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

Authors: Wen Qing Fu; Sheng Gang Li; Harish Garg; Heng Liu; Ahmed Mostafa Khalil; Jingjing Zhao
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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 arem-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 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 0,1 and 1,2 in the Euclidean plane R2 is 1 (resp., 2, 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 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 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 Example1).Example 1.
(1) The distance between two fuzzy setsA=0.1/x,0.3/y and B=0.6/u,0.8/v cannot be expressed accurately (or satisfactorily) in a real number but can be expressed satisfactorily in a 4-element set H=0.1−0.6,0.1−0.8,0.3−0.6,0.3−0.8=0.3,0.5,0.7, or an interval number d=0.3,0.7. d 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 bodiesA=x,y,z∈R3|x2+y2+z2≤ϵ2 (i.e., the closed ball in the 3-dimensional Euclidean space R3 with center 0,0,0=0⟶ and radius ϵ2) and B=x,y,z∈R3|x−ϵ2−ε2+y2+z2≤ε2 (i.e., the ball in R3 with center ϵ+ε,0,0 and radius ϵ) cannot be expressed accurately (or satisfactorily) in a real number but can be expressed satisfactorily in several numbers (where 0≤ϵ≤ε). For instance, it can be expressed in a 3-element set a,a˙,a¨ (in this way information loss can be almost avoided) or be expressed by any 2-element subset of a,a˙,a¨ (in this way much more information loss can be avoided), where a=0=infρ2x⟶,y⟶|x⟶∈A,y⟶∈B (ρ2 is the ordinary Euclidean metric on R3), a˙=ϵ+ε is the Euclidean distance between the centroid 0,0,0=0⟶ of A and the centroid ϵ+ε,0,0 of B, and a¨=2ε is the Hausdorff distance between A and B (notice that a≤a˙≤a¨).(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 0,1, 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)ρ is to take it as a weighted mean of some well-known or usually used metrics ρ1,…,ρn, i.e., let ρ=k1ρ1+⋯+knρn, where k1,…,kn∈0,1 satisfies k1+⋯+kn=1. However, information lost in this way may be too much compared to a way based on interval numbers (unless the set k1,…,kn is chosen as the optimal one such that ρx,y is the fittest value for each x,y∈X2).
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 betweenA and B in Example 1 (2) can be expressed as one of the following six interval numbers: a,a (or a,a), a˙,a˙ (or a˙,a˙), a¨,a¨ (or a¨,a¨), a,a˙ (or a,a˙), a,a¨ (or a,a¨), a˙,a¨ (or a˙,a¨).
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.
Let0,1R be the set of all fuzzy sets on R (i.e., the set of all mappings from R to 0,1), where R is the set of all real numbers (also called the real line) and 0,1 is the closed unit interval of R. For each subset Y⊆R and each b∈0,1, we use bY to denote the fuzzy set on R taking value b on Y and 0 elsewhere; we will make no distinction between 1Y and Y. If a fuzzy set μ∈0,1R satisfies μr=1 for some r∈R and μa−,μa+ (i.e., a closed interval of R) for any a∈0,1, then we call μ a fuzzy number (we will regard μ− and μ+ as functions on 0,1, cf. [14, 24, 25]). The set of all fuzzy numbers is denoted by ℱ. A fuzzy number μ is called nonnegative if supp μ⊆0,+∞ (where supp μ=x∈R|μx>0 is called the support of μ). The set of all nonnegative fuzzy numbers is denoted by ℱ+. 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 0,1, which is commutative (or symmetric), associative, and nondecreasing coordinately, is said to be a t-norm on 0,1 if it satisfies a∗1=a∀a∈0,1; ∗ is said to be a t-conorm on 0,1 if it satisfies a∗0=a∀a∈0,1.Definition 1.
(see [2]). A fuzzy metric space is a quadruple X,d,L,R (in this case, d:X×X⟶ℱ+ is called a fuzzy metric) which satisfies the following two conditions ∀x,y,z∈X3:(1)d is positive, i.e., dx,y=0R⇔x=y and symmetric, (i.e., dx,y=dy,x)(2)(i) ifs≤λ1x,z,t≤λ1z,y, and s+t≤λ1x,y, then dx,ys+t≥Ldx,zs,dz,yt(ii) ifs≥λ1x,z,t≥λ1z,y, and s+t≥λ1x,y, then dx,ys+t≤Rdx,zs,dz,yt
Here,d:X×X⟶ℱ+ is a mapping, L and R are operations on 0,1 which are symmetric, nondecreasing coordinately and satisfy L0,0=0 and R1,1=1, and dx,ya=λax,y,νax,y∀x,y∈X,∀a∈0,1.Definition 2.
(see [31]). An interval number (i.e., a special kind of fuzzy number) is a point a=a−,a+ in the 2-dimensional Euclidean space R2 which satisfies a−≤a+. The set of all interval numbers (with the point-wise order ≤) is denoted by IR (Notice that a total order [32, 33] ≤ was also defined on IR: a−,a+≤b−,b+⇔0.5a−+a+<0.5b−+b+ or 0.5a−+a+=0.5b−+b+ but a+−a−≥b+−b−. Moreover, we will identify a closed interval a,b of R, a point a,b in R2, and a fuzzy number 1a,b since there exists a natural one-to-one correspondence between the set of all closed intervals of R and IR.). For any a−,a+,b−,b+∈IR and each nonnegative real number r, define a−,a+⊕b−,b+=a−+b−,a++b+, a−,a+⊖b−,b+=mina−−b−,a+−b+,maxa−−b−,a+−b+, and ra−,a+=ra−,ra+. When a−>b− and a+>b+, write a−,a+≫b−,b+ (or b−,b+≪a−,a+); when a−=a+=a, write a−,a+=a. For every subinterval J⊆R, write IJ=a−,a+∈IR|a−,a+∈J.


