# An Improved FFIP Method Based on Mathematical Logic and SysML
## 3 Failure Logic Based on Mathematical Logic
### Description Method of Failure Logic  
Failure logic analysis is the core of the extended FFIP method. It is important to analyze and express the failure logic of systems and components. Mathematical logic is a logical reasoning that uses mathematical symbols to describe objective objects [43]. In mathematical logic, generally, \"1\" and \"0\" are used to represent \"true\" and \"false\". Symbols such as \"and\", \"or\" and \"not\" are used to represent the operation rules, to realize the transformation from complex logical operation to simple numerical calculation. In this paper, mathematical logic is adopted to describe the failure logic of systems and components. Thus, the complex functions of a system can be expressed in a specific way, and the readability of information is increased with mathematical logic. Through the guideline on the specific mathematical expression in failure logic, the designer and the user can unify their expression for further communication and modification. Common connectives in mathematical logic are \(\
eg\), \(\land\), \(\lor\), \(\rightarrow\) and \(\leftrightarrow\). The method of using mathematical logic to express failure logic is shown as follows.  
First, the failure deviations are classified according to the failure modes that may occur in the system. It is described with the symbol of mathematical logic, as shown in Table 2.  
Then, according to the formal graphical representation method [44; 45], the conversion method of failure logic is expressed. As shown in Figure 2, we use circles to indicate the state of the component and arrows to indicate the transition relationship between states. The content on the arrow is the mathematical description of the failure logic using the failure deviation, and the conversion condition of the state is expressed in simple and clear language.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Primary Classification** & **Secondary Classification** & **Description** \\ \hline Logic & Reverse & Contrary to normal operation logic \\  & Wrong material & Input/output/use the wrong material \\ Time & Start too early & Earlier before normal beginning time \\  & Start too late & Later than normal beginning time \\  & End too early & Earlier before normal ending time \\  & End too late & Later than normal ending time \\  & Out of sequence & A sequential error between two actions \\ Value & Too high & The current value is greater than normal \\  & Too low & The current value is less than normal \\  & None & Numerical is zero \\  & Intermittent & Numerical fluctuation \\ \hline \hline \end{tabular}
\end{table}
Table 1: Classification of failure deviation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Primary Classification** & **Secondary Classification** & **Mathematical Description** \\ \hline Normal & / & [P, (t\({}_{1}\), t\({}_{2}\)), n\({}_{0}\)] \\ Logic & Reverse & \(\{\
eg\)P, (t\({}_{1}\), t\({}_{2}\)), n\({}_{0}\)] \\  & Wrong material & [Error, (t\({}_{1}\), t\({}_{2}\)), n\({}_{0}\)] \\ Time & Start too early & \(\{\)P, (t\({}_{0}\) \(<\) t\({}_{1}\), t\({}_{2}\)), n\({}_{0}\)] \\  & Start too late & \(\{\)P, (t\({}_{0}\) \(<\) t\({}_{1}\), t\({}_{2}\)), n\({}_{0}\)] \\  & End too early & \(\{\)P, (t\({}_{1}\), t\({}_{1}\) \(<\) t\({}_{2}\)), n\({}_{0}\)] \\  & End too late & \(\{\)P, (t\({}_{1}\), n\({}_{0}\) \(>\) t\({}_{2}\)), n\({}_{0}\)] \\  & Out of sequence & \(\{\)P, (t\({}_{0}\) = variable, t\({}_{n}\) = variable), n\({}_{0}\)] \\ Value & Too high & \(\{\)P, (t\({}_{1}\), t\({}_{2}\)), n \(>\) n\({}_{0}\)] \\  & Too low & \(\{\)P, (t\({}_{1}\), t\({}_{2}\)), n \(<\) n\({}_{0}\)] \\  & None & \(\{\)P, (t\({}_{1}\), t\({}_{2}\)), n = 0\(\}\)] \\  & Intermittent & \(\{\)P, (t\({}_{1}\), t\({}_{2}\)), n = variable\(\}\)] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mathematical description of failure deviation.