Essential Conflict Measurement in Dempster–Shafer Theory for Intelligent Information Fusion

Abstract

Dempster’s combination rule in Dempster–Shafer theory is a powerful and effective tool for multi-sensor data fusion. However, counterintuitive results are possible under the condition of a high conflict between pieces of evidence. This study demonstrates that existing conflict measurements cannot prevent such results and, thus, proposes a quantitative conflict measurement based on the concept of essential conflict. This work analyzes two characteristics of the essential conflict, namely belief absolutization and uncorrectable assertions. In addition, considering the desirable properties of the measurement, this study demonstrates that the measurement of essential conflict can reveal the essence of counterintuitive results in Dempster’s combination process. Finally, properties and examples are used to validate the proposed measurement.

1. Introduction

The Dempster–Shafer (D–S) theory is a powerful and flexible tool for representing and handling uncertainty and imprecise or incomplete information [1,2]. It has been widely used in multi-sensor data fusion fields, such as decision making [3,4,5], fault diagnosis [6,7,8], and pattern recognition [9,10]. Dempster’s combination rule undoubtedly plays a crucial role in the D–S theory. Namely, when multiple pieces of evidence for a set of states are accumulated, this rule is used to combine them and see how strongly they support the set of states together [11,12]. However, counterintuitive results could be obtained when the evidence is highly conflicting, such as for the example given by Zadeh (1986); therefore, researchers have challenged its validity and consistency in such cases [13,14].

In recent years, many methods have been proposed to improve Dempster’s combination rule, and they can be roughly categorized into three main types: (i) the first type replaces Dempster’s combination rule with a new rule [15,16]; (ii) the second type reconstructs the mass functions before applying Dempster’s combination rule [17]; (iii) the third type specifies the situations when it is safe to use Dempster’s combination rule with conflict measurement [18,19,20]. Nevertheless, alternative combination rules used in the existing methods still cannot achieve wide acceptance in real-world applications. The assumptions made to relax Dempster’s combination rule can have many limitations, and the conditions proposed to make Dempster’s combination rule safe to use are still controversial [21,22,23,24], as discussed in Section 3.

A detailed review of the aforementioned three categories has indicated a lack of systematic and comprehensive analyses of what exactly a conflict means based on the epistemic characterization of the counterintuitive result of the combination process [25,26]. To address this issue, this study analyzes the notion of conflict based on its relationship with the counterintuitive results of the combination process and proposes an innovative measure of conflict.

This study conducts an epistemic characterization of the essential conflict, defines it formally, and proposes a quantitative measurement. To prove that the essential conflict is indeed closely related to the counterintuitive result, this study analyzes two of its crucial properties: belief absolutization and uncorrectable assertion. Belief absolutization indicates that the existence of essential conflict between two pieces of evidence will change a possible state of the original mass function into a necessary true state or impossible state in the combination process with Dempster’s combination rule. Uncorrectable assertion indicates that the necessary true state or impossible state caused by the property of belief absolutization cannot be corrected by further adding and combining with additional pieces of evidence. Hence, after revealing the properties’ conflicts in the combination process, the study argues that the essential conflict represents the most important factor for demonstrating the essence and uncorrectable disagreement between evidence. In view of that, this study analyzes the desirable properties of the quantitative measurement, confirms the advantages of the proposed conflict measurement through a comparative property analysis with the existing conflict measurements, and provides examples to verify the proposed solution.

This study pushes forward the frontiers of D–S theory research in four main respects:

i. 

This study introduces an innovative type of conflict, the essential conflict, in the combination process that is related to the counterintuitive results obtained using Dempster’s combination rule;

ii. 

The essential conflict is formally defined, and the corresponding measurement method is developed;

iii. 

It is demonstrated that the essential conflict reveals the essence of counterintuitive results of the D–S combination process, and the properties of the degree of essential conflict are analyzed;

iv. 

It is demonstrated that the proposed conflict measurement can handle cases that the existing methods cannot address.

The rest of this paper is organized as follows. Section 2 reviews the background knowledge. Section 3 discusses the related work. Section 4 provides epistemic discussions of the essential conflict. Section 5 formally defines the essential conflict and introduces a measurement process. Section 6 evaluates the essential conflict and the corresponding measurement process. Finally, Section 7 concludes this work and suggests future work directions.

2. Preliminaries

Dempster [27] first proposed the foundational D–S theory to represent imprecise probabilities with upper and lower probabilities. Ref. [28] further developed a systematic theory of uncertain reasoning to handle imprecise and uncertain information. Some basic concepts of the D–S theory are introduced in the following.

Definition 1

([28]). Assume $\Theta =\{{\omega }_1,\dots ,{\omega }_n\}$ is a set of exhaustive and mutually exclusive elements (i.e., states of the world) called a frame of discernment (or simply a frame). Then, the function $m:2^{\Theta }\to [0,1]$ is a mass function if

$$\begin{array}{ccc}& & m(⌀)=0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & \sum _{A\subseteq \Theta }m\left(A\right)=1.\hfill \end{array}$$

(2)

where the mass value of $m\left(A\right)$ ($A\subseteq \Theta$) represents the degree to which the corresponding evidence supports A; a subset $A\subseteq \Theta$ satisfying the condition of $m\left(A\right)>0$ is called a focal element of m; $F_m$ is the focal element set of the mass function m if for any $A\in F_m,m\left(A\right)>0;$ The mass function with $m(\Theta )=1$ is called a vacuous mass function, which represents a decision maker’s total ignorance over a frame Θ.

Definition 2.

Let m be a mass function over a frame of discernment Θ. Then, a function $Bel:2^{\Theta }\to [0,1]$, defined as follows, is a belief function over Θ:

$$Bel\left(A\right)=\sum _{B\subseteq A}m\left(B\right).$$

(3)

Similarly, a function $Pl:2^{\Theta }\to [0,1]$, defined as follows, can be called a plausibility function over Θ:

$$Pl\left(A\right)=\sum _{B\cap A\ne ⌀}m\left(B\right).$$

(4)

The function $Bel\left(A\right)$ quantifies the degree of belief that the state set A is true. In particular, $Bel(⌀)=0$, $Bel(\Theta )=1$, and $Pl\left(A\right)$ represents the belief level of not denying A. The mass function and belief function are nonadditive. If $m\left(A\right)>0$ and $\sum m\left(A\right)=1$, where A is a singleton (i.e., $\left|A\right|=1$), then m denotes a probability function, and $Bel\left(A\right)=Pl\left(A\right)$. Also, it can be easily proven that $Bel\left(A\right)\le Pl\left(A\right)$, and $Pl\left(A\right)-Bel\left(A\right)$ can be used to represent ignorance.

The D–S theory allows for accumulating and combining evidence from multiple distinct sources using Dempster’s combination rule [28].

Definition 3

([28]). Assume $m_1$ and $m_2$ are two mass functions from independent and fully reliable sources over a frame of discernment Θ. Then, the combined mass function of $m_1$ and $m_2$ obtained following Dempster’s combination rule is defined as follows:

$$m_{1,2}\left(x\right)=\left\{\begin{array}{cc}0\hfill & if x=⌀,\hfill \\ {\displaystyle \frac{1}{1-k_{1,2}}}({\displaystyle \sum _{A\bigcap B=x}}{m}_1\left(A\right)m_2\left(B\right))\hfill & if x\ne ⌀,\hfill \end{array}\right.$$

(5)

with normalization constant

$$k_{1,2}=\sum _{A\bigcap B=⌀}{m}_1\left(A\right)m_2\left(B\right).$$

(6)

Each belief value is divided by $(1-k_{1,2})$ to avoid assigning a non-zero probability to an empty set. The normalization constant $k_{1,2}$ is also called a classical conflict coefficient because it is generally considered capable of measuring the degree of conflict between pieces of evidence. In particular, $k_{1,2}=0$ means $m_1$ is consistent with $m_2$, while $k_{1,2}=1$ implies there is total conflict between them. It should be noted that Dempster’s combination rule is applicable only when $k_{1,2}<1$; hence, it cannot be used when two mass functions are in total conflict.

The D–S theory can directly express uncertainty by allocating the mass of belief to subsets consisting of multiple elements rather than to an individual element. Some methods can transform uncertain probability into certainty probability, and this process is called probability transformation. One of the classical probability transformation methods is the pignistic probability transformation method proposed by [29] in the transferable belief model.

Definition 4

([29]). Assume m is a mass function over Θ; then, its associated pignistic probability function $Bet{P}_m:\Theta \to [0,1]$ is defined by:

$$\begin{array}{c}\hfill Bet{P}_m\left(\omega \right)=\sum _{A\subseteq \Theta ,\omega \in A}{\displaystyle \frac{1}{\left|A\right|}}{\displaystyle \frac{m\left(A\right)}{1-m(⌀)}},m(⌀)<1,\end{array}$$

(7)

where $\left|A\right|$ is the cardinality of A.

In particular, when an initial mass function yields $m(⌀)=0$, ${\displaystyle \frac{m\left(\omega \right)}{1-m(⌀)}}$ is reduced to $m\left(\omega \right)$; $Bet{P}_m\left(\omega \right)$ indicates the total mass value that a state $\omega$ can carry for decision making based on the corresponding evidence from mass function m. Thus, $Bet{P}_m\left(\omega \right)$ can be referred to as a betting commitment to state $\omega$. In addition, $Bet{P}_m\left(\omega \right)>0$ means that mass function m considers that there is a certain opportunity for $\omega$ to be the real state, whereas $Bet{P}_m\left(\omega \right)=0$ indicates that it considers such a situation impossible.

Finally, although Dempster’s combination rule has been widely used in information fusion and other fields, it has been criticized for counterintuitive combination results, including Zadeh’s well-known counterexample.

