# 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
(1) Initial pattern  
First, put the multiset \(m^{6}n^{15}\) in the membrane M\({}_{1}\), execute the rule r\({}_{16}\) times to generate the multiset \(g^{6}\); execute \(r^{3}\) under the condition that both \(g\) and \(n\) exist, generate \(b^{9}\) in the membrane, and at the same time generate the multiset \(a^{9}Num^{9}\) to pass to the submembrane A\({}_{1}\); execute r\({}_{4}\) to generate the multiset \(a^{6}b^{6}Num^{6}\) into the submembrane A\({}_{1}\); at this time, the multiset in the membrane M\({}_{1}\) is \(b^{6}\), and the initialized multiset of the membrane A\({}_{1}\) is \(a^{15}b^{6}cNum^{15}\)CreateSubMem. (2) Calculate the sequence \(\{x_{i}\}\), \(\{y_{i}\}\), \(\{u_{i}\}\), \(\{z_{i}\}\)  
1. The rules available in membrane A\({}_{1}\) are executed in the following order: r\({}_{1}\to\) r\({}_{2}\to\) r\({}_{3}\to\) r\({}_{4}\to\) r\({}_{12}\to\)\(\{\)r\({}_{5}\), r\({}_{7}\),r\({}_{13}\}\to\)\(\{\)r\({}_{9}\),r\({}_{11}\}\). 1. Execute the rule r\({}_{1}\) 6 times in the membrane A\({}_{1}\) until \(b^{6}\) is completely consumed, generating the multiset \(x^{6}y^{6}\) and \(a^{9}\) remains; execute the rule r\({}_{2}\) under the condition that the object \(c\) exists, generating \(l^{9}u^{9}\). 2. At this point, only the rule r\({}_{3}\) can be executed: evolve \(c\) to the multisets \(k\) and \(f\) under the condition that the object \(u\) exists; execute the rule r\({}_{4}\) six times to generate the multiset \(v^{6}\), at which point, the multiset in the membrane A\({}_{1}\) is \(x^{6}u^{3}l^{9}v^{6}kf\). Execute the rule r\({}_{12}\), as described in Section 4.1.2, consuming the _CreateSubMem_ and a \(Num\) in this membrane to generate the membrane A\({}_{2}\). 3. Execute the rules \(\{\)r\({}_{5}\),r\({}_{7}\), r\({}_{13}\}\) within the same time slice. Execute the rule r\({}_{5}\) to generate a new object \(c\) from \(k\) into the control loop in membrane A\({}_{2}\); execute the rule r\({}_{7}\) so that the multiset \(u\) and \(f\) is consumed to generate object \(h\), and execute r\({}_{13}\) to pass the remaining \(Num\) into the submembrane for subsequent continuation of submembrane generation. 4. The next time slice executes \(\{\)r\({}_{9}\), r\({}_{11}\}\) at the same time. Execute the rule r\({}_{9}\) 6 times to evolve \(x^{6}\) into the multiset \(b^{6}\) and send \(b^{6}\) into the membrane A\({}_{2}\), execute \(r_{11}\) at the same time to evolve \(l^{9}\) into the multiset \(a^{9}\) and send it into the membrane A\({}_{2}\). 2. At this time, there are multiple \(a^{9}b^{6}cNum^{14}\) sets in membrane A\({}_{2}\). Execute the rules in the following order: r\({}_{1}\to\) r\({}_{2}\to\) r\({}_{3}\to\) r\({}_{4}\to\) r\({}_{12}\to\)\(\{\)r\({}_{5}\), r\({}_{6}\),r\({}_{13}\}\to\)\(\{\)r\({}_{8}\),r\({}_{10}\}\) 1. After executing the membrane rule in the order of r\({}_{1}\to\) r\({}_{2}\to\) r\({}_{3}\to\) r\({}_{4}\to\) r\({}_{12}\), the multiset within the membrane is \(l^{3}y^{3}x^{6}v^{3}kf\), and a new submembrane A\({}_{3}\) is generated. 2. As described in Section 4.1.2, the next time slice simultaneously executes the rule \(\{\)r\({}_{5}\),r\({}_{6}\),r\({}_{13}\}\): execute the rule r\({}_{5}\), object \(k\) evolves into object \(c\) to be sent to membrane A\({}_{3}\); while executing the rule r\({}_{6}\), object \(y\) and object \(f\) evolves into object \(i\) and sends \(z\) to membrane A\({}_{3}\), and executes the rule r\({}_{13}\) to pass the remaining \(Num\) to membrane A\({}_{3}\). 3. Execute the rules \(\{\)r\({}_{8}\),r\({}_{10}\}\) under the condition that object \(i\) exists. The rule r\({}_{8}\) is executed 6 times, and the multiset \(x^{6}\) evolves into the multiset \(a^{6}\) into the membrane A\({}_{3}\), while the rule r\({}_{10}\) is executed 3 times, and the \(l^{3}\) evolves into the multiset \(b^{3}\) into the membrane A\({}_{3}\). At this point, the membrane A\({}_{3}\) contains the multiset \(a^{3}b^{3}czNum^{13}\).