## 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.
LetX be a set, ρ:X×X⟶IR+ a mapping (where R+=0,+∞). If ρ−=p1∘ρ:X×X⟶0,+∞ is a pseudo-semi-metric (ρ− is called a pseudo-semi-metric on X if it satisfies ρ−x,x=0∀x∈X and ρ−x,y=ρ−y,x∀x,y∈X×X; ρ− is called a pseudo-metric on X if it is a pseudo-semi-metric on X satisfying triangle inequality; ρ− is called a semi-metric on X if it is a pseudo-semi-metric on X satisfying x,y∈X×X|ρ−x,y=0=ΔX, where ΔX=x,x|x∈X is the diagonal of X2.) (resp., a pseudo-metric, a semi-metric, a metric) on X and ρ+=p2∘ρ:X×X⟶0,+∞ is a pseudo-metric (resp., a pseudo-metric, a metric, a metric) on X, then ρ is called a WIVP-metric on X (resp., an interval-valued pseudo-metric on X, a weak interval-valued metric on X, an interval-valued metric on X) and X,ρ 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 p1:R2⟶R and p2:R2⟶R are the first coordinate projection and the second coordinate projection, respectively (For a WIVP-metric on X, write ℬ^1=B^x,a|x∈X,a∈0,+∞, ℬ^2=B^x,a−,a+|x∈X,a−,a+∈IR,a−,a+≫0,ℬ^ρx=B^x,a−,a+|a−,a+∈IR,a−,a+≫0x∈X, T^1,ρ=∪A|A⊆ℬ^1,T^2,ρ=∪A|A⊆ℬ^2, T^ρ=W∈2X|B^x,a−,a+⊆Wfor somea−,a+≫0wheneverx∈W , where B^x,a−,a+=y∈X|ρx,y≥a−,a+. Then, T^ρ=W∈2X|B^x,a⊆Wfor somea≫0wheneverx∈W⊆T^1,ρ⊆T^2,ρ, T^ρ is a topology on X, and both T^1,ρ and T^2,ρ are pre-topologies on X. It can be proved that T^ρ, T^1,ρ, and T^2,ρ have good topological properties.).Proposition 1.
LetR3 be the 3-dimension Euclidean space and ρ1, ρ2, ρ3, and ρ4 be the Chebyshev metric, the Euclidean metric, the Manhattan metric or the city block metric, and the river metric on R3 , respectively (We will consider them to be metrics on R2 by identifying a point a1,a2∈R2 with a point a1,a2,0∈R3.), defined by (where a⟶=a1,a2,a3∈R3 and b⟶=b1,b2,b3∈R3),(1)ρ1a⟶,b⟶=maxa1−b1,a2−b2,a3−b3,ρ2a⟶,b⟶=a1−b12+a2−b22+a3−b32,ρ3a⟶,b⟶=a1−b1+a2−b2+a3−b3,ρ4a⟶,b⟶=ρ2a⟶,b⟶,a1=b1,a22+a32+b22+b32+a1−b1,a1≠b1.Again, letρ0=1/3ρ3. Then, the following premises hold:(1)ρ0 is also a metric on R3, ρ0≤ρ1≤ρ2≤ρ3, and ρ3a⟶,b⟶≤ρ4a⟶,b⟶ if a⟶=b⟶ or a1≠b1 but a2−b2a3−b3−a2b2−a3b3≤a22+a32b22+b32 (particularly, a2=b2).(2) The mappingρi,j:R6⟶IR+, defined by ρi,ja⟶,b⟶=ρia⟶,b⟶,ρja⟶,b⟶, are interval-valued metrics on R3 for 0≤i≤j≤3, interval-valued metrics on 0,+∞3 for 0≤i≤j≤4, and interval-valued metrics on −∞,03 for 0≤i≤j≤4.(3) The mappingρe:R6⟶IR+, defined by ρea⟶,b⟶=e0ρ0a⟶,b⟶+e1ρ1a⟶,b⟶+e2ρ2a⟶,b⟶+e3ρ3a⟶,b⟶+e4ρ4a⟶,b⟶ei≥0,i=0,1,2,3,4 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 A and B with the centroid set a⟶,b⟶⊆R3.) on R3.(4) Letw1,w2∈0,12 satisfying w1+w2=1. Then, the mappings ρ,ρ.w,ρw.,:Ca,b×Ca,b⟶IR+, defined by ρh,g=hg¯,hg¯, ρ.wh,g=hg¯,w1hg¯+w2hg¯, and ρw.h,g=w1hg¯+w2hg¯,hg¯∀h,g⊆Ca,b, are interval-valued metric on Ca,b (the set of all continuous functions on a,b, a<b) which can be used to measure the difference between two functions h,g∈Ca,b, where(2)hg¯=1b−a∫abft−gtdt,(3)hg¯=maxt∈a,bft−gt.Proposition 2.
