# P System with Fractional Reduction
## 4 Fractional Simplification P System
### P System for Fractional Simplification of More Phase Derogation Algorithm
#### 4.1.2 Membrane Rules for the More Phase Derogation Algorithm
Execute \(\tau_{5}\) and \(\tau_{6}\): object \(k\) generates a new \(c\) and sends it to the submembrane; if \(y_{i+1}!=y_{i}\), then \(y\) and \(f\) evolves to \(i\) and generates a new object \(z\) and sends it to the submembrane; then, execute \(\tau_{13}\) after consuming a \(Num\); the remaining \(Num\) is passed into the submembrane for the submembrane to dynamically generate membranes. 5. If object \(i\) exists, it is shown that both \(a\) and \(b\) values of the submembrane need to be changed, for which rules \(\tau_{8}\) and \(\tau_{10}\) are executed: \(\tau_{8}\) passes the subtractive backup \(x\) to the \(a\) of the submembrane (i.e., \(x_{i+1}=y_{i}\)) and \(\tau_{10}\) passes the differential backup \(l\) to the \(b\) of the submembrane (i.e., \(y_{i+1}=u_{i}\)). If \(u\) is left in the environment, rules \(\tau_{5}\) and \(\tau_{7}\) are executed with object \(v\) as the catalyst. When \(h\) is generated in the environment, it is proved that \(y_{i+1}=y_{i}\) in the submembrane \(A_{i+1}\). Therefore, \(\tau_{9}\) and \(\tau_{11}\) are executed to pass the subtractive backup to \(x\) to the submembrane \(y_{i+1}\) and the differential backup \(l\) to \(x_{i+1}\) in the submembrane, respectively. The above steps are executed several times until the dynamic generation ends when \(x_{i-1}=2y_{i-1}\). When the submembrane \(A_{t-1}\) is executed to \(x_{t-1}=2y_{t-1}\), the order of rule execution is: \(\tau_{1}\to\tau_{2}\to\tau_{3}\to\tau_{4}\to\tau_{12}\to\{\tau_{5}\),\(\tau_{13}\}\). The rule execution is the same as the parent membrane, at this time \(u_{t-1}=y_{t-1}\), so the rest of the rules in this part are not eligible for execution, and the innermost membrane \(A_{t}\) no longer executes any rules and starts the second part of the calculation. (3) Compute the sequence \(\{p_{i}\}\), \(\{q_{i}\}\)  
When both \(y\) and \(u\) in membrane \(A_{t-1}\) are consumed, i.e., when the condition of termination of the more phase derogation algorithm is reached (the decrement is equal to the difference), the calculation of object \(z\) is ended and no new submembrane is generated in membrane \(A_{t}\). At this point, the sequence \(\{p_{i}\}\), \(\{q_{i}\}\) is started, i.e., the execution of the improved part of the more phase derogation algorithm is started. According to the description of the improved step of the more phase derogation algorithm in Section 3.1.2, it is known that, by several calculations, object \(z\) is generated in part of the membrane, and object \(z\) is a marker of whether the value of \(q\) in the parent membrane is changed or not. By the idea of the more phase derogation algorithm, it is known that the subtraction operation ends when the subtraction and difference are equal, and based on this conclusion, the following rule is designed:  
\[ \begin{array}{ll}\tau_{14}:(f\to\wp^{2}\,q\mid!\,y\&\&\,!\,u\&\,!\,a\&\,!\,b,6)&\tau_{15}:(p\to p1(r,\ out),\ 5)\\ \tau_{16}:(q\to q1(w,\ out),5)&\tau_{17}:(z\to(o,\ out),\ 5)\\ \tau_{18}:(r\to p\,q\mid_{0},5)&\tau_{19}:(w\to p\mid_{0},\ 5)\\ \tau_{20}:(w\to p\,q,6)&\tau_{21}:(r\to p,6)\end{array} \]  
1. The order of the execution of the rules in membrane \(A_{t-1}\) is \(\tau_{14}\to\{\tau_{15},\tau_{16},\tau_{17}\}\). To avoid confusion, the following explanation is given: in the membrane rule, \(\wp^{2}\) meansthat the membrane contains 2 objects \(p\), and \(p1\) means that the System contains an object named \"\(p1\)\". **Figure 4. Flowchart of the more phase derogation algorithm example. **The following details the execution process of rules in this P System.