# P System with Fractional Reduction
## 5 Experimental Results
### Comparative Experiment of Three Methods  
We will substitute all 100 sets of data into the combination of the more phase derogation algorithm, the division algorithm, and the optimal parameter \(\xi=8\) obtained from the experiment in Section 5.2 for experiments. The experimental results will be tabulated for statistical analysis.  
Figure 16: Six-digit parameter comparison line chart.  
According to the discount Figure 17 made based on the experimental results, it can be seen that out of the 100 randomly selected sets of data, 99 sets of data required the least number of steps when using the more phase subtraction method for the experiment, and only one set of data consumed the least number of steps when using the division algorithm. Ninety-nine percent of the data steps reached the highest value when using the division algorithm for the experiment. The number of steps consumed by \(\xi,=8\) is mostly in the middle of the steps consumed by the more phase derogation algorithm and the division algorithm, and in a few cases, it overlaps with the division algorithm.  
Analyzing the case of \(\xi=8\) in conjunction with Section 5.2.4: In the method of combining the two, regardless of the data values of the numerator and denominator, when the difference is less than eight, the P System needs to convert from a more phase derogation algorithm to a division algorithm for calculation. However, the implementation rules of a class A membrane in the first part are actually consistent with those of a class A membrane in the more phase derogation algorithm P System. If the maximum common divisor of the numerator and denominator is not less than eight, the final result can be obtained directly by the more phase derogation algorithm without passing out the intermediate value in the innermost membrane. At this point, the steps of combining the two in the P System are the same as the steps of reducing the loss of the P System, so it is necessary to focus on the impact of other rules on the time slices. However, the rule that multiset in all membranes realizes one-way transmission between membranes is only executed once, which has little impact on the total steps in the whole System. Therefore, the main reason for the increase in time slices is the rolling division.  
According to the execution process of the Section 4.2.2 division membrane rules, it can be seen that our designed film rule actually converts division into subtraction: the process of dividing a larger value by a smaller value to obtain a remainder is converted into the process of subtracting a larger value by an integer multiple of the smaller value. When the P System determines that subtraction cannot continue, the remaining value is the remainder. Repeat this step until the maximum common divisor is found when there is no remainder. Therefore, only one subtraction is needed for each layer of class A membrane in the more phase derogation P System, and then the object is directly transferred to the submembrane, while the division P System is to subtract an integer multiple in each layer until the rules of this layer cannot be executed, then the Multiset can be transferred to the inner membrane, and then the same rules are executed in the inner membrane until the  
Figure 17: Step number comparison of three methods in a line chart.  
maximum common divisor is found. For this reason, in theoretical applications, every time a division is performed, it actually requires multiple subtractions in the P System, resulting in a much higher cumulative number of subtractions compared to using the more phase derogation method alone.  
Therefore, the final experimental conclusion is drawn: in the process of fractional reduction, the efficiency is higher when using a more phase derogation algorithm P System. Or, to put it another way, it is also correct: in a combined P system, when the parameter \(\xi\), is set to 0, the efficiency of the system is the highest.