LetY≠∅ and w⟶=w1,w2 a weight vector (i.e., it satisfies w1,w2∈0,12 and w1+w2=1). Then,(1) Consider three mappingsd¯1,w,d¯w,2,d¯1,2 from Y12=0,1Y×0,1Y to IR+, defined by Φ,Ψ∈Y12d¯1,wΦ,Ψ=d1Φ,Ψ,dwΦ,Ψ,d¯w,2Φ,Ψ=dwΦ,Ψ,d2Φ,Ψ,d¯1,2Φ,Ψ=d1Φ,Ψ,d2Φ,Ψ, whered1Φ,Ψ=infΦy−Ψy|y∈Y,d2Φ,Ψ=supΦy−Ψy|y∈Y,anddwΦ,Ψ=w1d1Φ,Ψ+w2d2Φ,Ψ. The last two are WIVP-metrics on Y1 (but the first is not in general); all of them can be used to measure the difference between any two fuzzy sets Φ,Ψ∈Y1.(2) For eachd∈ρ0,ρ1,ρ2,ρ3,ρ4,ρe, consider three mappings d¯1,w,d¯w,2,d¯1,2 from Y22=0,12Y×0,12Y to IR+, defined by Φ,Ψ∈Y22d¯1,wΦ,Ψ=d1Φ,Ψ,dwΦ,Ψ,d¯w,2Φ,Ψ=dwΦ,Ψ,d2Φ,Ψ,d¯1,2Φ,Ψ=d1Φ,Ψ,d2Φ,Ψ, whered1Φ,Ψ=∧dΦy,Ψy|y∈Y,d2Φ,Ψ=∨dΦy,Ψy|y∈Y,dwΦ,Ψ=w1d1Φ,Ψ+w2d2Φ,Ψ, and the infimum and supremum are taken in IR,⩽. The last two are WIVP-metrics on Y2 (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] Φ,Ψ∈Y2.(3) Analogously, for eachd∈ρ0,ρ1,ρ2,ρ3,ρ4,ρe, consider three mappings d¯1,w,d¯w,2,d¯1,2 from Y3=0,13Y×0,13Y to IR+, defined by Φ,Ψ∈Y32d¯1,wΦ,Ψ=d1Φ,Ψ,dwΦ,Ψ,d¯w,2Φ,Ψ=dwΦ,Ψ,d2Φ,Ψ,d¯1,2Φ,Ψ=d1Φ,Ψ,d2Φ,Ψ, whered1Φ,Ψ=∧dΦy,Ψy|y∈Y,d2Φ,Ψ=∨dΦy,Ψy|y∈Y,dwΦ,Ψ=w1d1Φ,Ψ+w2d2Φ,Ψ, and the infimum and supremum are taken in IR,⩽ (Of course, the infimal and supremal in (2) and (3) can also be taken in IR,≤). The last two are WIVP-metrics on Y3 (but the first is not in general); all of them can be used to measure the difference between any two q-rung orthopair fuzzy sets q≥1 [35–40] or 3-polar fuzzy sets [34] Φ,Ψ∈Y3.Proof.
We only show thatdw in (1) does not satisfy the triangle inequality in general. Actually, take U=x,y, A=1x,z, B=1x,y, C=1y, and w⟶=0.6,0.4. Then, dwA,C=0.6×1+0.4×1=1≤0.8=0.6×0+0.4×1+0.6×0+0.4×1=dwA,B+dwB,C.Example 2.
(1) Let (The data needed here can also be taken from Example5.) A=0.1/x,0.2/y,0.1/z and B=1/x,0.9/y,1/z be feature fuzzy sets representing two physical entities (e.g., patients), and C=0.6/x,0.5/y,0.6/z the third.If we take a weight vectorw⟶=0.5,0.5, as d¯1,2C,A=0.3,0.5≤0.4=d¯1,2C,B, C belongs to class A (with respect to d¯1,2); as d¯1,wC,A=0.3,0.4≤0.4=d¯1,2C,B, C belongs to class A (with respect to d¯1,2); as d¯w,2C,B=0.4≤0.4,0.5=d¯w,2C,A, C belongs to class B (with respect to d¯w,2).If we take two weight vectorsw⟶=0.5,0.5 and v=0.4,0.6, then dwC,A=0.5×0.3+0.5×0.5=0.4, dvC,A=0.4×0.3+0.6×0.5=0.42, dwC,B=0.5×0.4+0.5×0.4=0.4, and dvC,B=0.4×0.4+0.6×0.4=0.4. Thus, dwvC,A=0.4,0.42 and dwvC,B=0.4. As d¯1,wvC,A=0.3,0.4,0.42≤0.4=d¯1,wvC,B, C belongs to class A (with respect to d¯1,wv); as d¯w,2C,B=0.4≤0.4,0.42,0.5=d¯w,2C,A, C belongs to class B (with respect to d¯w,2), where ≤ is the point-wise order on IR+×IR+ induced by ≤ on IR+.(2) LetA=0.1,0.2/x,0.3,0.4/y,0.1,0.2/z and B=0.9,1/x,0.8,0.9/y,0.9,1/z be feature interval-valued fuzzy sets representing two physical entities (e.g., patients), C=0.6/x,0.4,0.6/y,0.6/z the third, and w⟶=0.5,0.5.Asρ0¯1,2C,A=0.1,0.3≤7/30=ρ0¯1,2C,B, C belongs to class A (with respect to ρ0¯1,2); as ρ0¯1,2C,A=0.1,0.2≤7/30=ρ0¯1,wC,B, C belongs to class A (with respect to ρ0¯1,w); as ρ0¯w,2C,B=7/30≤0.2,0.3=ρ0¯w,2C,A, C belongs to class B (with respect to ρ0¯w,2).Asρ1¯1,2C,A=0.2,0.5≤0.4=ρ1¯1,2C,B, C belongs to class A (with respect to ρ1¯1,2); as ρ1¯1,wC,A=0.2,0.35≤0.4=ρ1¯1,wC,B, C belongs to class A (with respect to ρ1¯1,w); as ρ1¯w,2C,B=0.4≤0.35,0.5=ρ1¯w,2C,A, C belongs to class B (with respect to ρ1¯w,2).Asρ2¯1,2C,A=0.05,0.41≤0.5=ρ2¯1,2C,B, C belongs to class A (with respect to ρ2¯1,2); as ρ2¯1,wC,A=0.05,0.50.05+0.41≤0.5=ρ2¯1,wC,B, C belongs to class A (with respect to ρ2¯1,w); as ρ2¯w,2C,B=0.5≤0.50.05+0.41,0.41=ρ2¯w,2C,A, C belongs to class B (with respect to ρ2¯w,2).Asρ3¯1,2C,A=0.3,0.9≤0.7=ρ3¯1,2C,B, C belongs to class A (with respect to ρ3¯1,2); as ρ3¯1,wC,A=0.3,0.6≤0.7=ρ3¯1,wC,B, C belongs to class A (with respect to ρ3¯1,w); as ρ3¯w,2C,B=0.7≤0.6,0.9=ρ3¯w,2C,A, C belongs to class B (with respect to ρ3¯w,2).Asρ4¯1,2C,A=1.1,1.3≤1.9=ρ4¯1,2C,B, C belongs to class A (with respect to ρ4¯1,2); as ρ4¯1,wC,A=1.1,1.2≤1.9=ρ4¯1,wC,B, C belongs to class A (with respect to ρ4¯1,w); as ρ4¯w,2C,A=1.2,1.3≤1.9=ρ4¯w,2C,B, C belongs to class A (with respect to ρ4¯w,2).It follows thatρ¯e1,2C,A≤ρ¯e1,2C,B and ρ¯e1,wC,A≤ρ¯e1,wC,B for each e0,e1,e2,e3,e4∈R5, thus C belongs to class A (with respect to ρ¯e1,2 and ρ¯e1,w). Take e0=e1=e2=e3=e4=0.2. Then, ρ¯ew,2C,A≤ρ¯ew,2C,B, thus C belongs to class A (with respect to ρ¯ew,2).