practical applications of metrics. Balopoulos et al. [11] defined a new family of distance measures (based on matrix norms) for binary operators on , which can also be used to measure the difference between two fuzzy sets on two finite sets. Bustince et al. [12] constructed distance measures, proximity measures, and fuzzy entropies by aggregating restricted dissimilarity functions in a special way. Liu [13] gave systematically an axiom definition of entropy, distance measure, and similarity measure of fuzzy sets; he also discussed basic relations between these measures. Fan et al. [14] defined a new divergence measure, study the relations between fuzzy entropy, fuzzy Hamming distance, and divergence measures defined. They obtain quite a general conclusion and solve a problem proposed by Liu [13]. Dissimilarity measure and distance measure have already found wide applications for their intuitiveness (see [15–21]).
As far as data are concerned, fuzzy metrics mentioned above have some shortcomings. It seems not very convenient to apply these fuzzy metrics in practical problems because axioms satisfied by these metrics are complex; implementation of computing related to these fuzzy metric spaces is obviously difficult because operations (even if addition or subtraction) or sorting of fuzzy numbers cannot be realized in computers unless the fuzzy numbers involved are extremely simple (for example, they are triangle fuzzy numbers or interval numbers). Dissimilarity measures and distance measures mentioned above also have some faults. For example, they have few nice topological properties (see [8, 11–13, 16–21]) because they do not satisfy the triangle inequality in general; axioms they satisfy are also complex. These naturally urge people to find a kind of metrics (or their weaker forms) which are down to earth, i.e., they are theoretically good, intuitive (because intuition and examples always provide inspirations for both theoretical study itself and its applications in the real world), and easy-to-use. For example, try to find a metric (or its weaker form) such that the complexity of the distance value of two points is between that of a real number and that of a fuzzy number.
An immediately thought way to construct such a metric (or its weaker form) is to take it as a weighted mean of some well-known or usually used metrics , i.e., let , where satisfies . However, information lost in this way may be too much compared to a way based on interval numbers (unless the set is chosen as the optimal one such that is the fittest value for each ).
Interval numbers or interval data appear in many cases (such as numerical analysis-handling rounding errors, computer-assisted proofs, global optimization, particularly, modeling uncertainty) because the data involved there cannot be accurately expressed in real numbers but can be expressed in interval numbers. The measure values of true data can be looked as an interval data, i.e., one can use an interval number to denote the measure values of true data. For example, the distance between and in Example 1 (2) can be expressed as one of the following six interval numbers: (or ), (or ), (or ), (or ), (or ), (or ).
Motivated by Polya’s plausible reasoning [22], the present paper will consider a fusion of easy-to-obtain measure data (e.g., values of various easy-to-obtain pseudo-semi-metrics, pseudo-metrics, or metrics) by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners; of course, interval numbers and triangular fuzzy numbers can also be used. In Section 2, a distance-like notion, called weak interval-valued pseudo-metrics (briefly, WIVP-metrics), is defined by using known notions of pseudo-semi-metrics, pseudo-metrics, and metrics (this notion and its special cases are also characterized by using axioms and connections between these notions and other known and related notions); 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. Sections 3 and 4 are on theoretical applications of WIVP-metrics, where we demonstrate how to construct, by using some logic implication operators, some WIVP-metrics which may be useful in quantitative logic (cf. [23]) and quantitative reasoning (cf. [24]), and how to define well matched interval-valued metrics on interval-valued fuzzy graphs. Fixed point theorems and common fixed point theorems in WIVP-metrics are presented in Section 5. We end the paper with concluding remarks in Section 6.
Now, we present some preliminary notions needed in this paper.
Let be the set of all fuzzy sets on (i.e., the set of all mappings from to ), where is the set of all real numbers (also called the real line) and is the closed unit interval of . For each subset and each , we use to denote the fuzzy set on taking value on and 0 elsewhere; we will make no distinction between and . If a fuzzy set satisfies for some and (i.e., a closed interval of ) for any , then we call a fuzzy number (we will regard and as functions on , cf. [14, 24, 25]). The set of all fuzzy numbers is denoted by . A fuzzy number is called nonnegative if supp (where supp is called the support of ). The set of all nonnegative fuzzy numbers is denoted by . Fuzzy numbers are the most commonly used fuzzy sets on the real line (see [25–30] and references herein); one of their applications is that they can be used to define fuzzy metric space (an important notion in fuzzy analysis). A binary operation on , which is commutative (or symmetric), associative, and nondecreasing coordinately, is said to be a t-norm on if it satisfies ; is said to be a t-conorm on if it satisfies .

