# 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 \(A_{1}\) is \(\lambda^{15}{u^{9}}g^{15}f^{5}{h_{1}}{l_{1}}^{3}{k_{1}}^{5}\), and the condition is met to execute rule r\({}_{23},r_{24}\) to pass the multiset \({l_{1}}^{3}{k_{1}}^{5}\) into membrane \(A_{2}\), respectively. 2. When membrane \(A_{2}\) receives a multiset \({l_{1}}^{3}{k_{1}}^{5}\) from the parent membrane, the rules are executed in the order \(\{r_{25},r_{26}\}\rightarrow\{r_{27},r_{28}\}\). 1. First executing rule \(r_{25},r_{26}\) in membrane \(A_{2}\), converts the multisets \({k_{1}}^{5}\) and \({l_{1}}^{3}\) into \(p^{5}\) and \(q^{3}\), respectively; 2. Then executing rule \(r_{27},r_{28}\) converts the multiset \(p^{5}q^{3}\) into a multiset \(r^{5}w^{3}\) to pass directly into membrane \(A_{1}\); 3. The order of execution in membrane \(A_{1}\), at this time, is: \(\{r_{32},r_{33}\}\rightarrow\{r_{27},r_{28}\}\). 1. In the absence of object o\({}_{1}\), rule \(r_{32},r_{33}\) is executed at the same time, and the multiset \(r^{5}w^{3}\) evolves into \(p^{8}q^{3}\); 2. Execute rule \(r_{27},r_{28}\) to convert the multiset \(p^{8}q^{3}\) to \(r^{8}w^{3}\) and feed it into the membrane \(M_{1}\). 4. When the membrane \(M_{1}\) receives the multiset \(r^{8}w^{3}\), the object \(b_{2}\) is generated when the rule is executed because the numerator is smaller than the denominator, so the corresponding rule is executed to evolve the multiset \(r^{8}w^{3}\) into the multiset \({w_{1}}^{8}{r_{1}}^{3}\). At this time, the multiset in the membrane \(M_{1}\) is \({l_{1}}^{3}{k_{1}}^{5}f^{5}{l_{2}}^{5}{b_{2}}^{25}{w_{1}}^{8}{r_{1}}^{3}\), and there is no rule to execute in the System, so the whole System is stopped. The number of objects \(w_{1}\) and \(r_{1}\) represent the values of denominator and numerator, respectively, so the final reduction result of 15/40 is 3/8. Table 3 shows the partial process of simplifying 15/40 in a combinatorial P System \(\Pi_{3}\), including system initialization, obtaining the maximum convention \(x\), calculating the intermediate value reduction results \(k_{1}\), \(l_{1}\), and other key time slices. The complete execution process is shown in Appendix A. From the above process analysis, it is clear that the total time slice for this example is 61. \begin{table}
\begin{tabular}{c c c} \hline
**Time Slice** & **Rules for Implementation** & **Results of the Implementation** \\ \hline Initial Status & None & M\({}_{1}\): \(\lambda^{15}\), b\({}^{40}\), c, t; \\  & None & A\({}_{1}\): c, d\({}^{20}\), CreateSubMem; \\  & & B\({}_{1}\): c, CreateSubMem; \\ \hline  & & M\({}_{1}\): c, t, b\({}_{2}\)\({}^{25}\); \\  & & A\({}_{1}\): \(\lambda^{40}\), b\({}^{15}\), d\({}^{20}\), e, Num\({}^{40}\), j, k, CreateSubMem; \\  & & B\({}_{1}\): a\({}^{40}\), b\({}^{15}\), c, Num\({}^{40}\), CreateSubMem; \\ \hline  & & M\({}_{1}\): c, t, b\({}_{2}\)\({}^{25}\); \\  & & A\({}_{1}\): r\({}_{13}\), r\({}_{7}\), r\({}_{10}\); & A\({}_{1}\): Num\({}^{39}\), u\({}^{9}\), f\({}^{5}\), g\({}^{15}\), x\({}^{15}\), h, j, \(\lambda^{15}\), a\({}_{1}\)\({}^{25}\); \\  & & A\({}_{2}\): c, d\({}^{20}\), CreateSubMem; \\  & & B\({}_{1}\): c, Num\({}^{40}\), A\({}^{155}\), CreateSubMem; \\ \hline  & & M\({}_{1}\): c, t, m\({}^{25}\), n\({}^{15}\), b\({}_{2}\)\({}^{25}\); \\  & & A\({}_{1}\): u\({}^{9}\), f\({}^{5}\), h, j, \(\lambda^{15}\); \\  & & A\({}_{2}\): d\({}^{10}\), Num\({}^{39}\), e, u\({}^{10}\), f\({}^{5}\), j, k, CreateSubMem, o\({}^{10}\); \\  & & B\({}_{1}\): c, Num\({}^{40}\), A\({}^{155}\), CreateSubMem; \\ \hline  & & M\({}_{1}\): c, t, b\({}_{15}\), b\({}_{2}\)\({}^{25}\), a\({}_{125}\); \\  & & A\({}_{1}\): r\({}_{6}\), r\({}_{5}\); \\  & & A\({}_{1}\): r\({}_{6}\), r\({}_{5}\); \\  & & A\({}_{2}\): d\({}^{10}\), Num\({}^{39}\), e, u\({}^{10}\), f\({}^{5}\), j, k, CreateSubMem, o\({}^{10}\); \\  & & B\({}_{1}\): c, n\({}^{15}\), m\({}^{15}\), A\({}^{155}\), Num\({}^{40}\), CreateSubMem; \\ \hline \end{tabular}
\end{table} 
Table 3: Partial time slice of a combinatorial P System \(\Pi_{3}\).