(3)LetA=0.1,0.2,0.1/x,0.3,0.4,0.3/y,0.1,0.2,0.1/z and B=0.8,0.1,0.1/x,0.7,0.2,0.1/y,0.8,0.1,0.1/z be feature picture fuzzy sets representing two physical entities (e.g., patients), C=0.6,0.1,0.1/x,0.5,0.1,0/y,0.6,0.1,0.1/z the third, and w⟶=0.5,0.5.Asρ0¯1,2C,A=1/30.2,0.4≤1/30.6,0.8=ρ0¯1,2C,B, C belongs to class A (with respect to ρ0¯1,2); as ρ0¯1,wC,A=1/30.2,0.3≤1/30.6,0.7=ρ0¯1,wC,B, C belongs to class A (with respect to ρ0¯1,w); as ρ0¯w,2C,A=1/30.3,0.4≤1/30.7,0.8=ρ0¯w,2C,B, C belongs to class A (with respect to ρ0¯w,2).Asρ1¯1,2C,A=0.2≤0.3,0.5=ρ1¯1,2C,B, C belongs to class A (with respect to ρ1¯1,2); as ρ1¯1,wC,A=0.2≤0.3,0.4=ρ1¯1,wC,B, C belongs to class A (with respect to ρ1¯1,w); as ρ1¯w,2C,A=0.2≤0.4,0.5=ρ1¯w,2C,B, C belongs to class A (with respect to ρ1¯w,2).Asρ2¯1,2C,A=0.2,0.06≤0.22,0.26=ρ2¯1,2C,B, C belongs to class A (with respect to ρ2¯1,2); as ρ2¯1,wC,A=0.2,0.50.2+0.06≤0.22,0.50.22+0.26=ρ2¯1,wC,B, C belongs to class A (with respect to ρ2¯1,w); as ρ2¯w,2C,A=0.50.2+0.06,0.06≤0.50.22+0.26,0.26=ρ2¯w,2C,B, C belongs to class A (with respect to ρ2¯w,2).Asρ3¯1,2C,A=0.2,0.4≤0.6,0.8=ρ3¯1,2C,B, C belongs to class A (with respect to ρ3¯1,2); as ρ3¯1,wC,A=0.2,0.3≤0.6,0.7=ρ3¯1,wC,B, C belongs to class A (with respect to ρ3¯1,w); as ρ3¯w,2C,A=0.3,0.4≤0.7,0.8=ρ3¯w,2C,B, C belongs to class A (with respect to ρ3¯w,2).Asρ4¯1,2C,A=0.2+20.02,0.2+0.05+0.01≤0.5+20.02,0.9+0.05=ρ4¯1,2C,B, C belongs to class A (with respect to ρ4¯1,2); as ρ4¯1,wC,A=0.2+20.02,0.2+0.02+0.50.05+0.01≤0.5+20.02,0.45+0.02+0.50.05=ρ4¯1,wC,B, C belongs to class A (with respect to ρ4¯1,w); as ρ4¯w,2C,A=0.2+0.02+0.50.05+0.01,0.2+0.05+0.01≤0.45+0.02+0.50.05,0.9+0.05=ρ4¯w,2C,B, C belongs to class A (with respect to ρ4¯w,2).It followsρ¯e1,2C,A≤ρ¯e1,2C,B, ρ¯e1,wC,A≤ρ¯e1,wC,B, and ρ¯ew,2C,A≤ρ¯ew,2C,B for each e0,e1,e2,e3,e4∈R5, thus C belongs to class A (with respect to ρ¯e1,2, ρ¯e1,w, and ρ¯ew,2).(4)LetA=0.1,0.2,0.1/x,0.3,0.4,0.3/y,0.1,0.2,0.1/z and B=0.9,0.1,0.1/x,0.8,0.2,0.1/y,0.9,0.1,0.1/z be feature generalized pythagorean fuzzy sets (A generalized pythagorean fuzzy set is a special 3-polar fuzzy set A:X⟶0,13 satisfying ρ2Ax,0⟶≤1∀x∈X.) (see [38, 39] for a special case) representing two physical entities (e.g., patients), C=0.6,0.1,0.1/x,0.5,0.1,0/y,0.6,0.1,0.1/z the third, and w⟶=0.5,0.5. Similar to (3), as ρ0¯1,2C,B=1/30.3,0.5≤1/30.2,0.8=ρ0¯1,2C,A, C belongs to class B (with respect to ρ0¯1,2). Analogously, we can decide C belongs to class A or B with respect to ρ0¯1,w (resp., ρ0¯w,2, ρ1¯1,2, ρ1¯1,w, ρ1¯w,2, ρ2¯1,2, ρ2¯1,w, ρ2¯w,2, ρ3¯1,2, ρ3¯1,w, ρ3¯w,2, ρ4¯1,2, ρ4¯1,w, ρ4¯w,2).Example 3.
(1) Analogously, we can predictAn+1 from a given time series A1,A2,…,Ann>3 in the following way: Firstly, fix (maybe by experts) a natural number m<n (say, m=3) and a set w1,w2,w3⊆0,1 satisfying w1<w2<w3 and w1+w2+w3=1. Secondly, compute the differences rii=1,2,…,n−3:(4)ri=ρAi,Ai+1,Ai+2,An−2,An−1,An=w1An−2−Ai+w2An−1−Ai+1+w3An−Ai+2.Finally, takeAn+1=Ak+3 if rk≤rii=1,2,…,n−3. We can also compute each ri as Example 2.(2)(Continued to Proposition1 (4)) Let ft=t and gt=t3 be feature functions representing two physical entities, and ht=t2t∈0,1 the third. Then, hf¯=∫01t−t2dt=1/2−1/3=1/6, hg¯=∫01t2−t3dt=1/3−1/4=1/12, hf¯=1/4, and hg¯=4/27.