Example 1

(Zadeh’s counterexample [25]). Assume $m_1$ and $m_2$ are two mass functions defined on a frame of discernment $\Theta =\{a,b,c\}$, where:

$$\begin{array}{ccc}& & m_1\left(\left\{a\right\}\right)=0.9,m_1\left(\left\{b\right\}\right)=0.1,m_1\left(\left\{c\right\}\right)=0;\hfill \\ & & m_2\left(\left\{a\right\}\right)=0,m_2\left(\left\{b\right\}\right)=0.1,m_2\left(\left\{c\right\}\right)=0.9.\hfill \end{array}$$

It can be seen that the first piece of evidence strongly supports state a with $m_1\left(\left\{a\right\}\right)=0.9$, while the second piece of evidence completely denies state a. Similarly, the second piece of evidence strongly supports state c with $m_2\left(\left\{c\right\}\right)=0.9$, while the first piece of evidence completely denies state c. However, both pieces of evidence weakly support state b with a mass value of $0.1$. By combining these two pieces of evidence with Dempster’s combination rule, a combined mass value can be obtained as follows:

$$m_{1,2}\left(\left\{a\right\}\right)=0,m_{1,2}\left(\left\{b\right\}\right)=1,and{m}_{1,2}\left(\left\{c\right\}\right)=0.$$

It is clear that in the combined mass $m_{1,2}$, state b becomes fully supported after the combination, despite rarely being supported by each of the two pieces of evidence. Nevertheless, states a and c both become absolutely denied in the combined mass, even if they were previously strongly supported by one piece of evidence with a very high degree of belief. Zadeh argued that such a result violates intuition about the combination.

For counterintuitive results such as Example 1, existing methods are usually based on the assessment of the reliability of the evidence, using discounting and other methods to deal with it. However, in practical applications, such as pattern recognition and fault diagnosis, model users often do not have specialized knowledge such as Dempster’s combination rule, lack the ability to judge the reliability of evidence, and make decisions directly based on the combination results. In such scenarios, the reliability of the evidence cannot be inferred inversely from the combination results. Therefore, it is necessary to establish a limitation so that non-expert users can clearly recognize that when this condition is met, they should not apply Dempster’s combination rule directly, and should seek experts to reevaluate or adjust the model function in order to avoid counterintuitive results. It is proposed that if an essential conflict exists between two mass functions, applying Dempster’s combination rule will produce counterintuitive results. The proposed degree of essential conflict can be utilized to delineate when Dempster’s combination rule is applicable.

3. Related Work

Since counterintuitive results are common when combining conflicting evidence, such as in Example 1, many methods have been proposed to improve Dempster’s combination rule, which can be roughly divided into three categories.

3.1. First Category

The first category proposes new evidential combination rules. Dempster’s combination rule includes two steps: the intersection of any two sets of states of two original mass functions and the assignment of the conflict mass value (i.e., the partial mass value for the intersection that is empty) to the focal elements of the combined mass function with a normalization constant k. The key challenge lies in determining how to reassign this conflict mass. Yager [30] considered that the mass value of a partial conflict represents ignorance and suggested assigning the conflicting mass value to the universal set of frames of discernment. Dubois and Prade’s rule [31] assigns the mass resulting from a combination of conflicting focal elements to the union of these subsets. Deng et al. [32] proposed other rules to combine evidence with adapted conflict. However, none of these rules is associative or quasi-associative (i.e., to combine three or more pieces of evidence, these rules cannot meet the property that the combination order changes but the combination result remains unchanged), and the desirable properties of Dempster’s combination rule make it easier to add to an already combined body of evidence, thus reducing the computational complexity. In addition, alternative combination rules are ad-hoc and not theoretically justified. As Smets [33] has pointed out, the pragmatic fact that “our rule works fine” is not a proper justification (and at most necessary) in some applications. Therefore, these new rules have not been widely accepted.

3.2. Second Category

The second category of methods assumes that counterintuitive results are caused by directly applying the original mass functions for the combination process in conflict scenarios. Therefore, the original mass function is reconstructed based on different assumptions, including weighted averaging, discounting, and the open-world assumption.

Murphy [34] suggested arranging equal weight for each piece of evidence and then combining the weighted averaging mass function using Dempster’s combination rule. Accordingly, various methods with different types of weight measurement have been proposed, such as Deng’s entropy [35]. However, in these methods, the original mass function is changed after it is combined with a vacuous mass function, and such combination results indicate that a total lack of evidence with no information can still change the original belief. Therefore, using these methods can lead to counterintuitive behavior [36].

The discounting mass function [28,31] considers the reliability of evidence to reconstruct the original mass function and combines the reconstructed mass function with Dempster’s combination rule. Recent research has introduced many techniques for constructing the discounted mass function, such as the classical method [28,31], inconsistent measurements [20], contextual discounting [37], static discounting coefficient [38] and the composite discount factors [39].

The open-world assumption-based method, which was proposed by Smets [40] and Smets and Kennes [41], assigns conflicting mass assignments to an empty set under the assumption that the real state could be none of the alternatives. A generalized evidence theory [35] extends the D–S theory to a more general form using the mentioned assumption, where $m(⌀)=0$ is unnecessary, and ⌀ consists of the focal elements outside of a frame, where the possible true state is outside the frame.

These cases make an additional assumption that the evidence is not fully reliable or that the frame of discernment cannot be exhaustive. However, in some cases, these assumptions could be too conservative, as they impose too strong restrictions on Dempster’s combination rule.

3.3. Third Category

The third category of methods attributes counterintuitive results to the improper use of Dempster’s rule and aims to clarify when the rule can be safely applied [12,18,19]. In addition, it is considered that it is necessary to quantify the degree of conflict and state conditions for the safe application of Dempster’s combination rule. Although the conflict coefficient k (the mass assigned to the empty set before normalization) was initially used, it was later found inadequate to reflect conflict accurately [12]. Consequently, various alternative conflict measurements have been proposed.

Jousselme [19] introduced the concept of evidence distance by converting the mass function to the vector form and considering conflict information from the non-intersecting part. Based on the pignistic probability transformation function [41], Liu [12] defined the pignistic probability distance. Zhou et al. [42] defined a new weighted dissimilarity measure based on the pignistic probability function and Jousselme’s distance measure. Daniel [43] proposed a plausibility conflict based on the plausibility function to deal with the conflict between mass functions.

In recent years, many researchers have measured evidence consistency from the perspective of correlation coefficients. Song et al. [44] used a positive definite matrix $\mathbf{D}$ to preprocess $BPA$ in the vector form and adopted the cosine similarity as a correlation coefficient. Jiang [18] defined a correlation degree, considering the differences between non-intersecting parts and focal elements. Furthermore, Pan and Deng [45] defined a correlation coefficient based on Deng entropy. Xiao, Cao, and Jolfaei [46] defined a cosine angle function as a correlation coefficient. Liu [47] proposed a conflict coefficient on the basis of the Sørensen coefficient.

Finally, some researchers adopted strategies based on divergence measurements, including Jensen-Shannon divergence [48] and improved belief Hellinger divergence [49].

This study advocates the third category of methods. However, there has been little research and interpretation of the origin of conflicts and their meaning. Namely, since the concept of conflict is based on criticism of counterintuitive results in the D–S theory (i.e., when a counterintuitive result occurs, the pieces of evidence that are combined are considered highly conflicting), there is at least an indispensable type of conflict that should be highly related to the counterintuitive result.

Because such a conflict has a close relationship with counterintuitive results, it is denoted as essential conflict in this study, and it is considered that it has the following relationship with the counterintuitive combination result.

Claim 1.

Between the result of two mass functions obtained using Dempster’s combination rule, there is always an essential conflict, such that the value of a result that is counterintuitive should be higher than that of the result that is not counterintuitive.

4. Epistemic Discussions

Before proceeding to epistemic discussions, it is necessary to clarify what is meant by an intuitive combination result. Formally, a combination result is regarded as intuitive if and only if the following two conditions are satisfied:

(i)

Possible states remain possible after combination;

(ii)

The combination result remains correctable.

These requirements can be expressed mathematically as follows. Assume that $m_1$ and $m_2$ are two mass functions defined on a frame of discernment $\Theta$. Let $m_{1,2}$ denote their combination result obtained by Dempster’s combination rule. $Bet{P}_{m_1}$, $Bet{P}_{m_2}$, and $Bet{P}_{m_{1,2}}$ denote the pignistic probability functions of $m_1$, $m_2$, and $m_{1,2}$, respectively. If a combination result is intuitive, then it satisfies the following conditions:

(i)

For every state $\omega \in \Theta$, if $Bet{P}_{m_1}\left(\omega \right)>0\mathrm{or}Bet{P}_{m_2}\left(\omega \right)>0$, then

$$Bet{P}_{m_{1,2}}\left(\omega \right)>0.$$

(ii)

For every state $\omega \in \Theta$, there exists an additional mass function $m_3$ with $m_3(\Theta )\ne 1$ such that

$$Bet{P}_{m_{1,2,3}}\left(\omega \right)=Bet{P}_{m_i}\left(\omega \right),i\in \{1,2\},$$

where $m_{1,2,3}$ denotes the combination of $m_1$, $m_2$, and $m_3$.

In other words, a combination result is regarded as reasonable and epistemically intuitive if the following properties hold. First, states that are possible in the original mass functions remain possible in the combination result and are not transformed into necessarily true or impossible states. Second, the belief assigned to each state in the combination result remains correctable by incorporating additional pieces of evidence.

Based on this definition, we next examine several representative examples to assess whether their combination results satisfy these epistemic requirements.

This study discusses two real-world multi-sensor data fusion applications and shows that classical conflict measurement methods cannot reflect such intuition; namely, they cannot characterize the essential relationship between conflict and the counterintuitive combination results.

Consider an airport security surveillance scenario [50] where Dempster’s combination rule produces a counterintuitive combination.

Example 2.

Suppose an airport intelligent surveillance system has locked down three suspects with potential threats of a, b, and c. In addition, assume that suspect a is a normal person, suspect b is a thief, and suspect c is a bomb attacker. A series of cameras in the foreign currency exchange (FCE) shop observe that suspect a is looking around, and suspect b appears more than once. The metal detector senses that suspect a carries a fruit knife and suspect c has a lighter.

