Studies Of Dynamical Mechanism For Self-Reproduction In Cellular Automata

Cellular Automata are a kind of discrete mathematical models in physical space. It can be analyzed in mathematics precisely. So the computer resource can be used fully to compute its evolution as time precisely. Even if its form appears simple, it can show complex and unpredicted behavior, such as life game and Wolfram's four classes.Definition of Cellular Automata: a cellular automaton is a D-dimensional lattice with a finite automaton residing at each lattice site. Each automaton consists of a finite set of cell states. It takes as input the states of the automata within some finite local region of the lattice (neighborhood) and output the state of itself next time. The input and output satisfies transition function. Time progresses uniformly in discrete steps with all cells updating synchronously.In the CA space, two typical two-dimensional neighborhood templates are: Von Neumann neighborhood and Moore neighborhood. The states of cells are divided into quiescent and active ones. The active cells adjacent to each other can form a configuration, or a pattern. The CA showing complex behavior must have a non-quiescent initial configuration.Self– reproducing system: a kind of system in which reproducing process is directed by its own information. It can go smoothly without the influence of the external world. In the CA space, the behavior of self– reproducing is only caused by the rules for local interaction and self- organization. The'copying'is always taken for granted, being realized by centralized control program, whose task is just to make copies of the genome. An initial configuration is required to be in the CA space for Langton's self- reproducing process. The configuration is called Adam loop. Though the concept is not alike Von Neumann's, the Codd's data- path capable of extending is called construction arm, the cap is growing point.This paper makes the Langton's loop reappear and explores the dynamical mechanism of its realization in two- dimensional space with 8 states, Von Neumann neighborhood and strong rotational symmetry conditions.First, this paper investigates the components in the CA space. The essential role of components is found in the process of reproducing. Signals can make the components behave variously.1. Codd's data- path is the foundation of construction of all structures. Its realization is the first step of realizing signals'existence and flow.2. T- junction is a device of duplicating signals while they pass by.3. Periodic emitter is a timing device and storage element. It supplies various signals for the reproducing process.4. Codd's data- path with a cap can behave variously due to the signal in it, or it will keep quiet. When the signal meets cap, it may disappear, return, extend the path straight, to the left or right or retract the path. Additionally, signals causing the path extending straight, or to the left or right, may be one or two respectively. If a signal returned meets the one in the path, the two may disappear or cause new ones to come into being.5. While the Codd's data- path with a cap is combined with T- junction or cross, some interesting structures or behaviors may appear. Signals'duplicating and mergence can turn up in such structures.6. If the path extension sequence is put into the periodic emitter with a cap, a machine extending its path indefinitely will emerge.7. While two paths get close to each other, they will fuse together and form a new T- junction. It may be a T- junction where signals merge or new signals come into being.8. The structure formed from fusing will be cut and form two loops.Then, based on the above, this paper adopts Langton's initial configuration, adding the path's extending straight and to the left or right, and then realizes the reproduction of the loop. But note that the rules used here are differed from Chapter 2, though the number of signals is the same for certain function.1. In Chapter 3, two signals are used to extend the path to the left, but the way is different from that in Chapter 2 which not only builds a corner, but extends the path by one cell. However, the way used here only builds a corner with the help of state"3". In the loop,"3"appears and then disappears in the sheath.2. While two paths fuse, intermediary state"3"appears. And then a new T- junction is formed, where new signals will come into being for cutting the path and forming new loops.3. The loops'growing to adults is directed by new signals"5"and"6"respectively, even if for another way in Chapter 3.Moreover, the roles of the states of cells and signals are summarized in the process of seeking rules for reproduction.1."0"exists as the tail of a signal or the quiescent cells in the CA space.2."1"is the wire by which the signals propagate, and it can make a path without cap add a cap.3."2"is the sheath cell and growing point.4."3"plays a role of intermediary state. It is more active than signals and less than sheath cell.5."4"makes the path extend to the left.6."5"cuts the umbilical cord and initiates the construction of a new construction arm for parent loop.7."6"initiates the construction of a new construction arm for offspring loop.8."7"extends data- path by one cell.Finally, there are 36% rules different from Langton's in the new set of rules for self- reproduction.