## 3 Capability Gaps Computing Based on the Conditional Evidential Network
### Calculation of the Belief Distribution of the Requirement-Satisfaction Degree for Capabilities  
To calculate the belief distribution of the requirement-satisfaction degree for capabilities, first we need to establish the conditional-evidential-network identification framework and the conditional-belief-parameter table.  
We use a causality diagram along with other methods in order to construct the evidential network of weapon system of systems' capabilities. In order to model it, we use the conditional belief function, based on relevant knowledge and experts' experience. Following this, we obtain the evidential-network structure-chart and conditional-belief-function table. The conditional belief function in this paper is defined by each node and expressed in \(m(C|Pa(C))\), where C represents the state of the capability sub-node and \(Pa(C)\) represents the combined state of all of the node's parent nodes.  
Next, we calculate the belief distribution of the requirement-satisfaction degree of the bottom capability.  
Let \(c_{h}\) and \(c_{l}\) respectively denote the ideal value and the lowest capability-requirement value. \(c\) represents the actual value of the capability, \(\varepsilon(c)\) represents the credibility of this actual value, and \(\varphi(c)\) represents the degree to which the actual and ideal values match.  
According to the characteristics of the bottom capabilities, the matching degree between the actual and the ideal capabilities can each be calculated according to the indicators of benefit and cost.  
(1) The matching degree of the benefit index and the requirement's ideal value is:  
\[\varphi(c)=\left\{\begin{array}{cc}0&c\leq c_{l}\\ \frac{c-c_{l}}{c_{h}-c_{l}}&c_{l}<c<c_{h}\\ 1&c\geq c_{h}\end{array}\right. \tag{15}\]  
(2) The matching degree of the cost index and the requirement's ideal value is:  
\[\varphi(c)=\left\{\begin{array}{cc}0&c\geq c_{l}\\ \frac{c_{l}-c}{c_{l}-c_{h}}&c_{h}<c<c_{l}\\ 1&c\leq c_{h}\end{array}\right. \tag{16}\]  
According to the matching degree of the bottom index and the ideal value of the demand, the belief distribution of the requirement satisfaction can be calculated as follows:  
\[\left\{\begin{array}{c}m_{0}(c=H)=\varphi(c)\varepsilon(c)\\ m_{0}(c=L)=(1-\varphi(c))\varepsilon(c)\\ m_{0}(\Theta)=1-\varepsilon(c)\end{array}\right. \tag{17}\]  
where \(m_{0}(c=H)\) represents the belief that the degree of requirement satisfaction of the bottom indicators is high, \(m_{0}(c=L)\) represents the belief that it is low, and \(m_{0}(\Theta)\) represents the belief assigned to cognitive uncertainty.  
Then we calculate the belief distribution of the requirement-satisfaction degree for the upper layer capability.  
According to the conditional-belief-parameter table established using the method in Section 3.1 and the belief distribution of the bottom capability's requirement satisfaction, the belief distribution of the capability requirement satisfaction for each capability is calculated by the forward reasoning formula:  
\[m(C)=\sum m_{0}(Pa(C))m(C|Pa(C)) \tag{18}\]  
\[m_{0}(Pa(C))=\prod_{C_{l}\subseteq Pa(C)}m(C_{i}) \tag{19}\]