If we take a weight vectorw⟶=0.5,0.5, as ρh,g=1/12,4/27≤1/6,1/4=ρh,f, h belongs to class g (with respect to ρ); as ρ.wh,g=1/12,25/216≤1/6,5/24=ρ.wh,f, h belongs to class g (with respect to ρ); as ρw.h,g=25/216,4/27≤5/24,1/4=ρw.h,f, h belongs to class g (with respect to ρ).
If we take two weight vectorsw⟶=0.5,0.5 and v⟶=0.6,0.4, then ρ.vh,f=1/6,1/5, ρ.vh,g=1/12,59/540, ρv.h,f=1/5,1/4, and ρv.h,g=59/540,4/27. Therefore,ρ.vwh,f=0.51/6,1/5⊕0.51/6,5/24=1/6,49/240, ρ.vwh,g=0.51/12,59/540⊕0.51/12,25/216=1/12,89/1296, ρvw.h,f=0.51/5,1/4⊕0.55/24,1/4=49/240,1/4, ρvw.h,g=0.559/540,7/24⊕0.525/216,4/27=89/1296,11/54. As ρ.vwh,g≤ρ.vwh,f, h belongs to class g (with respect to ρ.vw); as ρvw.h,g≤ρvw.h,f, h belongs to class g (with respect to ρ.vw).Proposition 3.
(1) LetA⊆2X−∅X≠∅,A≠∅. If ρ− (resp., ρ+) is a pseudo-semi-metric (resp., a pseudo-metric) on A satisfying ρ−A,B≤ρ+A,BA,B⊆A, then ρ˜:A2⟶IR+, defined by ρ˜A,B=ρ−A,B,ρ+A,BA,B⊆A, is a WIVP-metric on A.(2) Ifϱ1 and ϱ2 are pseudo-metrics on XX≠∅, then ϱ1∧ϱ2, defined by ϱ1∧ϱ2x,y=minϱ1x,y,ϱ2x,y, is a pseudo-semi-metric on X, ϱ1∨ϱ2, defined by ϱ1∨ϱ2x,y=maxϱ1x,y,ϱ2x,y, is a pseudo-metric on X, and thus ϱ1Δϱ2, defined by(5)ϱ1Δϱ2x,y=ϱ1∧ϱ2x,y,ϱ1∨ϱ2x,y,is a WIVP-metric onX.(3) LetA=A∈2R3−∅|A is connected and bounded with respect to the ordinary Euclidean metric on R3, and Lebesgue measurable. Then, d0, defined by d0A,B=infρ2x⟶,y⟶|x⟶∈A,y⟶∈B (ρ2 is the ordinary Euclidean metric on R3), is a pseudo-semi-metric; d1, defined by d1A,B=ρ2cA,cB, is a pseudo-metric; and d2, defined by(6)d2A,B=maxsupa⟶∈Ainfb⟶∈Bρ2a⟶,b⟶,supb⟶∈Binfa⟶∈Aρ2a⟶,b⟶,is a metric (called the Hausdorff metric induced byρ2 and written also as ρ2H). Therefore, we obtain three WIVP-metrics di:A2⟶IR+i=1,2,3 on A each of which can be used to measure the difference between any two heavenly bodies A and B in A with the centroids cA,cB⊆R3, where diA,B=d0A,B,diA,Bi=1,2 and d3A,B=d1Δd2A,B=mind1A,B,d2A,B,maxd1A,B,d2A,B. As d0A,B≤mind1A,B,d2A,B, d1A,B∨d2A,B≤d3A,B∀A,B∈A2.(4)Letρ be a pseudo-metric on A⊆2X−∅X≠∅,A≠∅, A=Φ∈LX|Φd∈A∀d∈L (where L,≤ is a poset), and a,c⊆L. Then, the mapping ρac:A2⟶IR+, defined by Φ,Ψ⊆AρacΦ,Ψ=ρΦa,Ψa∧ρΦc,Ψc,ρΦa,Ψa∨ρΦc,Ψc, is a WIVP-metric on A.Proof.
(1) follows from Definition3, (2) can be proved by definition, and (3) follows from (2). (4) Defined ϱ1,ϱ2:A2⟶R+ by putting ϱ1Φ,Ψ=ρΦa,Ψa and ϱ2Φ,Ψ=ρΦc,ΨcΦ,Ψ⊆A. Then, ρac=ϱ1Δϱ2 and ϱ1 and ϱ2 are pseudo-metrics on A, thus ρac is a WIVP-metric on A by (2).Example 4.
(1) Consider the two bodiesA and B in Example 1 (2). Then, the differences between A and B can be taken as (the first three are real numbers and the last three are interval numbers): d0A,B=0, d1A,B=ϵ+ε, d2A,B=2maxϵ,ε, d1A,B=0,ϵ+ε, d2A,B=0,2maxϵ,ε, or d3A,B=ϵ+ε,2maxϵ,ε.(2) LetA be the set of all nonempty compact sets in the ordinary Euclidean plane R2,ϱ, ρ=ϱH be the Hausdorff metric on A induced by ϱ, A=Φ∈0,1R2|Φd∈A∀d∈0,1, a,c=0.4,0.6, and Φ,Ψ∈A be two clouds (in which Φx,y stands for the degree of the thickness of cloud in vertical line through the point x,y in the plane R2), defined by Φx,y=φx∧φy (i.e., minφx,φy) and Ψx,y=ψx∧ψyx,y∈R2, where φx=1−xx∈R and ψx=1−x−9x∈R. Then, Φa=−0.6,0.62, Ψa=8.4,9.62, Φc=−0.4,0.42, Ψc=8.6,9.42, ρΦa,Ψa=9.6−0.62+9.6−0.62=92, and ρΦc,Ψc=9.4−0.42+9.4−0.42=92. Therefore, ρacΦ,Ψ=92,92=92.(3)Analogously, letA be the set of all nonempty compact sets in the ordinary Euclidean space R3,ϱ, ρ=ϱH be the Hausdorff metric on A induced by ϱ, A=Φ∈0,1R3|Φd∈A∀d∈0,1, a,c=0.4,0.6, and Φ,Ψ∈A be two clouds (in which Φx,y,z stands for the degree of concentration of cloud at the point x,y,z), defined by Φx,y,z=φx∧φy∧φz and Ψx,y,z=ψx∧ψy∧ψzx,y,z∈R3, where φx=1−xx∈R and ψx=1−x−9x∈R. Then Φa=−0.6,0.63, Ψa=8.4,9.63, Φc=−0.4,0.43, Ψc=8.6,9.43, ρΦa,Ψa=39.6−0.62=93, and ρΦc,Ψc=39.4−0.42=93. Therefore, ρacΦ,Ψ=93,93=93.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 Table1), 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):(7)P1=39asthma,48sorethroat,18cough,37.8fever,67age,P2=41asthma,51sorethroat,41cough,38fever,41age,…P7=60asthma,57sorethroat,37cough,38.5fever,54age.