Next, let $m_3$ and $m_4$ be two mass functions on the frame of discernment $\Theta =\{a,b,c\}$, indicating the potential threat degree of the suspects assessed by the cameras and the metal detector, respectively, defined as follows:

$$\begin{array}{ccc}& & m_3\left(\left\{a\right\}\right)=0.8,m_3\left(\left\{b\right\}\right)=0.2;\hfill \\ & & m_4\left(\left\{a\right\}\right)=0.8,m_4\left(\left\{c\right\}\right)=0.2.\hfill \end{array}$$

These two pieces of evidence can be combined with Dempster’s combination rule to obtain a combined mass value of $m_{3,4}\left(\left\{a\right\}\right)=1.$

State a has become fully supported, and states b and c have become denied states. Hence, the combined result suggests that a is a potential threat and excludes the possibility that suspect b and c can be a threat. However, it is known that suspect a is not a threat, suspect b is loitering and hoping to steal money, and suspect c wants to use the lighter to ignite a bomb.

Even if there exists further evidence that indicates that suspect b or c is a potential threat, the result will be neglected; namely, for any mass function $m_5$, the combination will still fully support state a and deny states b and c because $m_{3,4,5}\left(\left\{a\right\}\right)=1$.

Next, consider the example given below.

Example 3.

This example relates to a multi-sensor fault diagnosis system, where vibration acceleration sensors are installed in two locations. Two types of faults are defined on a frame of discernment $\Theta =\{a,b\}$, where a and b stand for rotor unbalance and rotor misalignment, respectively. The signal data of Sensor 1 indicate that the rotor fault has a $90\%$ chance of being caused by $rotorunbalance$ and a $10\%$ chance of being caused by $rotormisalignment$. In addition, data obtained from Sensor 2 indicate that rotor fault has a $90\%$ chance of being caused by $rotormisalignment$ and a $10\%$ chance of being caused by $rotorunbalance$. Then, the two following mass functions can be defined:

$$\begin{array}{ccc}& & m_6\left(\left\{a\right\}\right)=0.9,m_6\left(\left\{b\right\}\right)=0.1;\hfill \\ & & m_7\left(\left\{a\right\}\right)=0.1,m_7\left(\left\{b\right\}\right)=0.9,\hfill \end{array}$$

and the result of Dempster’s combination rule is $m_{6,7}\left(\left\{a\right\}\right)=m_{6,7}\left(\left\{b\right\}\right)=0.5$.

From the respective mass functions, it can be concluded that the two pieces of evidence provide significantly different judgments. It should be noted that if only these two mass functions are used, it can be determined neither which sensor is more reliable nor which of them draws a more accurate conclusion. Therefore, according to the original mass function and combination result, states a and b are equally supported.

The original two mass functions assume that the two states have certain probabilities of being true, and their combination retains the possibility of the two states, which obviously does not violate the intuition.

This phenomenon occurs often in practice. For instance, suppose that two experts judge whether Plan A or Plan B is more reasonable. One expert strongly supports Plan A and weakly supports Plan B, and the other does the opposite. Then, a combined result that balances their views is consistent with the intuition.

Moreover, considering the reliability of a sensor, if Sensor 1 strongly supports state a, and all other sensors strongly support state b, such as Sensor 2, this indicates that Sensor 1 might produce a piece of fault evidence for some reasons (e.g., signal interference, a sudden jarring, or jerking). If $m_8$ is a mass function of Sensor 3 with $m_8\left(\left\{a\right\}\right)=0.1$ and $m_8\left(\left\{b\right\}\right)=0.9$, and if the combination result $m_{6,7}$ is further combined with $m_8$, a new combination result $m_{6,7,8}\left(\left\{a\right\}\right)=0.1,m_{6,7,8}\left(\left\{b\right\}\right)=0.9$ will be obtained, which is actually the same as the original mass function $m_7$. Therefore, even if there exists a piece of evidence to make a wrong judgment, the combined result can be further corrected by an additional evidence combination. Thus, it can be considered that the deviation caused by $m_1$ is corrected by an additional evidence combination.

Finally, various conflict measurements are applied to Examples 1–3, including the classical conflict coefficient [28], $k_{1,2}$; Jousselme distance [19], $d(m_1,m_2)$; pignistic probability distance [12], $difBet{P}_{m_1}^{m_2}$; plausibility conflict [43], $P{l}_{-}C\left(m_1,m_2\right)$; Song et al. [44], $1-cor\left(m_1,m_2\right)$; conflict coefficient [18], $1-r_{BPA}\left(m_1,m_2\right)$; Xiao’s reinforced divergence measure [48], $\mathcal{RB}\left(m_1,m_2\right)$; association coefficient [45], $1-a_{BPA}\left(m_1,m_2\right)$; evidential conflict coefficient [46], $1-ECC\left(m_1,m_2\right)$; dissimilarity measure [42], $Diss\left(m_1,m_2\right)$; belief conflict coefficient [47], $K_{\mathcal{BSC}}\left(m_1,m_2\right)$. The results are shown in

Table 1. The results of various conflict measurements in Examples 1–3.

Conflict Measurement	Example 1 (It Violates the Intuition)	Example 2 (It Violates the Intuition)	Example 3 (It Satisfies the Intuition)
$k_{1,2}$ [28]	0.99	0.36	0.82
$d(m_1,m_2)$ [19]	0.90	0.20	0.80
$difBet{P}_{m_1}^{m_2}$ [12]	0.90	0.20	0.80
$P{l}_{-}C\left(m_1,m_2\right)$ [43]	0.90	0	0.80
$1-cor\left(m_1,m_2\right)$ [44]	0.73	0.04	0.60
$1-r_{BPA}\left(m_1,m_2\right)$ [18]	0.99	0.06	0.78
$\mathcal{RB}\left(m_1,m_2\right)$ [48]	0.95	0.45	0.73
$1-a_{BPA}\left(m_1,m_2\right)$ [45]	0.99	0.06	0.78
$1-ECC\left(m_1,m_2\right)$ [46]	0.99	0.11	0.95
$Diss\left(m_1,m_2\right)$ [42]	0.98	0.23	0.82
$K_{\mathcal{BSC}}\left(m_1,m_2\right)$ [47]	0.99	0.11	0.95

, where it can be seen that the values of conflict measurements in Examples 1 and 3 are large, which indicates that they assume that the degree of conflict between the two pieces of evidence is relatively high, whereas they are relatively small in Example 2, especially the value of zero for plausibility conflict, which demonstrates that these measures assume little or no conflict.

Still, the combination result of Example 3 is intuitive, whereas that of Example 1 is obviously unacceptable. Similarly, the two conflict measurements assume little conflict between the two pieces of evidence in Example 2, but the result is obviously counterintuitive. Therefore, there is no direct relationship between the values of these conflict measurements and the counterintuitive results.

Further, according to Claim 1, the essential conflict value, which is highly related to a counterintuitive combination result, should be higher in Example 2 than in Example 3, but the values of the current conflict measurements in Example 2 are significantly less than those in Example 3. Thus, these conflict measurements cannot fully represent the essential conflict in a desirable way.

5. Formal Definition and Measurement

This section formally defines the essential conflict, illustrates it with examples, and proposes a quantitative measurement.

5.1. Essential Conflict

By analyzing the focal element set of the mass function corresponding to the two pieces of evidence in Examples 1–3, it can be found that for Examples 1 and 2, there is at least one focal element of one mass function whose intersection with any focal elements of another mass function represents an empty set, while for Example 3, there is no such focal element. This conclusion is critical to the definition of essential conflict.

Definition 5.

Assume that $m_1$ and $m_2$ denote two mass functions over a frame of discernment Θ, with focal element sets $F_1$ and $F_2$, respectively. Then, ${\mathrm{{\rm Y}}}_{1,2}\subseteq \Theta$ is a set of essential conflict elements with mass functions $m_1$ and $m_2$ if and only if for any $\omega \in {\mathrm{{\rm Y}}}_{1,2}$, there exists $A\in F_i\wedge \omega \in A$, such that for any $B\in F_j$, it holds $A\cap B=⌀$ ($i\ne j$ and $i,j\in \{1,2\}$).

In addition, if ${\mathrm{{\rm Y}}}_{1,2}\ne ⌀$, then $m_1$ and $m_2$ are in essential conflict.

Corollary 1.

Assume $m_1$ and $m_2$ are two mass functions over a frame of discernment Θ, with focal element sets $F_1$ and $F_2$, respectively, and ${\mathrm{{\rm Y}}}_{1,2}$ is a set of essential conflict elements with mass functions $m_1$ and $m_2$. Then, if there is $A\in F_i$ such that for any $B\in F_j$, it holds that $A\cap B=⌀$ ($i\ne j$ and $i,j\in \{1,2\}$), and for any $\omega \in A$, by Definition 5, it holds $\omega \in {\mathrm{{\rm Y}}}_{1,2}$. Thus, $A\subseteq {\mathrm{{\rm Y}}}_{1,2}$.

According to Definition 5 and Corollary 1, this study proposes Algorithm 1 to determine the set of essential conflict elements of two pieces of evidence.

Algorithm 1 Determining the set of essential conflict elements ${\mathrm{{\rm Y}}}_{1,2}$

Require: The two mass functions $m_1$, $m_2$ Ensure: The set of essential conflict elements ${\mathrm{{\rm Y}}}_{1,2}$  1: Determine the focal element sets of $m_1$ and $m_2$, denoted by $F_1$ and $F_2$ respectively  2: ${\mathrm{{\rm Y}}}_{1,2}\leftarrow ⌀$  3: ${\mathrm{{\rm Y}}}_{1,2}\leftarrow \mathrm{checkingFocalElement}(F_1,F_2,{\mathrm{{\rm Y}}}_{1,2})$  4: ${\mathrm{{\rm Y}}}_{1,2}\leftarrow \mathrm{checkingFocalElement}(F_2,F_1,{\mathrm{{\rm Y}}}_{1,2})$  5: return ${\mathrm{{\rm Y}}}_{1,2}$

Overview. Algorithm 1 includes the following three procedures: (1) determine the set of focal elements that correspond to each piece of evidence; (2) check the focal elements of a mass function $m_1$; (3) check the focal elements of a mass function $m_2$. The process of checking the focal elements is shown in Algorithm 2, where it can be seen that the algorithm iterates through all the focal elements of the first mass function. Suppose there is a focal element whose intersection with all focal elements of the second mass function is an empty set. Then, all the states contained in this element are added to the set of essential conflict elements.

