CELLULAR AUTOMATA WITH CHAOTIC RULE FORMATION AND ITS CHARACTERISTICS ABSTRACT

Transcription

1 CELLULAR AUTOMATA WITH CHAOTIC RULE FORMATION AND ITS CHARACTERISTICS Christopher Immanuel W * 1 Paul Mary Deborrah S 2 Samuel Selvaraj R 3 * Research and Development Centre, Bharathiar University, Coimbatore Department of Physics, Vel Tech High Tech Engineering College, Chennai, India. 2 Department of Physics, TheAmerican College, Madurai, India. 3 Department of Physics, Presidency College, Chennai, India. ABSTRACT The fractal procedure known as cellular automata are applied to the different field.cellular Automata are not only of theoretical interest. One of the main topics in CA research is their usage as models of phenomena which can be observed in real life, the practical applications of Cellular Automata can be seen in multiple fields of science and social science. They range from computer science, technology, physics, biology, and math, to economics, psychology, philosophy, and even art. Cellular Automata are discrete dynamical systems, of simple construction but complex and varied behavior.despite of the simplicity of their construction; Cellular Automata are capable of very complicated behavior with very little specification. The present paper introducesa cellular automata and their applicationwith some rules. It is very useful for future references. Keywords: Cellular Automaton, Fractal Dimension, Chaotic Rule. 97

2 1. Introduction This paper should give a short overview of what cellular automata are, how they work and what they can be used for. The next section starts with the history, how cellular automata were invented and briefly discuss some of their properties. Cellular automata are a relatively new development in modern science. It consists of simple progressions from algorithms or Rules over time.cellular automata (CA) are mathematical models for complex natural systems containing large numbers of simple identical componentswith local interactions. Cellular automata rules can be classified automatically for a spectrum of ordered, complex and chaotic dynamics, by a measure of the variance of input entropy overtime. They are discrete dynamical systems exhibiting a wide range of complex behavior;from on the one hand, point and cyclic behavior yielding structures of relative periodicity and, on the other hand, utterly chaotic attractors. Chaos and complexity theory teaches us that simple rules can lead to enormous levels of complexity. Cellular Automata or Cellular Spaces as they were called then were invented by John von Neumann half a century ago. 2. The basic idea 98

3 What are cellular automata? The word cellular here means consisting of cells a cellular automaton is made up of cells. Each of those cells contains an automaton, a finite state machine. The resulting construct is a space filled with cells, each of those cells containing a finite state machine. We say a cell has a state, meaning the state, the cells finite state machine is in. Usually the space is divided into a grid or lattice structure. In the cellular space, all the cells go from their current state to some next state according to the local rule, the switching rule of their state machine. The input for the local rule is the neighborhood of the cell, which is usually a small number of cells surrounding the cell. All the cells use the same local rule.the entire cells switch from their current state to the next state at the same time, the space is synchronous. A Cellular Automata consists of processors (cells), connected usually in an n-dimensional grid. Fig.1 One Dimensional CA 99

4 Fig.2 Two Dimensional CA Rules are formed through a definition of the 2 3 = 8 possible progressions of three cells (the cell, the cells left-hand neighbor, and the cells right-hand neighbor). Each of these progressions gives a single output, producing a new cell and creating a three to one mapping. A cell dies or lives according to some transition rule 100

5 Fig.3Cell transition from T=0 to T=1 3. Some terms The following terms will be used in the remainder of this paper. Cell A single element of a cellular space, the smallest unit of the space. Cellular Space A lattice space made up of cells, each of which is in one of several predefined states. This is what most people nowadays call a Cellular Automaton, but I find the distinction useful. Cellular Automaton A structure built in a cellular space, an automaton built out of cells. The term is often used as a synonym for Cellular Space, but to be clearer, I ll try not use it that way. CA is an abbreviation that is often used for these terms. Cellular Automata Plural of Cellular Automaton, sometimes used instead of the singular form. 101

6 Local Rule The rule governing the transition between states. The definition of a cell s finite state machine. It s called local, because it only uses the Neighborhood as its input. Neighborhood The cells surrounding a cell that influencesits next state. The choice of neighborhood influences the behavior of the cellular space. Configuration A snapshot of all cell states, representing a single point in time. When we talk about a configuration, it s usually the starting point or a result of running a cellular space. GenerationOne step in the evolution of a cellular space, an intermediate configuration. Passage of time in a cellular space is measured in generations. 4. History of CA In the 1950s, John von Neumann tried to construct self-replicating machines. He wanted to find the simplest machines that could build copies of themself, but to be useful, they also had to be universal computers. According to [John von Neumann 1951] started out working with differential equations until his colleague Stanislaw Ulam suggested a different model, similar to the lattice spaces Ulam used to study crystals, which lead to a significant simplification Von Neumann s automata Von Neumann realized that a 2-dimensional model would be sufficient. In his 2-dimensional model, a cellular space was formed of cells that could be in one of several possible states. Each 102