Definition 1. (see [2]). A fuzzy metric space is a quadruple (in this case, is called a fuzzy metric) which satisfies the following two conditions : (1) is positive, i.e., and symmetric, (i.e., ) (2) (i) if , and , then (ii) if , and , then Here, is a mapping, and are operations on which are symmetric, nondecreasing coordinately and satisfy and , and .

Definition 2. (see [31]). An interval number (i.e., a special kind of fuzzy number) is a point in the 2-dimensional Euclidean space which satisfies . The set of all interval numbers (with the point-wise order ) is denoted by (Notice that a total order [32, 33] was also defined on : or but . Moreover, we will identify a closed interval of , a point in , and a fuzzy number since there exists a natural one-to-one correspondence between the set of all closed intervals of and .). For any and each nonnegative real number , define , , and . When and , write (or ); when , write . For every subinterval , write .

2. Definition and Examples of WIVP-Metric

In this section, we will define the notion of weak interval-valued pseudo-metrics (shortly, WIVP-metrics), 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-semi-metrics, pseudo-metrics, 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.

Definition 3. Let be a set, a mapping (where ). If is a pseudo-semi-metric ( is called a pseudo-semi-metric on if it satisfies and ; is called a pseudo-metric on if it is a pseudo-semi-metric on satisfying triangle inequality; is called a semi-metric on if it is a pseudo-semi-metric on satisfying , where is the diagonal of .) (resp., a pseudo-metric, a semi-metric, a metric) on and is a pseudo-metric (resp., a pseudo-metric, a metric, a metric) on , then is called a WIVP-metric on (resp., an interval-valued pseudo-metric on , a weak interval-valued metric on , an interval-valued metric on ) and is called a WIVP-metric space (resp., an interval-valued pseudo-metric space, a weak interval-valued metric space, an interval-valued metric space), where and are the first coordinate projection and the second coordinate projection, respectively (For a WIVP-metric on , write , , , , where . Then, , is a topology on , and both and are pre-topologies on . It can be proved that , , and have good topological properties.).

Proposition 1. Let be the 3-dimension Euclidean space and , , , and be the Chebyshev metric, the Euclidean metric, the Manhattan metric or the city block metric, and the river metric on , respectively (We will consider them to be metrics on by identifying a point with a point .), defined by (where and ),

Again, let . Then, the following premises hold: (1) is also a metric on , , and if or but (particularly, ). (2) The mapping , defined by , are interval-valued metrics on for , interval-valued metrics on for , and interval-valued metrics on for . (3) The mapping , defined by is an interval-valued metric (actually, a metric) (Each of these interval-valued metrics can be used to measure the difference between any two bodies and with the centroid set .) on . (4) Let satisfying . Then, the mappings , defined by , , and , are interval-valued metric on (the set of all continuous functions on , ) which can be used to measure the difference between two functions , where

Proposition 2. Let and a weight vector (i.e., it satisfies and ). Then, (1) Consider three mappings from to , defined by , , , where , , and . The last two are WIVP-metrics on (but the first is not in general); all of them can be used to measure the difference between any two fuzzy sets . (2) For each , consider three mappings from to , defined by , , , where , , , and the infimum and supremum are taken in . The last two are WIVP-metrics on (but the first is not in general); all of them can be used to measure the difference between any two interval-valued fuzzy sets, intuitionistic fuzzy sets, or bipolar fuzzy sets [34] . (3) Analogously, for each , consider three mappings from to , defined by , , , where , ,, and the infimum and supremum are taken in (Of course, the infimal and supremal in (2) and (3) can also be taken in ). The last two are WIVP-metrics on (but the first is not in general); all of them can be used to measure the difference between any two -rung orthopair fuzzy sets [35–40] or 3-polar fuzzy sets [34] .