Algorithm 2 checkingFocalElement

Require: The focal element sets $F_1$, $F_2$; the set of essential conflict elements ${\mathrm{{\rm Y}}}_{1,2}$ Ensure: The set of essential conflict elements ${\mathrm{{\rm Y}}}_{1,2}$   1: for each focal element $f_1\in F_1$ do   2:       $\mathit{Flag}\leftarrow 0$   3:       for each focal element $f_2\in F_2$ do   4:             if $f_1\cap f_2\ne ⌀$ then   5:                $\mathit{Flag}\leftarrow 1$   6:                break   7:             end if   8:       end for   9:       if $\mathit{Flag}=0$ then 10:             for each state $\omega \in f_1$ do 11:                ${\mathrm{{\rm Y}}}_{1,2}\leftarrow {\mathrm{{\rm Y}}}_{1,2}\cup \left\{\omega \right\}$ 12:             end for 13:       end if 14: end for 15: return ${\mathrm{{\rm Y}}}_{1,2}$

For clarity, Figure 1 presents a flowchart that visually summarizes the decision-making process for identifying essential conflict between two pieces of evidence, as defined in Definition 5 and and Corollary 1.

Next, apply Algorithm 1 to Example 1. It can be found that focal element sets $F_1$ and $F_2$ corresponding to $m_1$ and $m_2$, respectively, where $F_1$=$\left\{\right\{a\},\{b\left\}\right\}$, $F_2=\{\left\{b\right\},\left\{c\right\}\}$. All the focal elements in $F_1$ and $F_2$ are checked, and it is found that:

• $\left\{a\right\}$ is in $F_1$, and the intersection of $\left\{a\right\}$ and all the focal elements in $F_2$ is an empty set;
• $\left\{c\right\}$ is in $F_2$, and the intersection of $\left\{c\right\}$ and all the focal elements in $F_1$ is an empty set.

Add states a and c to the set of essential conflict elements ${\mathrm{{\rm Y}}}_{1,2}$, so ${\mathrm{{\rm Y}}}_{1,2}=\{a,c\}$. It is clear that ${\mathrm{{\rm Y}}}_{1,2}$ is not empty. According to Definition 5, $m_1$ and $m_2$ are in essential conflict.

Similarly, for Example 2, the set of essential conflict elements is ${\mathrm{{\rm Y}}}_{1,2}=\{b,c\}$, and according to Definition 5, $m_1$ and $m_2$ in Example 2 are in essential conflict. However, for Example 3, the set of essential conflict elements is ${\mathrm{{\rm Y}}}_{1,2}=⌀$. Therefore, according to Definition 5, $m_1$ and $m_2$ in Example 3 are not in essential conflict. Consequently, counterintuitive results occur in Examples 1 and 2, but not in Example 3.

It should be noted that if the number of focal elements is large, determining the set of essential conflict elements and deciding whether the two mass functions are in essential conflict can be time-consuming. In view of that, this study proposes the following equivalence theorem.

Theorem 1.

Assume ${\mathrm{{\rm Y}}}_{1,2}\subseteq \Theta$ is a set of essential conflict elements with mass functions $m_1$ and $m_2$, and the corresponding focal element sets are $F_1$ and $F_2$, respectively; also, $U_1$ and $U_2$ are the union sets of all states of all focal elements belonging to $F_1$ and $F_2$, respectively. Then, it holds that

$${\mathrm{{\rm Y}}}_{1,2}\subseteq (U_1\cup U_2)-(U_1\cap U_2).$$

Proof. 

If ${\mathrm{{\rm Y}}}_{1,2}=⌀$, then ${\mathrm{{\rm Y}}}_{1,2}\subseteq (U_1\cup U_2)-(U_1\cap U_2)$.

If ${\mathrm{{\rm Y}}}_{1,2}\ne ⌀$, according to Definition 5, for any $\omega \in {\mathrm{{\rm Y}}}_{1,2}$, there exists $A\in F_i\wedge \omega \in A$ for $i\in 1,2$. Because $U_i$ represents a union set of all states of all focal elements belonging to $F_i$ for $i\in 1,2$, it holds that $A\subseteq U_1\cup U_2$. In addition, since $\omega \in A$, it holds that $\omega \in U_1\cup U_2$. Because $\omega \in U_1\cup U_2$ for any $\omega \in {\mathrm{{\rm Y}}}_{1,2}$, it holds that ${\mathrm{{\rm Y}}}_{1,2}\subseteq U_1\cup U_2$.

Next, prove that ${\mathrm{{\rm Y}}}_{1,2}\cap (U_1\cap U_2)=⌀$. Because $U_i$ represents a union set of all states of all focal elements belonging to $F_i$ for $i\in 1,2$, for any ${\omega }^{\prime }\in U_1\cap U_2$, there exists a focal element $A\in F_1$∧${\omega }^{\prime }\in A$ and a focal element $B\in F_2\wedge {\omega }^{\prime }\in B$. Thus, it holds that $A\cap B={\omega }^{\prime }$ (i.e., $A\cap B\ne ⌀$). In addition, since for any ${\omega }^{\prime }\in U_1\cap U_2$ and for any focal element A that satisfies the condition of ${\omega }^{\prime }\in A\wedge A\in F_1$, there exists a focal element B that satisfies the condition of ${\omega }^{\prime }\in B\wedge B$ in $F_2$, then $A\cap B$$\ne ⌀$. Thus, according to Definition 5, it holds that ${\omega }^{\prime }\notin {\mathrm{{\rm Y}}}_{1,2}$. Further, due to the fact that for any ${\omega }^{\prime }\in U_1\cap U_2$, ${\omega }^{\prime }\notin {\mathrm{{\rm Y}}}_{1,2}$, it holds that ${\mathrm{{\rm Y}}}_{1,2}\cap (U_1\cap U_2)=⌀$. Further, since ${\mathrm{{\rm Y}}}_{1,2}\subseteq U_1\cup U_2$ and ${\mathrm{{\rm Y}}}_{1,2}\cap (U1\cap U2)=⌀$, it holds that ${\mathrm{{\rm Y}}}_{1,2}\subseteq (U_1\cup U_2)-(U_1\cap U_2)$. □

Based on Theorem 1, this study introduces a simpler way to find the conflict element set. First, find different elements between the union set of all states of all focal elements by $(U_1\cup U_2)-(U_1\cap U_2)$. Then, check whether each focal element of each mass function is a subset of $(U_1\cup U_2)-(U_1\cap U_2)$. If there is such a focal element, then all states contained in it are added to the set of essential conflict elements.

5.2. A Quantity Measurement of Essential Conflict

This study proposes a quantity measurement of essential conflict based on Definition 5, which is called the degree of essential conflict.

Definition 6.

For a frame Θ, assume ${\mathrm{{\rm Y}}}_{1,2}\subseteq \Theta$ is a set of essential conflict elements with mass functions $m_1$ and $m_2$. Then, the degree of essential conflict between these two mass functions is defined by:

$$\kappa (m_1,m_2)=\sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}(m_1\left(A\right)+m_2\left(A\right))-\sum _{A,B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\left(m_1\left(A\right)m_2\left(B\right)\right),$$

(8)

where ${\displaystyle \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}}(m_1\left(A\right)+m_2\left(A\right))$ represents the total support degree of mass function $m_i$ for states ruled out in the combination process. Moreover, since the mass value of ${\displaystyle \sum _{A,B\subseteq {\mathrm{{\rm Y}}}_{1,2}}}{m}_1\left(A\right)m_2\left(B\right)$ is double counted, $-{\displaystyle \sum _{A,B\subseteq {\mathrm{{\rm Y}}}_{1,2}}}{m}_1\left(A\right)m_2\left(B\right)$ is required.

Unlike existing conflict measurements, which require setting a subjective threshold to distinguish conflict situations, in this study, according to Definition 6, mass functions are considered to be in conflict if their degree of essential conflict is greater than zero. Moreover, based on Definition 5 and Corollary 1, for any focal element that is a subset of the set of essential conflict elements, regardless of the mass value assigned to it by one of the original mass functions, its combined mass value will be zero according to Dempster’s combination rule. Therefore, all states contained in such a focal element are ruled out in the combination process. Consequently, when a focal element is a subset of the set of essential conflict elements, the larger the mass value assigned to the focal element by one of the original mass functions, the more the combination result will violate intuition and the greater the degree of essential conflict will be.

Finally, by further comparing the formulas of $\kappa (m_1,m_2)$ and $k_{1,2}$, it can be found that when mass functions $m_1$ and $m_2$ are in essential conflict, $\kappa (m_1,m_2)$ has a certain quantitative relationship with $k_{1,2}$.

Theorem 2.

Assume $m_1$ and $m_2$ are two mass functions over a frame of discernment Θ that are in essential conflict, having a degree of essential conflict $\kappa (m_1,m_2)$ and a classical conflict coefficient $k_{1,2}$; then, $\kappa (m_1,m_2)\le k_{1,2}$.

Proof. 

