# P System with Fractional Reduction
## 4 Fractional Simplification P System
### P System for Fractional Simplification of More Phase Derogation Algorithm
#### 4.1.1 Definition of the P System for More Phase Derogation Algorithm  
The more phase derogation P System is formalized as:  
\[\varPi_{1}=\big{(}V,O,H,\mu,\omega_{M_{1}},\omega_{A_{1}},\ldots,\omega_{A_{t} },R_{M_{1}},R_{A_{1}},\ldots,R_{A_{t}},i_{o}\big{)}, \tag{2}\]  
Among them,  
(1) \(V=\{m,n,c,CreateSubMem,Num,a,b,g,x,y,l,u,k,f,v\}\);  
(2) \(H=\{M_{1},A_{1},\ldots,A_{t}\}\), in this article, for clarity, membrane labels are expressed in the form of a combination of membrane type and subscripts. (3) \(\mu=\big{\{}\big{[}\big{[}\big{[}\big{[}\big{[}\big{[}\big{[}\big{]}_{A_{t}} \big{]}_{-}\big{]}_{A_{1}}\big{]}_{M_{1}}\big{]}_{M_{1}}\big{\}}\), in the initial state, the P System only contains membrane \(M_{1}\) and membrane \(A_{1}\), and membrane \(A_{1}\) is included in membrane \(M_{1}\). \(A_{i}(2\leq i\leq t)\) is dynamically generated during the execution of membrane rules, and any \(A_{i}\) always includes \(A_{i-1}\) in;  
(4) \(\omega_{M_{1}}=m^{x}n^{y}\omega_{A_{1}}=cCreateSubMem\), and \(x\) and \(y\) represent the numerator and denominator of fractions, respectively;  
(5) The \(R_{M_{1}},R_{A_{1}},\ldots,R_{A_{t}}\) evolution rule finite set will be explained in Section 4.1.2;  
(6) \(i_{o}=M_{1}\) indicates that when the entire System stops, the final result can be obtained in the membrane \(M_{1}\);  
For the convenience of description, there is the following agreement in the remaining content of this section: the newly created class A membrane is named \(A_{2},A_{3},\ldots,A_{t}\), and the rules within these membranes are consistent with those within membrane \(A_{1}\). Therefore, there will be no further explanation in the remaining content of this section. The initial pattern of the fraction simplification P System based on the more phase derogation algorithm is shown in Figure 2.  
Figure 2 depicts the initial pattern of the P System using the more phase subtraction method for fractional reduction: membrane \(M_{1}\) is mainly used to compare the sizes of molecule \(x\) and denominator \(y\), transfer the larger and smaller values into object \(a\) and object \(b\) of membrane \(A_{1}\), respectively, and save the final calculation results and dissolve other objects transferred from membrane \(A_{1}\). Membrane \(A_{1}\) and its dynamically generated sub membranes \(A_{2}\), \(A_{3}\), \(\ldots\), \(A_{t}\) are used for more phase reduction operations, and the values of variables such as sequence \(\{x_{i}\}\), \(\{y_{i}\}\), \(\{u_{i}\}\), \(\{z_{i}\}\), \(\{p_{i}\}\), \(\{q_{i}\}\) are calculated (where \(\mathrm{i}=1,2,\ldots,\mathrm{t}\)). Except for membrane \(M_{1}\) and membrane \(A_{1}\), other required membrane structures are dynamically created during the reduction process based on the actual conditions of the fraction, and the newly created membrane rules are completely consistent with the rules in membrane \(A_{1}\). Figure 2: Initial Pattern of the More Phase Derogation Algorithm P System. In Figure 2, objects \(m\) and \(n\) are used to represent the numerator and denominator, respectively; \(x\) and \(y\) are the numbers of objects \(m\) and \(n\), and object \(c\) is used to trigger the execution of rules in membrane \(A_{1}\) and determine whether a new membrane needs to be generated based on different conditions. Membrane \(A_{1}\) and other newly created submembranes are used together to calculate variables in the rules. These newly created membranes are nested layer by layer: membrane \(A_{2}\) is created within membrane \(A_{1}\), membrane \(A_{3}\) is created within membrane \(A_{2}\), and finally, membrane \(A_{t}\) is created within membrane \(A_{t-1}\). The process of calculating sequences \(\{x_{i}\}\), \(\{y_{i}\}\), \(\{u_{i}\}\) is described below, and the specific execution process of membrane rules will be explained in detail in Section 4.1.2. 1.