# P System with Fractional Reduction
## 4 Fractional Simplification P System
### P System for Fraction Simplification in the Division Algorithm
#### 4.3.3 Examples of Fractional Simplification by Combining the Two Methods
At this point, the multiset in membrane \(A_{2}\) is \(y^{15},u^{10},o^{10},d^{10},k,e,j,Num^{39}\), \(CreateSubMem\), and the multiset in membrane \(A_{1}\) is \(m^{25},n^{15},\lambda^{15},u^{9},f^{5},g^{15}j,h\); execute the rules \(r_{21},r_{22}\) simultaneously without object \(o\); the multiset \(m^{25}n^{15}\) is fed into the membrane \(M_{1}\). (3) Division algorithm  
(i): The multiset in membrane \(B_{1}\) is \(m_{1}{}^{25}n_{1}{}^{15}cNum^{40}CreateSubMem\), and the rules are executed in the order \(r_{1}\to r_{2}\to r_{3}\to r_{4}\to r_{5}\). (a): Implementation rules \(r_{1}\), multiset \(m_{1}{}^{15}n_{1}{}^{15}\) evolves to \(x^{15}\); then execute the rule \(r_{2}\) that evolves object \(c\) to \(y\), subject to the existence of object \(x\); (b): Implementation rules \(r_{3}\), multiset \(m_{1}y\) evolves to \(m_{1}d\); execute the rule \(r_{4}\) under the condition that object \(d\) exists, so that \(x^{15}\) evolves into the multiplicity set \(n_{1}{}^{15}\); then execute the rule \(r_{5}\), and the object \(d\) evolves to object \(c\); (ii): At this time, the multiset in membrane \(B_{1}\) is \(m_{1}{}^{10}n_{1}{}^{15}cNum^{40}CreateSubMem\), and the rules are executed in the order \(r_{1}\to r_{2}\to r_{6}\to\{r_{7},r_{9}\}\to\{r_{8},r_{10},r_{11}\}\to r_{12}\). (a): Execute the rule \(r_{1}\) that evolves \(m_{1}{}^{10}n_{1}{}^{10}\) to the multiset \(x^{10}\); execute the rule \(r_{2}\) to evolve \(c\) into \(y\) under the condition that object \(x\) exists; then execute the rule \(r_{6}\) to evolve the multiset \(n_{1},y\) to \(n_{1},e,g\); (b): Execution rules \(r_{7},r_{9}\), consume \(CreateSubMem\) and a \(Num\) to generate a sub-membrane \(B_{2}\), while the object \(g\) evolves to \(h\); (c): Under the condition that the multiset \(h\) exists, execute the rule \(r_{8},r_{10},r_{11}\) that transfers the multiset \(Num^{39}n_{1}{}^{10}m_{1}{}^{5}\) to the membrane \(B_{2}\); (d): Execute the rule \(r_{12}\), and the multiset \(h\) evolves into \(c\) and feeds into the membrane \(B_{2}\). (iii): The initialized multiset in membrane B\({}_{2}\) is \(m_{1}{}^{5}n_{1}{}^{10}cNum^{39}CreateSubMem\). The rules are executed in the same order \(r_{1}\to r_{2}\to r_{6}\to\{r_{7},r_{9}\}\to\{r_{8},r_{10},r_{11}\}\to r_{12}\). After the rules are executed, the remaining multiset in membrane \(B_{2}\) is \(e\). The multiset obtained from membrane \(B_{3}\) is \(m_{1}{}^{5}n_{1}{}^{5}cNum^{38}\). (iv): At this point, the initialized multiset in \(B_{3}\) is \(m_{1}{}^{5}n_{1}{}^{5}cNum^{38}CreateSubMem\). And the rules are executed in the order \(r_{1}\to r_{2}\to r_{13}\to r_{14}\). (a): Implementation rules \(r_{1}\), multiset \(m_{1}{}^{5}n_{1}{}^{5}\) evolves to \(x^{5}\), and then, execute the rule \(r_{2}\) to evolve \(c\) into the object \(y\); (b): Under the condition that there are no objects \(m_{1}\) and \(n_{1}\), execute rules \(r_{13}\) that evolves \(y\) to \(z\); at this point, the condition is met to execute \(r_{14}\), the multiset \(x^{5}\) evolves to \(z^{5}x^{5}\) and is fed into membrane \(B_{2}\);  
In membrane \(B_{2}\), continue to execute the rule \(r_{14}\) to generate the multiset \(z^{5}x^{5}\) in membrane \(B_{1}\); membrane \(B_{1}\) also executes the rule \(r_{14}\), and finally sends the multiset \(z^{5}x^{5}\) into the membrane \(M_{1}\); at this point, the number of \(x\) is the maximum number of conventions \(x_{1}\). (4) Calculate the median simplification result \(k_{1},l_{1}\)  
At this point, the multiset in membrane \(M_{1}\) is \(a_{1}{}^{25}b_{1}{}^{15}b_{2}{}^{25}x^{5}ct\). The execution process of the rule is the same as the approximate partitioning process of the example in Section 4.2.3, and will not be described too much here. After the rule is executed, the multiset in membrane \(M_{1}\) is \({l_{1}}^{3}{k_{1}}^{5}f^{5}{l_{2}}^{5}{b_{2}}^{25}\), and the multiset passed into membrane \(A_{1}\) is \({l_{1}}^{3}{k_{1}}^{5}\). (5) Calculate the final result  
1.