# 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
Similarly in A\({}_{2}\), the available rules are executed in order r\({}_{1}\to\) r\({}_{2}\to\) r\({}_{3}\to\) r\({}_{4}\to\) r\({}_{12}\), and after execution, the multiset in membrane A\({}_{3}\) is \(l^{3}x^{3}v^{3}kfzNum^{12}\). This result indicates that the subtraction and difference are equal, and the first part of the more phase derogation algorithm is over. (3) Compute the sequence \(\{p_{i}\}\), \(\{q_{i}\}\)  
1. At this time, the multiset in the membrane A\({}_{3}\) is \(l^{3},x^{3},v^{3},k,f,z\). The P System executes the rules in the following order: r\({}_{14}\to\)\(\{\)r\({}_{15}\),r\({}_{16}\),r\({}_{17}\}\). 1. Execute the rule r\({}_{14},f\) generate the multiset \(p^{2}q\); at this point, the multiset in the membrane A\({}_{3}\) is \(l^{3},x^{3},v^{3},k,z,p^{2},q\); 2. The next time slice simultaneously executes the rules \(\{\)r\({}_{15}\),r\({}_{16}\),r\({}_{17}\}\): execute r\({}_{15}\) to evolve \(p^{2}\) into the multiset \(p_{1}{}^{2}\), and generate the multiset \(r^{2}\) into the membrane A\({}_{2}\); execute the rule r\({}_{16}\) to evolve \(q\) into the multiset \(q_{1}\), generate the object \(w\) into the membrane A\({}_{2}\), and execute the rule r\({}_{17}\) to generate the multiset \(o\) of object \(z\) to output into A\({}_{2}\). 2. At this time, the multiset in membrane A\({}_{2}\) is \(o,i,y^{2},v^{3},r^{2},w\) and the order of execution of the rules is {r\({}_{18}\),r\({}_{19}\)}\(\rightarrow\){r\({}_{15}\),r\({}_{16}\)}. 1. Under the condition that the object \(o\) exists, execute the rules {r\({}_{18}\),r\({}_{19}\)}: \(w\) generates the multiset \(p\), \(r^{2}\) generates the multiset \(p^{2}q^{2}\), at which point, the multiset in A\({}_{2}\) is \(o,i,y^{2},v^{3},p^{3},q^{2}\); 2. Next, execute the rule {r\({}_{15}\),r\({}_{16}\)} to generate the multiset \({p_{1}}^{3}{q_{1}}^{2}\) within the membrane A\({}_{2}\), and also generate the multiset \(r^{3}w^{2}\) to pass into the membrane A\({}_{1}\). 3. At this time, the multiset in membrane A\({}_{1}\) is \(r^{3},w^{2},h,u^{2},v^{6}\), and the order of execution of the rules is {r\({}_{20}\),r\({}_{21}\)}\(\rightarrow\){r\({}_{15}\),r\({}_{16}\)}. 1. Execute the rule {r\({}_{20}\),r\({}_{21}\)}, the multiset \(r^{3}\) generates the multiset \(p^{3}\), and the multiset \(w^{2}\) generates the multiset \(p^{2}q^{2}\). At this time, the multiset in the membrane is \(p^{5},q^{2},h,u^{2},v^{6}\). 2. Then execute {r\({}_{15}\),r\({}_{16}\)} to pass out \(r\) and \(w\), respectively. In the membrane, M\({}_{1}\) obtains \(r^{5}w^{2}\), corresponding with the numerator and denominator, and then converts it to \(r_{1}w_{1}\)\({}^{5}\). After that there is no rule to execute, so the whole System stops. So the final simplification of 6/15 results in 2/5. The specific execution process is divided according to the time slice, as shown in Table 1. To facilitate experimental verification, all the membranes generated do not dissolve within the system. The number of membranes is large and the substances are complex, so only multiset that have changed is listed in the table. If a membrane is not listed in one of the rows of the table, it means that the material in this membrane has not changed during that time slice and the multiset in that membrane is the same as the last time it was listed.