By Definition 1 and Equation (8), we have

$$\begin{array}{cc}\hfill \kappa (m_1,m_2)=& \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B\in F_2}{m}_1\left(A\right)m_2\left(B\right)+\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{A\in F_1}{m}_1\left(A\right)m_2\left(B\right)-\sum _{A,B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\left(m_1\left(A\right)m_2\left(B\right)\right)\hfill \\ \hfill =& \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right).\hfill \end{array}$$

Next, according to Definition 5, for any $A\in F_i\wedge A\subseteq {\mathrm{{\rm Y}}}_{1,2}$, it holds that $A\cap B=⌀$ for any $B\in F_j$ ($i\ne j$ and $i,j\in \{1,2\}$). Thus, based on Equation (6),

$$\begin{array}{cc}\hfill k_{1,2}=& \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)\hfill \\ & +\sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{A\cap B=⌀\wedge A⊈{\mathrm{{\rm Y}}}_{1,2}\wedge B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right).\hfill \end{array}$$

By Definition 1, for any $A,B\subseteq \Theta$, it holds that $m_1\left(A\right)\in [0,1]$, $m_2\left(B\right)\in [0,1]$. Thus

$$\begin{array}{c}\hfill \sum _{A\cap B=⌀\wedge A⊈{\mathrm{{\rm Y}}}_{1,2}\wedge B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)>0.\end{array}$$

Thus, $\kappa (m_1,m_2)\le k_{1,2}$. □

According to Theorem 2, $k_{1,2}$ will never be smaller than $\kappa (m_1,m_2)$, which arises two problems related to $k_{1,2}$. First, when the combination result of the two mass functions satisfies the intuition, the conflict value of $k_{1,2}$ might be high, such as in Example 3, where $k_{1,2}$ reaches 0.82. Therefore, $k_{1,2}$ might over-evaluate the counterintuitive combination results. Second, since $\kappa (m_1,m_2)$ precisely describes the relationship between conflict and the counterintuitive combination result, $\kappa (m_1,m_2)$ can determine the essential conflict directly based on whether it is greater than zero. However, since $k_{1,2}$ always exceeds $\kappa (m_1,m_2)$, a subjective threshold is necessary to distinguish between conflict and non-conflict situations.

6. Evaluation

This study analyzes the two properties of essential conflict and shows that they denote direct and essential causes of counterintuitive combination results. The essential conflict is compared with other conflict measurements in Examples 1–3 to show that the proposed conflict measurement can more effectively avoid counterintuitive results with Dempster’s combination rule than the current existing measurements. The rationality and validity of the proposed measurement are verified by revealing several desirable properties that cannot be fully satisfied by the existing conflict measurements.

6.1. Properties of Essential Conflict

It is demonstrated that the two properties of essential conflict, namely belief absolutization and uncorrectable assertion, can lead to counterintuitive results when mass functions are in essential conflict and Dempster’s combination rule is used.

Theorem 3

(Belief Absolutization). Assume $m_1$ and $m_2$ are two mass functions over a frame of discernment Θ that are in essential conflict, with a combination result $m_{1,2}$ according to Dempster’s combination rule; $Bet{P}_{m_1}$, $Bet{P}_{m_2}$, and $Bet{P}_{m_{1,2}}$ denote the pignistic probability functions of $m_1$, $m_2$, and $m_{1,2}$, respectively. If $m_1$ and $m_2$ are in essential conflict, then there is $\omega \in \Theta$, such that $Bet{P}_{m_1}\left(\omega \right)>0$ or $Bet{P}_{m_2}\left(\omega \right)>0$ but $Bet{P}_{m_{1,2}}\left(\omega \right)=0$.

Proof. 

Suppose $F_1$ and $F_2$ are the focal element sets of $m_1$ and $m_2$, respectively; ${\mathrm{{\rm Y}}}_{1,2}$ is a set of essential conflict elements. According to Definition 5, for any ${\omega }^{\prime }\in {\mathrm{{\rm Y}}}_{1,2}$, there is $A\in F_i\wedge {\omega }^{\prime }\in A$, such that for any $B\in F_j$, it holds that $A\cap B=⌀$ ($i\ne j$ and $i,j\in \{1,2\}$). Without loss of generality, it can be assumed that $A\in F_1$. Then, according to Definition 1, it holds that $m_1\left(A\right)>0$.

Further, for any $\omega \in A$, according to Definition 3, it holds that $Bet{P}_{m_1}\left(\omega \right)>0$. Since $\forall B\in F_2,A\cap B=⌀$ and $\omega \in A$, then for any $C\in F_1$ and any $B\in F_2$, it holds that $\omega \notin C\cap B$. Thus, for any $E\in F_1$ that satisfies the condition of $\omega \in E$, it holds that ${\displaystyle \sum _{X\bigcap Y=E}}{m}_1\left(X\right)m_2\left(Y\right)=0$. Furthermore, according to Definitions 3 and 4, for any $E\in F_1$ that satisfies the condition of $\omega \in E$, it holds that $m_{1,2}\left(E\right)=0$ and $Bet{P}_{m_{1,2}}\left(\omega \right)=0$. □

Corollary 2.

If $m_1$ and $m_2$ are in essential conflict, then for any $\omega \in {\mathrm{{\rm Y}}}_{1,2}$, it holds that $Bet{P}_{m_1}\left(\omega \right)>0$ or $Bet{P}_{m_2}\left(\omega \right)>0$, but $Bet{P}_{m_{1,2}}\left(\omega \right)=0$.

In this study, $Bet{P}_m\left(\omega \right)$ is referred to as betting commitment to a state $\omega$ by a mass function m. For any $\omega \in A$, $Bet{P}_{m_i}\left(\omega \right)>0$ for $i\in 1,2$ indicates that, according to one of the mass functions, there is a certain chance that $\omega$ is the real state, and $Bet{P}_{m_{1,2}}\left(\omega \right)=0$ indicates that this scenario is impossible. Based on the above-presented proof and Corollary 2, it can be concluded that for two pieces of evidence that are in essential conflict, there exists a focal element A of certain evidence regardless of how much mass value the evidence assigns to A, as long as set A is a subset of the set of essential conflict elements (i.e., for any $\omega \in A,\omega \in {\mathrm{{\rm Y}}}_{1,2}$); then, the result obtained based on Dempster’s combination rule still excludes it and its more special subsets from the set of possible states. In other words, the combined mass value of any subset of any $A\subseteq {\mathrm{{\rm Y}}}_{1,2}$ is zero based on Dempster’s combination rule.

Thus, Theorem 3 indicates that when two pieces of evidence that are in essential conflict with Dempster’s combination rule are combined, at least one possible state will change into an impossible state. This combination result is counterintuitive. Particularly, from Corollary 2 implies that, after combination, all states are contained in the set of essential conflict elements that will change from a possible state to an impossible state.

Furthermore, if only one possible state of a frame does not belong to the set of essential conflict elements, then it will become necessary after the combination process.

Theorem 4.

Assume $m_1$ and $m_2$ are two mass functions over a frame of discernment Θ that are in essential conflict, $m_{1,2}$ is a combination result of $m_1$ and $m_2$ obtained based on Dempster’s combination rule, ${\mathrm{{\rm Y}}}_{1,2}\subseteq \Theta$ is a set of conflict elements with mass functions $m_1$ and $m_2$, and $U_1\cup U_2$ is a union set of all states of all focal elements of mass functions $m_1$ and $m_2$. Then, if there is $\omega \in \Theta$ such that $\left\{\omega \right\}=(U_1\cup U_2)-{\mathrm{{\rm Y}}}_{1,2}$, it holds that $m_{1,2}\left(\left\{\omega \right\}\right)=1$.

Proof. 

Based on Corollary 2, for any ${\omega }^{\prime }\in {\mathrm{{\rm Y}}}_{1,2}$, it holds that $Bet{P}_{m_1}\left({\omega }^{\prime }\right)>0$ or $Bet{P}_{m_2}\left({\omega }^{\prime }\right)>0$, but $Bet{P}_{m_{1,2}}\left({\omega }^{\prime }\right)=0$. Thus, according to Definition 4, for any subset $A\subseteq \Theta$ that satisfies ${\omega }^{\prime }\in A\wedge {\omega }^{\prime }\in {\mathrm{{\rm Y}}}_{1,2}$, it holds that $m_{1,2}\left(A\right)=0$, which yields ${\sum }_{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_{1,2}\left(A\right)=0$. Further, according to Definition 1, it holds that ${\sum }_{E\subseteq \Theta }{m}_{1,2}\left(E\right)=1$. Then, if there is $\omega \in \Theta$ such that $\left\{\omega \right\}=(U_1\cup U_2)-{\mathrm{{\rm Y}}}_{1,2}$, it holds that ${\sum }_{B\subseteq \Theta \wedge \omega \in B}{m}_{1,2}\left(B\right)=1$. It should be noted that if a subset B contains other states in addition to state $\omega$, the other states must belong to ${\mathrm{{\rm Y}}}_{1,2}$, and $m_{1,2}\left(B\right)=0$; thus, $m_{1,2}\left(\left\{\omega \right\}\right)=1$. □

Theorem 4 represents the reason why a possible state in Examples 1 and 2 is a necessary true state after the combination process. Therefore, the property of belief absolutization and Theorem 4 are claimed.

Claim 2.

If two mass functions are in essential conflict, then a possible state of one will always turn into a necessary true state or an impossible state after the combination process.

Definition 7.

Assume $m_1$ and $m_2$ are two mass functions over a frame of discernment Θ; $m_{1,2}$ is a combination result of $m_1$ and $m_2$ obtained based on Dempster’s combination rule; $Bet{P}_{m_1}$ and $Bet{P}_{m_2}$ the pignistic probability functions of $m_1$ and $m_2$, respectively. Then, the combination result $m_{1,2}$ is correctable if for any $\omega \in \Theta$ and $Bet{P}_{m_i}\left(\omega \right)$$(i\in \{1,2\left\}\right)$, and an additional mass function $m_3$ such that

$$\begin{array}{c}\hfill Bet{P}_{m_{1,2,3}}\left(\omega \right)=Bet{P}_{m_i}\left(\omega \right)\end{array}$$

can be defined for a combination result $m_{1,2,3}$ of $m_1$, $m_2$, and $m_3$.

In addition, $Bet{P}_{m_{1,2,3}}\left(\omega \right)=Bet{P}_{m_i}\left(\omega \right)$ for $i\in \{1,2\}$ indicates that a combination result $m_{1,2,3}$ shares the same judgment with $m_i(i\in \{1,2\})$ about the chance that $\omega$ represents a true state. This means that regardless of what $m_{1,2}$ is, as long as the combined result is correctable, it is possible to find additional evidence to combine it with it further and obtain the same combination result as $m_i(i\in \{1,2\})$. Therefore, even if $m_i$ is faulty evidence, as long as the combined result is correctable, the effects of $m_i$ can be eliminated in the subsequent combination process with additional evidence.

