# P System with Fractional Reduction
## 1 Introduction
[10] designed a multi-layer membrane P System to implement unsigned quadratic operations, which reduced the computational complexity; Ref. [11] designed a single-layer membrane P System to implement arithmetic operations, further simplifying the membrane structure and improving computational efficiency; Ref. [12] designed a multi-layer membrane P System to implement arithmetic operations with signed numbers, improving the application range and execution efficiency of basic operations; Refs. [13; 14; 15] designed a single-layer membrane P System to implement expression evaluation in the domain of integers; Ref. [16] used the P System implemented basic arithmetic operations in the domain of rational numbers, expanding the scope of application of arithmetic operations in P System to further enhance the computing power of biological computers. Since all numbers in the Rational number field can be expressed as fractions, the results of arithmetic operations in the Rational number field may not be in the simplest form. If these results continue to be used in the new operation process, the fraction needs to be initialized first and reduced to its simplest form. In addition, from the perspective of mathematical rigor, the numerical meanings of \(1/2\) and \(1349/2698\) are the same. In the P System, the latter uses much more objects than the former. If used in other arithmetic P Systems, the large number calculation process will also be more complex, reducing system efficiency. In precise calculations, using \(1/3\) will be more accurate than \(0.33333\) and easier to represent in the P System. For this reason, scholars have conducted research on the problem of fractional simplification in order to simplify the process and obtain more accurate results when applying the simplification results to other arithmetic operations [17]. In [18], the authors studied a hybrid code-based arithmetic P System based on a cell-like P System and used fractional simplification as an example to simplify the arithmetic operations on large operands in the P System and to achieve fractional simplification after arithmetic operations on real numbers expressed as fractions in the P System. Its method used multi-layer membrane nesting, and the objects [19] inside each layer of membrane are related to the objects generated by its parent membrane, so this is a highly serialized P System that does not fully utilize the maximum parallelism of membrane computing. However, the solution to the problem of fractional simplification using membrane computing is not sufficient, and currently, there is no other literature that can fully utilize the characteristics of membrane computing to design a more suitable P System. In response to this phenomenon, we also study the issue of fractional reduction in this article, considering partial improvements on the basis of the more phase derogation method and attempting to find suitable solutions in the two traditional reduction methods of more phase derogation and division to further improve the parallelism of the P System, thereby improving the overall efficiency of arithmetic operations. To this end, this paper designs a fraction simplification P System based on rule priorities, and the specific article is structured as follows:  
Section 1 first introduces research on membrane computation in arithmetic operations, then presents the purpose and significance of this paper, and finally gives the organization of this paper. Section 2 begins with an introduction to the biological basis of membrane computation, followed by a definition of the cell-like P System. Section 3 introduces the principles and algorithms of two methods, namely, the more phase derogation algorithm and the division algorithm, respectively, and improves some parts of the more phase derogation algorithm by combining them with applications in the P System.