Proof. We only show that in (1) does not satisfy the triangle inequality in general. Actually, take , , , , and . Then, .

Example 2. (1) Let (The data needed here can also be taken from Example 5.) and be feature fuzzy sets representing two physical entities (e.g., patients), and the third. If we take a weight vector , as , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). If we take two weight vectors and , then , , , and . Thus, and . As , belongs to class (with respect to ); as , belongs to class (with respect to ), where is the point-wise order on induced by on . (2) Let and be feature interval-valued fuzzy sets representing two physical entities (e.g., patients), the third, and . As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). It follows that and for each , thus belongs to class (with respect to and ). Take . Then, , thus belongs to class (with respect to ). (3)Let and be feature picture fuzzy sets representing two physical entities (e.g., patients), the third, and . As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). As , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ). It follows , , and for each , thus belongs to class (with respect to , , and ). (4)Let and be feature generalized pythagorean fuzzy sets (A generalized pythagorean fuzzy set is a special 3-polar fuzzy set satisfying .) (see [38, 39] for a special case) representing two physical entities (e.g., patients), the third, and . Similar to (3), as , belongs to class (with respect to ). Analogously, we can decide belongs to class or with respect to (resp., , , , , , , , , , , , , ).

Example 3. (1) Analogously, we can predict from a given time series in the following way: Firstly, fix (maybe by experts) a natural number (say, ) and a set satisfying and . Secondly, compute the differences : Finally, take if . We can also compute each as Example 2. (2)(Continued to Proposition 1 (4)) Let and be feature functions representing two physical entities, and the third. Then, , , , and .If we take a weight vector , as , belongs to class (with respect to ); as , belongs to class (with respect to ); as , belongs to class (with respect to ).
If we take two weight vectors and , then , , , and . Therefore,, , , . As , belongs to class (with respect to ); as , belongs to class (with respect to ).

Proposition 3. (1) Let . If (resp., ) is a pseudo-semi-metric (resp., a pseudo-metric) on satisfying , then , defined by , is a WIVP-metric on . (2) If and are pseudo-metrics on , then , defined by , is a pseudo-semi-metric on , , defined by , is a pseudo-metric on , and thus , defined by is a WIVP-metric on . (3) Let is connected and bounded with respect to the ordinary Euclidean metric on , and Lebesgue measurable. Then, , defined by ( is the ordinary Euclidean metric on ), is a pseudo-semi-metric; , defined by , is a pseudo-metric; and , defined by is a metric (called the Hausdorff metric induced by and written also as ). Therefore, we obtain three WIVP-metrics on each of which can be used to measure the difference between any two heavenly bodies and in with the centroids , where and . As , . (4)Let be a pseudo-metric on , (where is a poset), and . Then, the mapping , defined by , is a WIVP-metric on .

Proof. (1) follows from Definition 3, (2) can be proved by definition, and (3) follows from (2). (4) Defined by putting and . Then, and and are pseudo-metrics on , thus is a WIVP-metric on by (2).

Example 4. (1) Consider the two bodies and in Example 1 (2). Then, the differences between and can be taken as (the first three are real numbers and the last three are interval numbers): , , , , , or . (2) Let be the set of all nonempty compact sets in the ordinary Euclidean plane , be the Hausdorff metric on induced by , , , and be two clouds (in which stands for the degree of the thickness of cloud in vertical line through the point in the plane ), defined by (i.e., ) and , where and . Then, , , , , , and . Therefore, . (3)Analogously, let be the set of all nonempty compact sets in the ordinary Euclidean space , be the Hausdorff metric on induced by , , , and be two clouds (in which stands for the degree of concentration of cloud at the point ), defined by and , where and . Then , , , , , and . Therefore, .

