# P System with Fractional Reduction
## 1 Introduction  
Membrane computing (also known as P System) is a new branch of natural computing [1] introduced by Prof. [PERSON] in a 1998 research paper and is an abstraction of a new model of computing based on the study of the function and structure of living cells in living organisms. The skin of a whole organism is equivalent to an entire computational System, which contains individual computational units. The membrane model of computing divides each biological cell into a hierarchy of regions, each of which is called a membrane; i.e., regions correspond to membranes. Each region contains a multiset of objects that evolve according to the evolutionary rules within the region, and the evolutionary rules vary from region to region. Membrane computation is based on the abstraction of living cells, and each computational unit is able to perform its own computation. Due to the extremely large number of cell membranes and the small amount of energy required to drive them, one of the greatest advantages of membrane computation is that it allows for maximum parallelism in the computation. Although computers nowadays have a profound impact on all aspects of human society, they still have limitations in their ability to process information. Due to their shortcomings, such as small storage capacity, slow computing speed and low intelligence, electronic computers are powerless to deal with many real-life challenges, such as NP problems. In contrast, membrane computing brings a finite, discrete, distributed, parallel, and hierarchical or mesh-based model of information processing to computer science [2]. [PERSON] has demonstrated in the literature [3] that membrane computing has the equivalent computing power to Turing machines, and its powerful parallel computing capability can effectively solve the bottlenecks currently faced by electronic computers. There is a proliferation of international research in the field of membrane computing, which has greatly facilitated the intersection and penetration of mathematics, computer science and biology. The research on membrane computing and membrane computers is developing rapidly, and great progress has been made in such areas as membrane computing applied to the solution of some NP-complete problems, membrane computing in biological laboratories, language Systems for membrane computers, associative memory problems, and the application of membrane computing to cryptography. For example, in solving NP problems, [4] proposed a new solution based on membrane computing and the dynamic window approximation algorithm, which enables the model to calculate the optimal motion command of the robot in logarithmic time and avoid obstacles in real time. Membrane computing is abstracted from living cells in organisms and is also applied to biology. In [5], it improved the implementation of biological processes, including Mitogen-activated protein kinase (MAPK), by virtue of its biological characteristics. In [6], it introduced the application of membrane computing in systems and Synthetic biology. Ref. [7] provided a detailed explanation of the extensive applications of membrane computing in biology, computer science, and linguistics for scholars to conduct in-depth learning and explore new ideas. However, to achieve a general-purpose computer, it must also be able to implement basic arithmetic operations, such as the four operations of addition, subtraction, multiplication and division. Arithmetic operations are considered the basic operations for many complex operations, and therefore, to solve more complex problems, arithmetic operations need to be fully investigated first. Ref. [8] implemented arithmetic operations based on a designed arithmetic P System, but its membrane System structure was complex and did not make full use of the maximum parallelism of membrane computation; literature [9] designed a natural coding-based arithmetic P System to implement arithmetic operations, which greatly simplified the membrane System structure; Ref.