The property of being correctable will be illustrated in the following example. Assume that in Example 3, Sensor 1 strongly supports state a, with $m_1\left(\left\{a\right\}\right)=0.9$ and $m_1\left(\left\{b\right\}\right)=0.1$, while Sensor 2 strongly supports state b, with $m_2\left(\left\{a\right\}\right)=0.1$ and $m_2\left(\left\{b\right\}\right)=0.9$.

In addition, assume that b is a real state; namely, Sensor 1 provides incorrect evidence. A combination with such incorrect evidence equally supports states a and b, that is, $m_{1,2}\left(\left\{a\right\}\right)=m_{1,2}\left(\left\{b\right\}\right)=0.5$. Sensor 2 judges that $Bet{P}_{m_2}\left(a\right)=0.1$ and $Bet{P}_{m_2}\left(b\right)=0.9$, but the combination result is $Bet{P}_{m_{1,2}}\left(a\right)=Bet{P}_{m_{1,2}}\left(b\right)=0.5$. However, if Sensor 3 has a mass function $m_3$, with $m_3\left(\left\{a\right\}\right)=0.1$ and $m_3\left(\left\{b\right\}\right)=0.9$, and $m_{1,2}$ is combined with $m_3$, a new combination result with $Bet{P}_{m_{1,2,3}}\left(a\right)=0.1$ and $Bet{P}_{m_{1,2,3}}\left(b\right)=0.9$ can be obtained. Hence the deviation caused by $m_1$ can be corrected by an additional evidence combination whose result $m_{1,2}$ is correctable. The property of being correctable is desirable because certain sources of information might provide incorrect judgments.

However, not all combination results are correctable. For instance, for Zadeh’s counterexample in Example 1, the combination result obtained by combining mass functions $m_1$ and $m_2$ is $m_{1,2}\left(\left\{b\right\}\right)=1$. According to Definition 3, for any mass function $m_3$ with $k_{1,2,3}<1$, even if the new mass function strongly supports state a or c, the combination result of $m_{1,2}$ and $m_3$ is $m_{1,2,3}\left(\left\{b\right\}\right)=1$. Therefore, according to Definition 7, the combination result $m_{1,2}$ in Example 1 is uncorrectable.

For Example 1, with mass functions $m_1$ and $m_2$ of Sensors 1 and 2, respectively, assume the real state is a. Sensor 1 makes a correct judgment with $Bet{P}_{m_1}\left(a\right)=0.9$ and $Bet{P}_{m_1}\left(b\right)=0.1$, while for Sensor 2, $Bet{P}_{m_2}\left(a\right)=0$, and the combination yields $Bet{P}_{m_{1,2}}\left(a\right)=0$. Then, if all subsequent mass functions of the other sensors strongly support state a, the combinations are consistent with $m_{1,2}$, excluding the possibility that a is the real state, and the error caused by Sensor 2 cannot be corrected.

The following theorem shows that the result of combining two pieces of evidence in essential conflict must be uncorrectable.

Theorem 5

(Uncorrectable Assertion). Assume $m_1$ and $m_2$ are mass functions over a frame of discernment Θ that are in essential conflict, with a combination result $m_{1,2}$ obtained based on Dempster’s combination rule. Also, assume that $Bet{P}_{m_1}$ and $Bet{P}_{m_2}$ are pignistic probability functions of $m_1$ and $m_2$, respectively. Then, there is a state $\omega \in \Theta$ and $Bet{P}_{m_i}\left(\omega \right)$ ($i\in \{1,2\}$) such that for any mass function $m_3$ with $m_3(\Theta )\ne 1$, it holds that:

$$\begin{array}{c}\hfill Bet{P}_{m_{1,2,3}}\left(\omega \right)\ne Bet{P}_{m_i}\left(\omega \right).\end{array}$$

Proof. 

Suppose $F_1$ and $F_2$ are the focal element sets of $m_1$ and $m_2$, respectively; $m_{1,2}$ is their combination result obtained based on Dempster’s combination rule; ${\mathrm{{\rm Y}}}_{1,2}$ is a set of essential conflict elements. According to Definition 5, for any ${\omega }^{\prime }\in {\mathrm{{\rm Y}}}_{1,2}$, there exists $A\in F_i\wedge {\omega }^{\prime }\in A$ such that for any $B\in F_j$, it holds that $A\cap B=⌀(i\ne j$ and $i,j\in \{1,2\})$. Without loss of generality, this study assumes $A\in F_1$. According to Definition 1, it holds that $m_1\left(A\right)>0$, and for any $\omega \in A$, based on Definition 4, it holds that $Bet{P}_{m_1}\left(\omega \right)>0$. Further, since ${\displaystyle \sum _{X\bigcap Y=A}}{m}_1\left(X\right)m_2\left(Y\right)=0$, based on Definitions 3 and 4, it holds that $m_{1,2}\left(A\right)=0$ and $Bet{P}_{m_{1,2}}\left(\omega \right)=0$. Furthermore, according to Definition 4 and $Bet{P}_{m_{1,2}}\left(\omega \right)=0$, for any $T\subseteq \Theta$ that satisfies the condition of $\omega \in T$, it holds that $m_{1,2}\left(T\right)=0$. Afterward, according to Definition 3, for any mass function $m_3$, the mass value of the combination result obtained based on Dempster’s combination rule with mass functions $m_{12}$ and $m_3$ is given by:

$$\begin{array}{c}\hfill m_{1,2,3}\left(T\right)={\displaystyle \frac{{\displaystyle \sum _{A\bigcap B=T}}{m}_{1,2}\left(A\right)m_3\left(B\right)}{1-{\displaystyle \sum _{A\bigcap B=⌀}}{m}_{1,2}\left(A\right)m_3\left(B\right)}}={\displaystyle \frac{{\displaystyle \sum _{A\bigcap B=T}}0·m_3\left(B\right)}{1-{\displaystyle \sum _{A\bigcap B=⌀}}{m}_{1,2}\left(A\right)m_3\left(B\right)}}=0.\end{array}$$

Thus, based on Definition 4,

$$\begin{array}{c}\hfill Bet{P}_{m_{1,2,3}}\left(\omega \right)=\sum _{T\subseteq \Theta ,\omega \in T}{\displaystyle \frac{1}{\left|T\right|}}{\displaystyle \frac{m_{1,2,3}\left(T\right)}{1-m_{1,2,3}(⌀)}}=0.\end{array}$$

Finally, since $Bet{P}_{m_1}\left(\omega \right)>0$ and $Bet{P}_{m_{1,2,3}}\left(\omega \right)=0$, it holds that $Bet{P}_{m_1}\left(\omega \right)\ne Bet{P}_{m_{1,2,3}}\left(\omega \right)$, and the theorem is proven. □

Theorem 3 states that if two mass functions are in essential conflict, then at least one possible state $\omega$ is impossible after their combination, regardless of how much correct evidence is subsequently combined. The property of uncorrectable assertions can be claimed as follows.

Claim 3.

If two mass functions are in essential conflict, the betting commitments of some states for their combination result cannot be corrected by further combination with any new mass function.

Analyzing Claims 2 and 3 together, it can be found that if two mass functions are in essential conflict, when Dempster’s combination rule is applied to combine them, at least one possible state will become impossible or necessary true and will not be corrected by further combination, which is obviously counterintuitive.

Claims 2 and 3 reveal the relationship between the essential conflict and counterintuitive combination results as follows: the essential conflict will inevitably lead to a counterintuitive result when Dempster’s combination rule is adopted. Claims 2 and 3 state belief absolutization and uncorrectable assertions, respectively, and they represent direct and essential causes of essential conflict that yields counterintuitive combination results.

Hence, the following relationship between essential conflict and counterintuitive results can be stated.

Claim 4.

If two mass functions are in essential conflict, using Dempster’s combination rule will yield a counterintuitive result.

6.2. Applicability Analysis

According to Definition 6, the degree of essential conflict is zero when two mass functions are not in essential conflict, but it is larger than zero when they are in essential conflict. For the latter case, according to Claim 4, using Dempster’s combination rule will yield a counterintuitive result, so in such a case, this rule cannot be used safely.

The applicability of the degree of essential conflict can be explained in Examples 1–3, starting with Zadeh’s counterexample (Example 1). Use Algorithm 1 to obtain a set of essential conflict elements, ${\mathrm{{\rm Y}}}_{1,2}=\{a,c\}$. Then, according to Definition 6, it holds that:

$$\begin{array}{cc}\hfill \kappa (m_1,m_2)=& \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}(m_1\left(A\right)+m_2\left(A\right))-\sum _{A,B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\left(m_1\left(A\right)m_2\left(B\right)\right)\hfill \\ \hfill =& m_1\left(\left\{a\right\}\right)+m_2\left(\left\{c\right\}\right)-m_1\left(\left\{a\right\}\right)m_2\left(\left\{c\right\}\right)\hfill \\ \hfill =& 0.9+0.9-0.9*0.9\hfill \\ \hfill =& 0.99.\hfill \end{array}$$

Next, applying Algorithm 1 to Example 2 yields ${\mathrm{{\rm Y}}}_{1,2}=\{b,c\}$, and according to Definition 6, it holds that $\kappa (m_1,m_2)=0.36$. Further, apply Algorithm 1 to Example 3 to obtain ${\mathrm{{\rm Y}}}_{1,2}=⌀$; according to Definition 6, it holds that $\kappa (m_1,m_2)=0$.

The degree of essential conflict in Example 3 is zero, but it is larger than zero in Examples 1 and 2, which could imply that the mass function combination based on Dempster’s combination rule should be rejected. For Example 3, Dempster’s rule can be used without obtaining a counterintuitive result. Unlike the proposed measurement, the other conflict measurements decide on the safe use of Dempster’s combination rule by comparing the value of conflict measurement to a predefined threshold, but it is not directly specified whether using this rule will produce counterintuitive results.