Example 5. Now, we exemplify an application of the idea of WIVP-metrics in pattern recognition (i.e., COVID-19 diagnosis). Consider the following symptom data (see Table 1), involving 5 symptoms (asthma, sore throat, cough, fever, age), and diagnosis results, involving yes (briefly, Y) and no (briefly, N), of 7 patients given by expert physicians in a Chinese hospital in Nanjing:
From this, we can get the following 7 mappings (patients):Take other 2 patients in the same hospital for test:Notice that both and have two asthma values (we will take their center in the following computation for simplicity) because he/she ate a kind of TCL-like food before taking the two values. As the group of experts give two different weight vectors and for importance of vector of symptom (asthma, sore throat, cough, fever, age), we finish this decision-making problem in the following nine steps (see Figure 1). Step 1. Compute the difference between and those with diagnosis results Y using a metric rely on (there are many cases on choices of ). , , , , . Thus, we obtain the average . Step 2. Compute the difference between and those with diagnosis results N using a metric rely on . , . Thus, we obtain the average . Step 3. Compute the difference between and those with diagnosis results Y using a metric rely on . , , , , . Thus, we obtain an average . Step 4. Compute the difference between and those with diagnosis results N using a metric rely on . , . Thus, we obtain an average . Step 5. Compute the difference between and those with diagnosis results Y using a metric rely on . , , , , . Thus, we obtain an average . Step 6. Compute the difference between and those with diagnosis results N using a metric rely on . , . Thus, we obtain an average . Step 7. Compute the difference between and those with diagnosis results Y using a metric rely on . , , , , . Thus, we obtain an average . Step 8. Compute the difference between and those with diagnosis results N using a metric rely on . , . Thus, we obtain an average . Step 9. As , is N (which is the same as the diagnosis result given by expert physicians in that hospital); as , is Y (which is also the same as the diagnosis result given by expert physicians in that hospital).

Proposition 4. (1) For a mapping , consider the following conditions: (M0) (M1) if (M2) (M3) (M4) Then, is a WIVP-metric (resp., an interval-valued pseudo-metric, a weak interval-valued metric, an interval-valued metric) on if it satisfies (M0) + (M2) + (M4) (resp., (M0) + (M2) + (M3), (M1) + (M2) + (M4), (M1) + (M2) + (M3)). (2) Every interval-valued metric space can be looked as a special fuzzy metric space, where we define and as and , and set .

3. WIVP-Metrics Pertinent to Logic Theory

An logic implication operator is a mapping which satisfies some conditions (see [41]). There are many well-known logic implication operators (see [23, 24, 41]); some of them are defined as follows: (i) Boolean implication operator: (if ) or 1 (otherwise) (ii) Gaines-Rescher implication operator: (if ) or 0 (if ) (iii) Goguen implication operator: (if ) or (if ) (iv) Godel implication operator: (if ) or (if ) (v) Kleene-Dienes implication operator: (vi) Lukasiewicz implication operator: (vii) Mamdani implication operator: (vii) Product implication operator: (if ) or (if ) (ix) Wang implication operator: (if ) or (if ) (x) Reichenbach implication operator: (xi) Yager implication operator: (xii) Zadeh implication operator:

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

Definition 4. (cf. [42]). (1) Let be a countable set (where if ), a unary operator, and binary operators, and the free algebra of type generated by . Elements of are called atomic formulae (or atomic propositions), and elements of are called formulae (or propositions). (2) can be looked as an algebra of type if we define the unary operator and the two binary operators and as follows: , , , where is a logic implication operator. Furthermore, for each , there exists a unique finite subset (without loss of generality we assume ) such that can be represented uniquely by and (written as ). Substitute for in and keep the logic connectives in unchanged, then we obtain an expression and thus an -variable function which is called the -function corresponding to . For example, if , then . (3) A mapping which preserves operations and is called an -valuation on . The set of all -valuations is written as . (4)Let . If , then and are said to be -logically equivalent.