7 cell would change its state according to a local rule that calculated the new state as a function of the four nearest neighboring cells (the orthogonal neighbors, this type of neighborhood is now called a von Neumann neighborhood). Von Neumann called his creation a cellular space and the constructions in that space were the cellular automata. The cells in von Neumann s space could be in one of 29 states and there was a complicated set of rules for the movement between these states. For a detailed explanation of these states see e.g. [John von Neumann 1951]. But self-replication was only half of the problem; there were actually two properties von Neumann was looking for: universal computation and universal construction. The first was to make sure that the automata he constructed were not trivial; the second was the basis for self-replication. Universal computation meant that constructions in the cellular space were capable of computing any computable function. The simplest way of proving this property was showing that a Turing machine can be built in the cellular space. Universal construction meant that an automaton could be built that could create other constructions in the cellular space. A programmable construction machine was created by coupling a Turing machine with a universal constructor. Although von Neumann developed the core of his self-replicating machines, he didn t finish the work. The complete model was published after his death in 1966 by Arthur W. Burks. Following this publication, a small number of other works were published, including a monograph by E. F. Codd Codd s space 103

8 In his Monograph, Codd summarizes his research into the requirements for universal computation and construction. He shows that there exists no cellular space with 2 states and a local rule using the von Neumann neighborhood that exhibits universality. He then proceeds to develop a cellular space that does exhibit universality but has simpler rules than von Neumann s space. He shows how both universal computation and universal construction can be realized in his simpler space. Another first is the fact that he tries out his cellular automata on-line, ie. on a computer terminal as opposed to having simulations run in batch mode. In a section titled Why On-Line? he argues Rapid interaction between man and machine tends to develop the intuition and perceptivity of the experimenter. The immediacy of the machine s response has the effect of maintaining a high level of cerebral activity in the experimenter. He may have been the first to experience the fascination of cellular automata first hand! Codd was not the first to create a simpler cellular space, though. According to Sarkar, in 1966 Arbib created self-replicating universal computers in a space with 4 different states and von Neumann neighborhood. 4.3.Konradzuse s rechnenderraum KonradZuse described how the whole of space might be in reality a cellular space of the kind von Neumann described. He called this space Computing Space ( RechnenderRaum in German). The idea was later picked up by both Edward Fredkin [Hurd 1990] and StephenWolfram. Compelling as the idea may be, there are no practical applications. 104

9 4.4. Conway s game of life Short after Codd s experiments, the British mathematician John Horton Con-way developed another cellular space that he called Game of Life. The Game of Life was a breakthrough because it is one of the simplest cellular spaces to exhibit universal computation and universal construction, although that was not known when the Game of Life was first published. Martin Gardner wrote about the Game of Life in his column in Scientific American and it became very popular to run simulations of it on the mini computers of the time. Although Conway constructed the rules for his cellular space in a way that should allow for complex interaction, the constructions that are possible in this space were discovered after the rules for the space had been set. In contrast, most of the patterns used by Codd and von Neumann to construct their automata were developed at the same time as the rules governing their spaces, the rules are complex and it seems clear that they were crafted specifically to enable certain constructions. The rules for Conway s Game of Life are actually very different from the ones von Neumann and Codd used they are totalistic, which means that only the number of neighbors that are in a certain state is important, their positions are not taken into account. In Codd s and von Neumann s spaces, most of the rules had directional components only neighbors in certain positions are considered in each state. This difference means that Conway s rules are much simpler than Codd s and von Neumann s. 105

10 Around this time a trend started to call the cellular spaces cellular automata, the former distinction blurring, possibly because no one thought of the gliders and guns in Conway s Game of Life as automata, they seemed more like multi celled organisms that sprang to life in this foreign universe Wolfram s surveys of cellular space On a quest to figure out where the complexity in nature comes from, Stephen Wolfram encountered cellular automata (a colleague told him to have a look and look he did). In contrast to Codd, who thought that exhaustive search is out of the question ), he wanted to do an exhaustive survey of possible cellular automata (or spaces, as Codd would have called them). But there are too many 2-dimensional cellular spaces so he simplified the model even further.wolfram looked at the simplest possible cellular automata: 1-dimensional automata with two states and a three cell neighborhood. Since there are eight possible input configurations of three cells with two possible states, there are 256 possible local rules for these cellular automata.of these 256 possible rules, only 32 are legal under the restrictions he used in his initial survey.as a result of his experiments, he grouped the rules into three classes the first class that evolves from arbitrary initial conditions into a static pattern, like rules numbers 0 or 4; the second class that evolves into simple crystalline structures and the third class that evolves into complex self-similar patterns. 106