To further illustrate the applicability and scalability of the proposed essential conflict framework in multi-sensor fusion scenarios, a target recognition example involving multiple sensors is presented.

Example 4.

Consider a target recognition problem in which three types of aircraft are defined: airliner ($E_1$), bomber ($E_2$), and fighter ($E_3$). These hypotheses constitute the frame of discernment $\Theta =\{E_1,E_2,E_3\}$. Five independent radar sensors, denoted by $R_1$ to $R_5$, monitor an unknown aircraft and transform their observations into mass functions $m_1$ to $m_5$, respectively. The corresponding mass assignments are listed in

Table 2. The mass functions based on radars.

m	$\left\{{\mathit{E}}_1\right\}$	$\left\{{\mathit{E}}_2\right\}$	$\left\{{\mathit{E}}_3\right\}$	$\{{\mathit{E}}_1,{\mathit{E}}_2,{\mathit{E}}_3\}$
$m_1$	0.4	0.6	0	0
$m_2$	0	0.7	0.3	0
$m_3$	0.85	0	0	0.15
$m_4$	0.4	0.6	0	0
$m_5$	0.75	0	0	0.25

.

The essential conflict relationships among these pieces of evidence are first examined. According to Definition 5, it can be verified that $m_2$ is in essential conflict with $m_1$, $m_3$, $m_4$, and $m_5$. Specifically, the corresponding set of essential conflict elements is either $\mathrm{{\rm Y}}=\{E_1,E_3\}$ or $\mathrm{{\rm Y}}=\left\{E_1\right\}$.

Following the principle of preserving as much reliable information as possible, the mass functions $m_1$, $m_3$, $m_4$ and $m_5$ are first combined using Dempster’s combination rule. The resulting mass function is given by

$$\begin{array}{c}\hfill m_{1345}\left(\left\{E_1\right\}\right)=0.9222,m_{1345}\left(\left\{E_2\right\}\right)=0.0778.\end{array}$$

Next, the degree of essential conflict between $m_{1345}$ and $m_2$ is evaluated. The set of essential conflict elements is $\mathrm{{\rm Y}}=\{E_1,E_3\}$. According to Definition 6, the degree of essential conflict is computed as

$$\kappa (m_{1345},m_2)=m_{1345}\left(\left\{E_1\right\}\right)+m_2\left(\left\{E_3\right\}\right)-m_{1345}\left(\left\{E_1\right\}\right)·m_2\left(\left\{E_3\right\}\right)=0.94554.$$

This large value indicates a severe essential conflict between $m_{1345}$ and $m_2$. When $m_{1345}$ is directly combined with $m_2$ using Dempster’s combination rule, the resulting mass function becomes $m\left(\left\{E_2\right\}\right)=1.$ That is, hypothesis $E_2$ became necessarily true, even though it had been only weakly supported in $m_{1345}$, while the strongly supported hypothesis $E_1$ was completely excluded. This outcome is clearly counterintuitive and constitutes a manifestation of the belief absolutization property induced by essential conflict.

Moreover, if $m_1$ and $m_2$ are combined first, the resulting mass function is $m_{12}\left(\left\{E_2\right\}\right)=1$. This result cannot be altered by any subsequent combination with additional evidence, which directly reflects the uncorrectable assertion property of essential conflict.

This example demonstrates that: (i) essential conflict can arise in multi-sensor fusion scenarios; (ii) ignoring essential conflict inevitably leads to absolute and uncorrectable counterintuitive fusion results; and (iii) the essential conflict criterion (Definition 5) effectively identifies such conflicts, while the proposed degree of essential conflict (Definition 6) quantitatively characterizes their severity, thereby providing a principled basis for evidence pruning or reliability discounting prior to fusion.

Therefore, in practical applications such as target recognition, it is advisable to detect essential conflict in advance and to exclude or appropriately discount conflicting evidence, so as to ensure rational, interpretable, and reliable fusion outcomes.

6.3. Desirable Properties

Destercke and Burger [51] discussed the inherent properties of verifying the effectiveness of conflict measurement tools; therefore, we have found that the degree of essential conflict has the following desirable properties.

Theorem 6.

Assume that $m_1$ and $m_2$ are two mass functions over a frame of discernment Θ, with focal element sets $F_1$ and $F_2$, respectively; also, assume ${\mathrm{{\rm Y}}}_{1,2}\subseteq \Theta$ is a set of essential conflict elements with mass functions $m_1$ and $m_2$, and $\kappa (m_1,m_2)$ is their essential conflict measurement. Then, it holds that:

(i) 

Symmetry: $\kappa (m_1,m_2)=\kappa (m_2,m_1)$;

(ii) 

Boundedness: $\kappa (m_1,m_2)\in [0,1]$;

(iii) 

Extreme consistency: $\kappa (m_1,m_2)=0$ if mass functions $m_1$ and $m_2$ are identical, and $\kappa (m_1,m_2)=1$ if the two mass functions are in total conflict;

(iv) 

Ignorance: If $m_2(\Theta )=1$, then $\kappa (m_1,m_2)=0$;

(v) 

Insensitivity to refinement: If mass functions $m_1$ and $m_2$ are refined from Θ into ${\Theta }^{\prime }$, then $\kappa (m_1^{\Theta },m_2^{\Theta })=\kappa (m_1^{{\Theta }^{\prime }},m_2^{{\Theta }^{\prime }})$.

Proof. 

(i)

According to Definition 9,

$$\begin{array}{c}\hfill \kappa (m_1,m_2)=\sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}(m_1\left(A\right)+m_2\left(A\right))-\sum _{A,B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\left(m_1\left(A\right)m_2\left(B\right)\right)=\kappa (m_2,m_1).\end{array}$$

Thus, item (i) holds;

(ii)

According to Definition 2, for any $A,B\subseteq \Theta$, it holds that $m_1\left(A\right)\in [0,1]$, $m_2\left(B\right)\in [0,1]$, ${\sum }_{A\subseteq \Theta }{m}_1\left(A\right)=1$, and ${\sum }_{A\subseteq \Theta }{m}_2\left(A\right)=1$. Then, it can be written that:

$$\begin{array}{c}\hfill \sum _{A\in F_1}\sum _{B\in F_2}{m}_1\left(A\right)m_2\left(B\right)=1.\end{array}$$

Thus, it holds that:

$$\begin{array}{cc}& \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)\hfill \\ & +\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)=1.\hfill \end{array}$$

Therefore, according to the proof process of Theorem 2,

$$\begin{array}{cc}\hfill \kappa (m_1,m_2)=& \sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)+\sum _{A\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)\hfill \\ & +\sum _{B\subseteq {\mathrm{{\rm Y}}}_{1,2}}\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)\hfill \\ \hfill =& 1-\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right).\hfill \end{array}$$

Since ${\displaystyle \sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}}{\displaystyle \sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}}{m}_1\left(A\right)m_2\left(B\right)\in [0,1]$, then

$$\begin{array}{c}\hfill 1-\sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)\in [0,1].\end{array}$$

Thus, item (ii) holds;

(iii)

If mass functions $m_1$ and $m_2$ are identical, then there is no focal element whose intersection with all focal elements of another mass function is an empty set. Therefore, based on Definition 5, mass functions $m_1$ and $m_2$ are not in essential conflict, and ${\mathrm{{\rm Y}}}_{1,2}=⌀$. Thus, according to Definition 6, $\kappa (m_1,m_2)=0$. If mass functions $m_1$ and $m_2$ are in total conflict, that is, $A\cap B=⌀$ for any $A\in F_1$ and $B\in F_2$, then according to Definition 5, $\kappa (m_1,m_2)=1$. Based on the proof process of Theorem 3, it can be concluded that

$$\begin{array}{c}\hfill k_{1,2}=\kappa (m_1,m_2)+\sum _{A\cap B=⌀\wedge A⊈{\mathrm{{\rm Y}}}_{1,2}\wedge B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right).\end{array}$$

Since $A\cap B=⌀$ for any $A\in F_1$ and $B\in F_2,$ according to Definition 5, $A\subseteq {\mathrm{{\rm Y}}}_{1,2}$ and $B\subseteq {\mathrm{{\rm Y}}}_{1,2}$. Then, it can be written that:

$$\begin{array}{c}\hfill \sum _{A⊈{\mathrm{{\rm Y}}}_{1,2}}\sum _{B⊈{\mathrm{{\rm Y}}}_{1,2}}{m}_1\left(A\right)m_2\left(B\right)=0,\end{array}$$

and $\kappa (m_1,m_2)=1$. Thus, item (iii) holds;

(iv)

If $m_2(\Theta )=1$, then according to Definition 5, ${\mathrm{{\rm Y}}}_{1,2}=⌀$. In addition, according to Definition 6, $\kappa (m_1,m_2)=0$; thus, item (iv) holds;

(v)

If mass functions $m_1$ and $m_2$ are refined from $\Theta$ into ${\Theta }^{\prime }$, but the mass value of the focal element of each mass function remains unchanged, then according to Definition 5, ${\mathrm{{\rm Y}}}_{1,2}^{\Theta }={\mathrm{{\rm Y}}}_{1,2}^{{\Theta }^{\prime }}$. Similarly, according to Definition 6, $\kappa (m_1^{\Theta },m_2^{\Theta })=\kappa (m_1^{{\Theta }^{\prime }},m_2^{{\Theta }^{\prime }})$. Thus, item (v) holds.

□

This study first discusses the significance of the above-mentioned desirable properties and examines whether existing conflict measurements satisfy them. A comparative summary is provided in

Table 3. Different conflict measurements’ inconsistency with the properties identified.