Lemma 1. If the formulae and in are -logically equivalent, then (where is a logic implication operator).

Proof. Without loss of generality, we assume , , and . Let (where ), then . For each , there exists an -valuation on which satisfies since is a free algebra. As and are -logically equivalent, , which means .

Lemma 2. (1) For any , define if and only if and is -logically equivalent. Then, is an equivalence relation on . The equivalence class determined by is written , and the quotient set determined by is written . (2) The mapping , defined by , is a pseudo-metric on , where is the integral domain related . (3) For each , the mapping , defined by (if ) or (if ), is a pseudo-metric on . (4). Notice also that (5) is a family of metrics on which satisfy , where is defined by . (6)(cf. [42]). The mapping , defined by , is a metric on .

Remark 1. Lemma 2 (2) is Theorem 6 (without proof) in [41].

Theorem 1. (1) , is a WIVP-metric space (2) , is a weak interval-valued metric space (3) , is an interval-valued metric space

Proof. It follows from Lemma 2 (including its proof) and Remarks 1.

4. Interval-Valued Metrics on Interval-Valued Fuzzy Graphs

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 [[43], p.99–121] for details). In this section, we extend the notion of metrics on a graph to interval-valued metrics on an interval-valued fuzzy graph (particularly, on a fuzzy graph) and give some related examples.

An -graph (where is a completely distributive complete lattice with the least element 0) is a triple consisting of a nonempty finite set and a pair of mappings and which satisfies supp and . The underlying graph of is defined as , where . An -graph is said to be connected if its underlying graph is connected, i.e., for any 2-element subset , there exists an -element subset such that , , and, , are all in ; the word is called a path from to , and the set of all paths from to is denoted by .

Definition 5. Let be a -graph (called also fuzzy graph [42]), and . For each , let . For any , define and as follows: (1)If there is no path from to , define (the cardinality of ). (2)If there is a path from to , defineThen, the mappings are metrics on (called metrics on the fuzzy graph ).

Remark 2. It is easy to verify for all satisfying , thus we obtain a family of interval-valued metrics on (called interval-valued metrics on the fuzzy graph ).

Theorem 2. Let be an -graph (called also interval-valued fuzzy graph). For any , letwhereand . Then, is an interval-valued metric on (called interval-valued metric on the interval-valued fuzzy graph ).

5. Fixed Point Theorems and Common Fixed Point Theorems in WIVP-Metrics

Definition 6. (1) A sequence in a WIVP-metric space is said to be a Cauchy sequence if, for each , there exists a positive integer such that ; is said to be convergent in if there exists an such that converges to (i.e., for each , there exists a positive integer such that ). (2) A WIVP-metric space is said to be complete if every Cauchy sequence in is convergent.It is not difficult to verify the following.

Lemma 3. A WIVP-metric space is complete if and only if is complete.
The main results of this section are the following.

Theorem 3. Let be a complete WIVP-metric space. Then, has a unique fixed point in if it satisfies one of the following conditions: (1) , where is a constant. (2) , where is a constant. (3) , where is a constant. (4) , where are nonnegative numbers satisfying . (5) , where is a constant. (6) , where is a constant. (7) , where are nonnegative numbers satisfying . (8) , , where , , and is a fixed positive integer. (9) , where , , and is a fixed positive integer.

Theorem 4. Let be a complete weak interval-valued metric space. Then, have common fixed point in if they satisfy one of the following conditions (where and are fixed positive integers): (1) , where is a constant. (2) , where is a constant. (3) , where is a constant. (4) , where are nonnegative numbers satisfying . (5) , where is a constant. (6) , where is a constant. (7) , where , .

Theorem 3 may be proved based on Lemma 3 and results on contractive-type mappings satisfying , , in [44], and Theorem 4 may be proved based on Lemma 3 and results on contractive-type mappings satisfying (176), (179), (180), (182), (186), (187), (193) in [44].