Take other 2 patients in the same hospital for test:(8)P8=59,63asthma,60sorethroat,28cough,37.9fever,67age,P9=60,64asthma,47sorethroat,43cough,38.6fever,70age.
Notice that bothP8 and P9 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 e⟶=e1,e2,e3,e4,e5=2,1,5,30,1 and v⟶=v1,v2,v3,v4,v5=4,3,10,60,1 for importance of vector of symptom s⟶=(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 P8 and those with diagnosis results Y using a metric rely on e⟶ (there are many cases on choices of ρe). ρeP2,P8=e141−61+e251−60+e341−28+e438−37.9+e541−67=143, ρeP4,P8=e161−45+e260−53+e344−28+e437.8−37.9+e544−67=145, ρeP5,P8=e161−39+e262−60+e349−28+e440−37.9+e570−67=217, ρeP6,P8=e161−55+e255−60+e336−28+e438−37.9+e550−67=77, ρeP7,P8=e161−60+e257−60+e337−28+e438.5−37.9+e554−67=77. Thus, we obtain the average d8,e,Y=132.6.Step 2. Compute the difference between P8 and those with diagnosis results N using a metric rely on e⟶. ρeP1,P8=e139−61+e248−60+e318−28+e437.8−37.9+e567−67=109, ρeP3,P8=e152−61+e248−60+e332−28+e438.5−37.9+e544−67=91. Thus, we obtain the average d8,e,N=100.Step 3. Compute the difference between P9 and those with diagnosis results Y using a metric rely on e⟶. ρeP2,P9=103, ρeP4,P9=71, ρeP5,P9=133, ρeP6,P9=95, ρeP7,P9=63. Thus, we obtain an average d9,e,Y=93.Step 4. Compute the difference between P9 and those with diagnosis results N using a metric rely on e⟶. ρeP1,P9=199, ρeP2,P9=105. Thus, we obtain an average d9,e,N=152.Step 5. Compute the difference between P8 and those with diagnosis results Y using a metric rely on v⟶. ρvP2,P8=269, ρvP4,P8=252, ρvP5,P8=433, ρvP6,P8=128, ρvP7,P8=152. Thus, we obtain an average d8,v,Y=246.8.Step 6. Compute the difference between P8 and those with diagnosis results N using a metric rely on v⟶. ρvP1,P8=230, ρvP2,P8=171. Thus, we obtain an average d8,v,N=200.5.Step 7. Compute the difference between P9 and those with diagnosis results Y using a metric rely on v⟶. ρvP2,P9=181, ρvP4,P9=146, ρvP5,P9=281, ρvP6,P9=178, ρvP7,P9=120. Thus, we obtain an average d9,v,Y=181.2.Step 8. Compute the difference between P9 and those with diagnosis results N using a metric rely on v⟶. ρvP1,P9=396, ρvP2,P9=185. Thus, we obtain an average d9,v,N=290.5.Step 9. As d8,e,N,d8,v,N=100,200.5≤132.6,246.8=d8,e,Y,d8,v,Y, P8 is N (which is the same as the diagnosis result given by expert physicians in that hospital); as d9,e,Y,d9,v,Y=93,181.2≤152,290.5=d9,e,N,d9,v,N, P9 is Y (which is also the same as the diagnosis result given by expert physicians in that hospital).Table 1
Symptom data and diagnosis results.
-p1p2p3p4p5p6p7Asthma39415245395560Sore throat48514853625557Cough18413244493637Fever37.83838.537.8403838.5Age67414468705054ResultNYNYYYYFigure 1
A diagram for the proposed algorithm.Proposition 4.
(1) For a mappingρ:X2⟶IR+, consider the following conditions:(M0)ρx,x=0∀x∈X(M1)ρx,y=0 if x=y∀x,y⊆X(M2)ρx,y=ρy,x∀x,y⊆X(M3)ρx,y≤ρx,z⊕ρz,y∀x,y,z⊆X(M4)p2∘ρx,y≤p2ρx,z⊕ρz,y=p2∘ρx,z+p2∘ρz,y∀x,y,z⊆XThen,ρ is a WIVP-metric (resp., an interval-valued pseudo-metric, a weak interval-valued metric, an interval-valued metric) on X if it satisfies (M0) + (M2) + (M4) (resp., (M0) + (M2) + (M3), (M1) + (M2) + (M4), (M1) + (M2) + (M3)).(2) Every interval-valued metric spaceX,ρ can be looked as a special fuzzy metric space, where we define L and R as La,b=mina,b∀a,b∈0,1×0,1 and Ra,b=maxa,b∀a,b∈0,1×0,1, and set d=ρ.