	Property (i)	Property (ii)	Property (iii)	Property (iv)	Property (v)	Time Complexity
$k_{1,2}$ [28]	Yes	Yes	No	Yes	Yes	$O\left(n^2\right)$
$d(m_1,m_2)$ [19]	Yes	Yes	No	No	Yes	$O\left(n^3\right)$
$difBet{P}_{m_1}^{m_2}$ [12]	Yes	Yes	No	No	Yes	$O\left(n\right)$
$P{l}_{-}C\left(m_1,m_2\right)$ [43]	Yes	Yes	No	Yes	Yes	$O\left(n^2\right)$
$1-cor\left(m_1,m_2\right)$ [44]	Yes	Yes	No	No	No	$O\left(n^2\right)$
$1-r_{BPA}\left(m_1,m_2\right)$ [18]	Yes	Yes	Yes	No	Yes	$O\left(n^2\right)$
$\mathcal{RB}\left(m_1,m_2\right)$ [48]	Yes	Yes	Yes	No	Yes	$O\left(n^2\right)$
$1-a_{BPA}\left(m_1,m_2\right)$ [45]	Yes	Yes	Yes	No	Yes	$O\left(n^3\right)$
$1-ECC\left(m_1,m_2\right)$ [46]	Yes	Yes	Yes	No	Yes	$O\left(n\right)$
$Diss\left(m_1,m_2\right)$ [42]	Yes	Yes	Yes	No	Yes	$O\left(n^2\right)$
$K_{\mathcal{BSC}}\left(m_1,m_2\right)$ [47]	Yes	Yes	Yes	No	Yes	$O\left(n\right)$
Proposed measurement	Yes	Yes	Yes	Yes	Yes	$O\left(n^2\right)$

. In addition to the satisfaction of desirable properties, we further investigate the computational efficiency of different conflict measurements. Specifically, the worst-case time complexity of each method is analyzed and reported in Table 3 as an additional comparison criterion. According to the D-S theory, the computational complexity is evaluated with respect to the number of focal elements involved. Let $n=|F_1\cup F_2|$, where $F_1$ and $F_2$ denote the sets of focal elements of $m_1$ and $m_2$, respectively.

• Property (i): Symmetry
  The conflict measurement should be symmetric; namely, the value of conflict measurement should not depend on the order of the two mass functions;
• Property (ii): Boundedness
  The degree of conflict between two mass functions should be bounded; namely, the conflict measurement should have a maximum and minimum value;
• Property (iii): Extreme consistency
  The value of conflict measurement of two mass functions should be maximal if the two mass functions are in total conflict but minimal if they are identical;
• Property (iv): Ignorance
  The value of conflict measurement of the combination of the vacuous mass function and any other mass function should be minimal;
• Property (v): Insensitivity to refinement
  If two mass functions are refined only from one frame to another and the mass value of the focal element of each mass function remains unchanged, then the conflict measurement value should not be affected.

As shown in Table 3, the proposed conflict measurement satisfies all the desirable properties while maintaining a competitive computational complexity. In contrast, existing conflict measurements fail to satisfy one or more properties and may lead to counterintuitive results, which are further illustrated through representative cases in

Table 4. The evidence of different conflict measurements violating the properties.

	Mass Functions	Conflict Value	Violated Property	Our Measurement
$k_{1,2}$ [28]	Case 1	$k_{1,2}=0.64$	Violate property (iii): the value should be minimal.	$\kappa (m_1,m_2)=0$
$d(m_1,m_2)$ [19]	Case 2	$d(m_1,m_2)=0.92$	Violate property (iii): the value should be maximal.	$\kappa (m_1,m_2)=1$
	Case 3	$d(m_1,m_2)=0.71$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$difBet{P}_{m_1}^{m_2}$ [12]	Case 4	$difBet{P}_{m_1}^{m_2}=0.5$	Violate property (iii): the value should be maximal.	$\kappa (m_1,m_2)=1$
	Case 3	$difBet{P}_{m_1}^{m_2}=0.47$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$P{l}_{-}C\left(m_1,m_2\right)$ [43]	Case 2	$P{l}_{-}C\left(m_1,m_2\right)=0.9$	Violate property (iii): the value should be maximal.	$\kappa (m_1,m_2)=1$
$1-cor\left(m_1,m_2\right)$ [44]	Case 2	$1-cor\left(m_1,m_2\right)=0.74$	Violate property (iii): the value should be maximal.	$\kappa (m_1,m_2)=1$
	Case 3	$1-cor\left(m_1,m_2\right)=0.26$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
	Case 5	$1-cor\left(m_1^{\Theta },m_2^{\Theta }\right)=0.39$	Violate property (v): the value should not be affected.	$\kappa \left(m_1^{\Theta },m_2^{\Theta }\right)=0$
	Case 6	$1-cor\left(m_1^{{\Theta }^{\prime }},m_2^{{\Theta }^{\prime }}\right)=0.37$		$\kappa \left(m_1^{{\Theta }^{\prime }},m_2^{{\Theta }^{\prime }}\right)=0$
$1-r_{BPA}\left(m_1,m_2\right)$ [18]	Case 3	$1-r_{BPA}\left(m_1,m_2\right)=0.60$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$\mathcal{RB}\left(m_1,m_2\right)$ [48]	Case 3	$\mathcal{RB}\left(m_1,m_2\right)=0.45$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$1-a_{BPA}\left(m_1,m_2\right)$ [45]	Case 3	$1-a_{BPA}\left(m_1,m_2\right)=0.83$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$1-ECC\left(m_1,m_2\right)$ [46]	Case 3	$1-ECC\left(m_1,m_2\right)=0.84$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$Diss\left(m_1,m_2\right)$ [42]	Case 3	$Diss\left(m_1,m_2\right)=0.69$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$
$K_{\mathcal{BSC}}\left(m_1,m_2\right)$ [47]	Case 3	$K_{\mathcal{BSC}}\left(m_1,m_2\right)=0.97$	Violate property (iv): the value should be minimal.	$\kappa (m_1,m_2)=0$

. The six representative cases listed in Table 4 are discussed in detail as follows.

$$\begin{array}{cc}\hfill \mathrm{Case}1:& m_1\left(\left\{a\right\}\right)=0.4,m_1\left(\left\{b\right\}\right)=0.4,m_1\left(\left\{c\right\}\right)=0.2;\hfill \\ \hfill & m_2\left(\left\{a\right\}\right)=0.4,m_2\left(\left\{b\right\}\right)=0.4,m_2\left(\left\{c\right\}\right)=0.2.\hfill \\ \hfill \mathrm{Case}2:& m_1\left(\left\{a\right\}\right)=0.2,m_1\left(\left\{b\right\}\right)=0.8,m_1\left(\left\{c\right\}\right)=0;\hfill \\ \hfill & m_2\left(\left\{a\right\}\right)=0,m_2\left(\left\{b\right\}\right)=0,m_2\left(\left\{c\right\}\right)=1.\hfill \\ \hfill \mathrm{Case}3:& m_1\left(\left\{a\right\}\right)=0.2,m_1\left(\left\{b\right\}\right)=0.8,m_1\left(\left\{c\right\}\right)=0;\hfill \\ \hfill & m_2\left(\{a,b,c\}\right)=1.\hfill \\ \hfill \mathrm{Case}4:& m_1\left(\left\{a\right\}\right)=0.5,m_1\left(\left\{b\right\}\right)=0.5,m_1\left(\left\{c\right\}\right)=0,m_1\left(\left\{d\right\}\right)=0;\hfill \\ \hfill & m_2\left(\left\{a\right\}\right)=0,m_2\left(\left\{b\right\}\right)=0,m_2\left(\left\{c\right\}\right)=0.5,m_2\left(\left\{d\right\}\right)=0.5.\hfill \\ \hfill \mathrm{Case}5:& m_1^{\Theta }\left(\left\{a\right\}\right)=0.2,m_1^{\Theta }\left(\left\{b\right\}\right)=0.8;\hfill \\ \hfill & m_2^{\Theta }\left(\left\{a\right\}\right)=0.2,m_2^{\Theta }\left(\left\{b\right\}\right)=0.8;\Theta =\{a,b\}.\hfill \\ \hfill \mathrm{Case}6:& m_1^{{\Theta }^{\prime }}\left(\left\{a\right\}\right)=0.2,m_1^{{\Theta }^{\prime }}\left(\left\{b\right\}\right)=0.8;\hfill \\ \hfill & m_2^{{\Theta }^{\prime }}\left(\left\{a\right\}\right)=0.2,m_2^{{\Theta }^{\prime }}\left(\left\{b\right\}\right)=0.8;{\Theta }^{\prime }=\{a,b,c\}.\hfill \end{array}$$

7. Conclusions and Future Work

This study introduces the concept of essential conflict to improve the understanding and application of Dempster’s combination rule and to explain the origin of counterintuitive fusion results. The essential conflict is defined using the intersection relation of the focal elements of the original mass functions and illustrated with examples. In addition, the degree of essential conflict is defined. Furthermore, the two properties of essential conflict are explained, namely belief absolutization and uncorrectable assertion. It is demonstrated that when two mass functions are in essential conflict, their combination always produces extreme judgments that cannot be corrected in further combinations and yields counterintuitive results under Dempster’s combination rule. Moreover, the proposed conflict measure is shown to possess desirable properties that are absent in existing measures. Through different examples and analyses, it is noted that this method provides a better interpretation of the conflict relationship and counterintuitive outcomes arising from evidence combination compared to existing approaches.

Beyond theoretical analysis, the state output achieved by the proposed method has important practical implications. By explicitly revealing absolute and uncorrectable fusion states, the method provides valuable diagnostic information for system debugging, facilitates decision interpretation, and enhances user trust. These properties are particularly critical in high-risk application scenarios, such as medical diagnosis, autonomous driving, multi-sensor surveillance and other safety-critical decision-making systems.

Future work may further investigate correction mechanisms guided by the proposed essential conflict. In particular, when a high degree of essential conflict is detected, the direct application of Dempster’s combination rule may be inappropriate, and conflicting evidence could be excluded when feasible. In scenarios where exclusion is impractical, the proposed $\kappa$ metric may serve as a quantitative indicator for evidence discounting or weighted averaging, reflecting the relative reliability of different sources. In this sense, the essential conflict framework is intended to function as a principled diagnostic tool that can be integrated into existing evidence fusion pipelines.