6. Concluding Remarks

Since data from many real-world problems are not only from multi agents but also becoming more and more big and complex for vagueness and uncertainty, measurement by a single metric does 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 pseudo-metric (WIVP-metric for short), which, as a generalization of the notion of metric, is still topologically 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 precise, flexible, and compatible than single pseudo-semi-metric, pseudo-metric, 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 example, to fuse easy-to-obtain pseudo-semi-metrics, pseudo-metrics, 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 contain making full use of interval numbers and very special triangular fuzzy numbers. Of course, these (including also determination of weighted vector) are not always easy to practitioners. How to overcome this limitation will be one of our future works.

Our future work also 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 pseudo-metric spaces), and applications of weak interval-valued pseudo-metrics in medical diagnosis and decision-making problems (see related works [45–51] for details); for decision-making problems with data given by a fuzzy set (resp., interval-valued fuzzy sets, picture fuzzy sets, generalized pythagorean fuzzy sets), one can use the WIVP-metrics used in Example 2 to replace distance measures used in published papers and use the linear orders in [32, 33, 52] to replace the ordinary order in the real line.

Appendix

Proof. of Proposition 1. We only prove conditionally. If , then . Next, we assume satisfies and . There are only four cases which need to be considered. Case 1 . Then, , and thus . Case 2 and . As , , and thus . Since , the case that and cannot appear. Case 3 and . Write , , , . Then, , . Therefore, and , , where and are the heights of triangles and (with as the common base), respectively. As , . Case 4 and . Without loss of generality, we assume . If , then . Thus, . If , then from the proof of Case 3 we can see .

Proof. of Proposition 4. We only prove (1). Firstly, we prove satisfies condition (2) (i) of Definition 1. By (M3) and Definition 1, . Write , , and . Then, . It does not loose generality to assume (i.e., ), then and . Since is nondecreasing with and , (i.e., ). Hence, .
Now we prove satisfies condition (2) (ii) of Definition 1. By (M3) and Definition 3, . Write . If , , and . Then, . Hence, or . This implies that or and thus .

Proof. of Lemma 2. We only prove (2) (for the case of ) and (3). Write . As or , .
We first show that if and only if and are -logically equivalent (particularly, for all ). Actually, if (i.e., ), then since is continuous. Thus, and . It follows that and , i.e., . Since is arbitrary, and are -logically equivalent. Conversely, if and are -logically equivalent, then (and thus ) by Lemma 1. follows from definition of . It remains to prove the triangle inequality. Step 1., where . It is not loss of generality to assume . We consider the following three cases. (i). Then, , , and thus . (ii). Then, , , and thus . (iii). Then, , , and thus . Step 2. By the inequality in Step 1, we have (3) By definition of and Lemma 1, we know that if and only if (a.e), and that if and only if and are -logically equivalent . Particularly, holds. Since is obvious, we only need to prove the triangle inequality. It is not loss of generality to consider only the case . Step 1. , where . It is not loss of generality to assume . Then, , thus we only need to prove . We consider the following five cases. (i) . As , (ii) . As , (iii) . As , (iv) . As , (v) . As , Step 2. For the case , by Lemma 1 and the inequality in Step 1, we have For the case , take . We consider the following two cases (it is not loss of generality to assume and are not -logically equivalent). (i) Neither and are -logically equivalent nor and are -logically equivalent. By the inequality in Step 1, we have (ii) Either and are -logically equivalent or and are -logically equivalent. It is not loss of generality to assume that and are -logically equivalent). By Lemma 1, we have.

Proof. of Theorem 2We only show that satisfies (M3) in Remark 1. Suppose is a 3-element subset of . If , then or , and thus (which means that satisfies (M3)). If , then we consider the following two cases: Case 1 or . Then , which means that satisfies (M3). Case 2 and . For each path from to and each path from to , there exists a path from to such that . Therefore, , which means that satisfies (M3).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

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

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant nos. 11771263, 61967001, and 61807023) and the Fundamental Research Funds for the Central Universities (Grant nos. GK202105007 and GK201702011).

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Copyright © 2021 Wen Qing Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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