Discrete Dynamics in Nature and Society

Volume 2015, Article ID 123960, 11 pages

http://dx.doi.org/10.1155/2015/123960

## Research on P System with Chain Structure and Application and Simulation in Arithmetic Operation

School of Economics and Management, Beihang University, Beijing 100191, China

Received 19 August 2014; Accepted 24 December 2014

Academic Editor: Zhan Zhou

Copyright © 2015 Jing Luan and Zhong Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Considering the advantages of distribution and maximum parallelism of membrane computing and availability of discrete Morse theory to deal with discrete structure, in this paper, combining discrete Morse theory and membrane computing, a novel membrane structure—P system with chain structure, is proposed, which is constructed on the basis of discrete gradient vector path of the discrete Morse theory. At the theoretical level, due to its unique chain structure, compared with traditional P system, its structure, object, and rule are described in details. In the practical aspect, a specific application example, P chain system for arithmetic operation, is presented to demonstrate the superiority, computational efficiency, and ability of P system with chain structure. Moreover, a simulation system of arithmetic operations based on P chain system is designed, giving a visual display of the implementation of P chain system for arithmetic operation, and verifying the feasibility and effectiveness of P chain system.

#### 1. Introduction

Membrane computing is a new computational model proposed by Romanian scientist Păun in 1998, due to being introduced by Păun at the first time, it is also called P system [1]. Membrane computing abstracts cell as the computational unit, permitting every computational unit to calculate dependently and the whole system to operate in the way of maximum parallelism, whose computational efficiency has been improved obviously [2–5]. It has been proved that the computation capacity of membrane computing is equivalent with that of turing machines, so due to its strong parallel computing power it has been the highlight of the recent study.

Morse theory [6] is a useful tool in differential topology, applied for investigating the topology of smooth manifolds, particular for computer graphics, having been the focus of the research. Forman [7] extended it to the discrete aspect, which provides an effective tool to describe the topology of discrete object, such as simplex and simplicial complex and plays a vital role in pure mathematics and applied mathematics. Concepts in discrete Morse theory, such as simplex [8], discrete gradient vector path and so on, provide a useful tool to research the topology of discrete structure.

Generally, the structure of P system is abstracted as the nested structure of the cell wall and organelle, so it is a kind of nested and hierarchical structure. Of course, there are many other structures, such as reticular formation of neural network. Up to now, there are three main P systems, cell-like P system, tissue-like P system and neural P system [3], and the study of cell-like P system has been well-developed. In literature [4], Păun pointed out that the focus of the next stage of membrane computing study was the nonhierarchical arrangement of membranes. In literature [9], the author has proposed a P system based on simplicial complexes, which is an innovative try in the aspect of nonhierarchical membrane structure. Inspired from this, in this paper, a P system with chain structure is introduced, which combines membrane computing with discrete Morse theory and constructs a P system based on the discrete gradient vector path, forming a new kind of nonhierarchical membrane structure—P chain system. To try to arrange a novel nonhierarchical structure of P system not only makes a contribution to knowledge but also makes a clear methodological contribution.

The organization of the reminder in this paper is described as follows: Section 2 is the part of theoretical discussion, reviewing theories and properties of discrete Morse theory and P system, which are the foundation of P system with chain structure, and then giving the specific description of structure, object, and rule of P chain system. In Section 3, a practical application of P chain system, P chain system for arithmetic operation, is proposed, displaying four kinds of P chain systems. In Section 4, by a computer simulation, the specific implementation of P chain system for arithmetic operation is demonstrated, showing the computational efficiency and power of P system with chain structure. In Section 5, the summary and prospect are included.

#### 2. P System with Chain Structure

##### 2.1. Foundation of P System with Chain Structure

###### 2.1.1. Discrete Morse Theory

Here are some core definitions in discrete Morse theory, which are also essentials in this paper [6–9].

*Definition 1 (-simplex). *Assuming that are points which cover the widest position in -dimensional Euclidean space and , that is to say, vectors , are linearly independent relationship with each other, then defining , written as , and real number satisfying two factors: (1) ; (2) , the set of in is called -simplex, marked as , while are called vertices of the simplex. A simplex is uniquely denoted by its vertices; therefore it can be expressed as [] or simply.

*Definition 2 (simplex with orientation). *For a -simplex , there are permutations of different sequences for its vertices ; when , there are two kinds of permutations; any two permutations of the same kind differ in even commutations, while any two permutations in different group differ in odd commutations. These two kinds are called two orientations of . The simplex which has been given an orientation is called simplex with orientation, one denoted as and the other as .

*Definition 3 (-chain). *Suppose a fundamental constituent of -complex , for an integer , there is , and a linear combination with integer coefficient is called -chain of .

*Definition 4 (discrete gradient vector field). *For the gradient vector field on a -complex , it is a series of ordered pair sets denoted as and subjected to (1) , (2) each simplex of belongs to one pair in at most. Specifically, a gradient vector field is called discrete gradient vector field if and only if there is no nontrivial closed -path on it, which is an alternating sequence of simplexes , where and .

###### 2.1.2. P System Theory

Membrane computing is a novel computational model abstracted from biochemical reactions in living cells, whose merit is internal maximum parallelism. The essential components of P system are membrane structure, objects, and rules. The formalization definition of P system is shown as follows:

Here is the alphabet, representing the object; is the output alphabet and ; is the catalyst and ; is the set of membranes; is the object multisets, is the membrane label; is the set of evolutionary rules, some rules are applied to reflect the chemical reactions, such as rewriting rules, and others are employed to simulate biological processes, such as communication rules, and is the priority set of these rules.

##### 2.2. Description of the P Chain System

###### 2.2.1. Definitions and Properties of P Chain System

*Definition 5 (P system with chain structure). *Generalized P system with chain structure is written as , where is integer and represents the number of the membranes. Additionally, if , represents positive generalized P system with chain structure, especially if , denotes positive standard P chain system. Moreover, if , represents negative generalized P chain system, particularly if , denotes negative standard P chain system. Furthermore, if or , represents multiply generalized P chain system, especially if or , denotes multiply standard P chain system.

Generally, what we call P system with chain structure is referred to the standard P chain system. Here is an example for the above definition. There is a complex in Figure 1(a), and 0-simplex are these vertices , , , , , , , , 1-simplex are these segments , , , , , 2-simplex are these surfaces , 3-simplex are these bodies , and so on. There is a discrete gradient vector path on the 3-complex, composed by and , so there are 0-chain whose units come from and 1-chain whose units belong to . And then membranes from 0-chain or from 1-chain can compose a P system with chain structure. is an example of positive standard P chain system. can be called multiply generalized P chain system shown in Figure 1(b), where two-way arrow represents the repeated appearance of membrane, figure denotes the number of repeated appearance of its front membrane, that is to say, appears 5 times and 3 times, usually 1 is omitted.