## 3. WIVP-Metrics Pertinent to Logic Theory

An logic implication operator is a mappingI:0,1×0,1⟶0,1 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:IBa,b=0 (if a=1,b=0) or 1 (otherwise)(ii) Gaines-Rescher implication operator:IGRa,b=1 (if a≤b) or 0 (if a≤b)(iii) Goguen implication operator:IGoa,b=1 (if a=0) or 1∧b/a (if a≠0)(iv) Godel implication operator:IGa,b=1 (if a≤b) or b (if a≤b)(v) Kleene-Dienes implication operator:IKDa,b=1−a∨b(vi) Lukasiewicz implication operator:ILua,b=1−a+b∧1(vii) Mamdani implication operator:IMa,b=a∧b(vii) Product implication operator:IΠa,b=1 (if a≤b) or b/a (if a≤b)(ix) Wang implication operator:I0a,b=1 (if a≤b) or 1−a∨b (if a≤b)(x) Reichenbach implication operator:IRa,b=1−a+ab(xi) Yager implication operator:IYa,b=ba∀a,b∈0,1(xii) Zadeh implication operator:IZa,b=1−a∨a∧bIn 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) LetS=p1,p2,… be a countable set (where pi≠pj if i≠j), ¬ a unary operator, ∨ and ⟶ binary operators, and FS the free algebra of type ¬,∨,⟶ generated by S. Elements of S are called atomic formulae (or atomic propositions), and elements of FS are called formulae (or propositions).(2)0,1 can be looked as an algebra of type ¬,∨,⟶ if we define the unary operator ¬:0,1⟶0,1 and the two binary operators ∨:0,1×0,1⟶0,1 and ⟶:0,1×0,1⟶0,1 as follows: ¬a=1−a∀a∈0,1, a∨b=maxa,b∀a,b∈0,1, a⟶b=Ia,b∀a,b∈0,1, where I:0,1×0,1⟶0,1 is a logic implication operator. Furthermore, for each A∈FS, there exists a unique finite subset SA⊆FS (without loss of generality we assume SA=p1,p2,…,pn) such that A can be represented uniquely by SA and ¬,∨,⟶ (written as A=Ap1,p2,…,pn). Substitute xk for pk in Ap1,p2,…,pnk=1,2,…,n and keep the logic connectives in A unchanged, then we obtain an expression AI¯x1,x2,…,xn and thus an n-variable function AI¯:0,1n⟶0,1 which is called the I-function corresponding to A. For example, if A=¬p1⟶p2∨p3, then AI¯x1,x2,x3=I1−x1,maxx2,x3.(3) A mappingν:FS⟶0,1 which preserves operations ¬,∨ and ⟶ is called an I-valuation on FS. The set of all I-valuations is written as ΩI.(4)LetA,B∈FS. If νA=νB∀ν∈ΩI, then A and B are said to be I-logically equivalent.Lemma 1.
If the formulaeA and B in FS are I-logically equivalent, then AI¯=BI¯ (where I is a logic implication operator).Proof.
Without loss of generality, we assumeSA=p1,p2,…,pm, SB=p1,p2,…,pn, and m+1=n. Let A∗=A∧r⟶r (where r∈SB−SA), then SA∗=SB. For each x⟶=x1,x2,…,xn∈0,1n, there exists an I-valuation νx⟶:FS⟶0,1 on FS which satisfies νx⟶pi=xii=1,2,…,n since FS is a free algebra. As A and B are R-logically equivalent, AI¯x⟶=AI∗¯x⟶=νx⟶A∗=νx⟶A=νx⟶B=BI¯x⟶, which means AI¯=BI¯.Lemma 2.
(1) For anyA,B∈FS, define A∼B if and only if A and B is ILu-logically equivalent. Then, ∼ is an equivalence relation on FS. The equivalence class determined by A∈FS is written A, and the quotient set determined by ∼ is written FS/∼.(2) The mappingdn:FS×FS⟶0,+∞, defined by ∀A,B⊆FSdnA,B=1−∫△InAIn¯x⟶,BIn¯x⟶∧InBIn¯x⟶,AIn¯x⟶dx⟶, is a pseudo-metric on FSn∈Lu,Go,G,0, where △ is the integral domain related A,B.(3) For eachε∈0,1, the mapping dGRε:FS×FS⟶0,+∞, defined by 1−ε∫△IGRAIGR¯x⟶,BIGR¯x⟶∧IGRBIGR¯x⟶,AIGR¯x⟶dx⟶ (if A≁B) or dGRεA,B=0 (if A∼B), is a pseudo-metric on FS.(4)dGRε1≥dGRε20≤ε1<ε2≤1. Notice also that dGR0A,B=0,A∼B,1,A≁B.(5)ρGRεε∈0,1 is a family of metrics on FS/∼ which satisfy ρGRε1≥ρGRε20≤ε1<ε2<1, where ρGRεε∈0,1 is defined by ρGRεA,B=dGRεA,B∀A,B∈FS.(6)(cf. [42]). The mapping ρLu:FS/∼×FS/∼⟶0,+∞, defined by ρLuA,B=dLuA,B∀A,B∈FS, is a metric on FS/∼.Remark 1.
Lemma2 (2) is Theorem 6 (without proof) in [41].Theorem 1.
(1)FS,d1Δd2, is a WIVP-metric space d1,d2∈dG,dGo,dLu,d0∪dGRε|ε∈0,1(2)FS/∼ρ1Δρ2, is a weak interval-valued metric space ρ1,ρ2∈ρLu∪ρGRε|ε∈0,1(3)FS/∼ρ1Δρ2, is an interval-valued metric space ρ1,ρ2∈ρGRε|ε∈0,1Proof.
It follows from Lemma2 (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.AnJ-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 μx,y=μy,x≤σx∧σy∀x,y∈V×V. The underlying graph of G is defined as G=V,E, where E=x,y⊆V|μx,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 mx,y-element subset z1,z2,…,zmx,y⊆Vmxy≥2 such that x=z1, zmx,y=y, and, z1,z2,z2,z3,…,zmx,y−1,zmx,y, are all in E; the word P=z1,z2,…,zmx,y is called a path from x to y, and the set of all paths from x to y is denoted by Pxy.Definition 5.
LetG=V,σ,μ be a 0,1-graph (called also fuzzy graph [42]), and ε∈0,1. For each x∈V, let ρ−1x,x=ρεx,x=0. For any x,y∈Vx≠y, define ρ−1x,y and ρεx,y as follows:(1)If there is no path fromx to y, define ρ−1x,y=ρεx,y=V (the cardinality of V).(2)If there is a path fromx to y, define(9)ρ−1x,y=∧μz1,z2+μz2,z3+⋯+μzmx,y−1,zmx,y∧1≤i≤mx,yσzi|z1z2,…,zmx,y∈Pxy,ρεx,y=∧μz1,z2εσz1∨σz2+1−εσz1∧σz2+μz2,z3εσz2∨σz3+1−εσz2∧σz3+⋯+μzmx,y−1,zmx,yεσzmx,y−1∨σzmx,y+1−εσzmx,y−1∧σzmx,y|z1,z2,…,zmx,y∈Pxy.
Then, the mappingsρ−1,ρε:V×V⟶0,+∞ are metrics on V (called metrics on the fuzzy graph G).Remark 2.
It is easy to verifyρε1≤ρε2 for all ε1,ε2∈−1∪0,1 satisfying ε1≤ε2, thus we obtain a family ρε1Δρε2|ε1≤ε2,ε1,ε2∈−1∪0,1 of interval-valued metrics on V (called interval-valued metrics on the fuzzy graph G).Theorem 2.
LetG=V,σ,μ be an I0,1-graph (called also interval-valued fuzzy graph). For any x,y∈V, let(10)ρx,y=0,x=y,∧ℓP|P∈Pxy,x≠yandPxy≠∅,V2mμ,V2mμ,x≠yandPxy=∅,where(11)ℓP=1μz1,z2+1μz2,z3+⋯+1μzn−1,zn,∀P=z1,z2,…,zn∈Pxy,1a˙,a¨=1a˙∧1a¨,1a˙∨1a¨,∀a˙,a¨∈I0,+∞,and mμ=minμs,t|s,t∈E. Then, ρ is an interval-valued metric on V (called interval-valued metric on the interval-valued fuzzy graph G).


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

Definition 6.
(1) A sequencexn in a WIVP-metric space X,ρ is said to be a Cauchy sequence if, for each ε−,ε+≫0, there exists a positive integer nε such that ρxn,xm≪ε−,ε+∀n,m≥nε; xn is said to be convergent in X,ρ if there exists an x∈X such that xn converges to x (i.e., for each ε−,ε+≫0, there exists a positive integer nε such that ρxn,x≪ε−,ε+∀n≥nε).(2) A WIVP-metric spaceX,ρ is said to be complete if every Cauchy sequence in X,ρ is convergent.
It is not difficult to verify the following.Lemma 3.
A WIVP-metric spaceX,ρ is complete if and only if X,ρ+ is complete.
The main results of this section are the following.Theorem 3.
LetX,ρ be a complete WIVP-metric space. Then, f:X⟶X has a unique fixed point in X if it satisfies one of the following conditions:(1)ρ+fx,fy≤kρ+x,y∀x,y∈X, where k∈0,1 is a constant.(2)ρ+fx,fy≤kρ+x,fx+ρ+y,fy∀x,y∈X, where k∈0,1/2 is a constant.(3)ρ+fx,fy≤kmaxρ+x,fx,ρ+y,fy∀x,y∈X, where k∈0,1 is a constant.(4)ρ+fx,fy≤aρ+x,fx+bρ+y,fy+cρ+x,y∀x,y∈X, where a,b,c are nonnegative numbers satisfying a+b+c<1.(5)ρ+fx,fy≤kρ+x,fy+ρ+y,fx∀x,y∈X, where k∈0,1/2 is a constant.(6)ρ+fx,fy≤kmaxρ+x,fy,ρ+y,fx∀x,y∈X, where k∈0,1/2 is a constant.(7)ρ+fx,fy≤aρ+x,fy+bρ+y,fx+cρ+x,y∀x,y∈X, where a,b,c are nonnegative numbers satisfying a+b+c<1.(8)ρ+fx,fy≤k1ρ+x,fx+k2ρ+y,fy+k3ρ+x,fy+k4ρ+y,fx+k5ρ+x,y, ∀x,y∈X, where ki≥0i∈1,2,…,5, ∑i=15ki<1, and s is a fixed positive integer.(9)ρ+fsx,fsy≤k1ρ+x,fsx+k2ρ+y,fsy+k3ρ+x,fsy+k4ρ+y,fsx+k5ρ+x,y, ∀x,y∈Xwhere ki≥0i∈1,2,…,5, ∑i=15ki<1, and s is a fixed positive integer.Theorem 4.
LetX,ρ be a complete weak interval-valued metric space. Then, f,g:X⟶X have common fixed point in X if they satisfy one of the following conditions (where s and t are fixed positive integers):(1)ρ+fsx,gty≤kρ+x,y∀x,y∈X, where k∈0,1 is a constant.(2)ρ+fsx,gty≤kρ+x,fsx+ρ+y,gty∀x,y∈X, where k∈0,1/2 is a constant.(3)ρ+fsx,gty≤kmaxρ+x,fsx,ρ+y,gty∀x,y∈X, where k∈0,1 is a constant.(4)ρ+fsx,gty≤aρ+x,fsx+bρ+y,gty+cρ+x,y∀x,y∈X, where a,b,c are nonnegative numbers satisfying a+b+c<1.(5)ρ+fsx,gty≤kρ+x,gty+ρ+y,fsx∀x,y∈X, where k∈0,1/2 is a constant.(6)ρ+fsx,gty≤kmaxρ+x,gty,ρ+y,fsx∀x,y∈X, where k∈0,1/2 is a constant.(7)ρ+fsx,gty≤k1ρ+x,fsx+k2ρ+y,gty+k3ρ+x,gty+k4ρ+y,fsx+k5ρ+x,y∀x,y∈X, where ki≥0i∈1,2,…,5, ∑i=15ki<1.Theorem3 may be proved based on Lemma 3 and results on contractive-type mappings satisfying 1,4,5,7, 11,12, 14,18,43 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.



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