11 Wolfram refined his classification scheme later into four classes, splitting the class of rules with complex behavior intocomplex andchaoticrules. The rules with chaotic behavior can be used to generate randomness, while some of the rules with complex behavior are computation universal. 5. Cellular automata and the mechanisms of nature Cellular automata are examples of simple programs, that work by having the color of each cell in successive rows be determined by the same simple rule that certain simple rules like the one on the right can produce patterns and behavior of immense complexity. Rule 30 cellular automaton Rule 30 is a one-dimensional binary cellular automaton. current pattern new state for center cell Fig.4One-dimensional binary cellular automaton Rule

12 Rule 30 exhibits class 3 behaviors, meaning even simple input patterns such as that shown lead to chaotic, seemingly random histories. With the rule above, a simple pattern is produced. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 108

13 Step 7 Step 8 109

14 Fig.5Successive lines give configurations obtained on successive time steps, with white and black squares representing sites with values 0 and 1 respectively. The above figure showed patterns produced by evolution according to various Cellular automaton rules, starting from typical disordered initial conditions, in which the value of each site is randomly chosen to be zero or one. Such cellular automata can be used as models of randomness in nature. some class 3 cellular automata yield rather simple patterns, which are typically periodic or at least self-similar (almost periodic). There are never the less class 3 cellular automatawhich yield complex patterns, even from simple initial states. Their evolution can intrinsically produce apparent randomness, withoutexternal input of random initial conditions. Many class 3 cellular automata seem to perform very complicated trans-formations on their initial conditions. Their evolution thus corresponds to a complicated computation. But any predictions of the cellular automaton behavior must also be obtained through computations. Most class 3cellular automata are expected to be computationally irreducible. A few rules however 110

15 have special simplifying features which make predictions and analysis possible. Most class 3 cellular automata are however nonlinear. But the two rules that seem best as random sequence generators are nonlinear, and are given by s i n+1 =s i-1 nxor (s i nor s i+1 n) (1) or, equivalently, a i = (a i-1 + a i +a i+1 + a i a i+1 ) mod 2 (2) The cellular automaton rule 30 is essentially nonlinear. Any class 3 (chaotic) cellular automaton rule can be considered a candidate random sequence generator. Here, we shall restrict ourselves to the case of one-dimensional binary CA ofncells with neighborhood ofsize equal to 3, defined as elementaryby Wolfram. In these models, Lis a linear array where each celladopts one of two possible states ins={0,1}and thedynamical rule is the form s i n+1 =f (s i-1 n, s i n,s i+1 n) means that,n(i) s i n+1 =f (s i-1 n, s i n,s i+1 n) In this case, there are 256 possible rules f. If {bi}is a binary string, corresponding to the result of applying f to each different set of 3-tuples (s i-1 n, s i n, s i+1 n) sorted in descending order,it is possible to use the Wolfram s naming scheme to assign, to each rule, a label given by the decimal representation of the binary sequence. Rules of elementary automata were classified qualitatively by [Wolfram 1984] according to its asymptotic behavioras Class I to IV, if the automata evolution 111

16 asymptoticallygoes to: a constant state, an isolated periodic segments,chaotic regime or isolated chaotic segments, respectively.in contrast with the continuous case, the elements ofthe Class III can be linear or nonlinear CA. The linear arethe rules that can be written as In the case of elementary CA this class contains 16 elementsand the rest are nonlinear. 6. Main characteristics of cellular automata a. Space is discrete, and there is a regular array of sites (cells). Each cell has a finite set of values. b. Time is discrete, and the value of each site is up-dated in a sequence of discrete time steps. c. The Rule for the new value of a site depends only on the values of a local neighborhood of sites near it. d. The variables at each site are up-dated simultaneously ( synchronously ) based on the values of the variables at the previous time step. Conclusion The model studied in this paper was developed in an attempt to increase our understanding of complex behavior in dynamical systems. It is well known that dynamical systems differ dramatically in several important aspects of their behavior, such as periodicity, predictability, 112

17 stability, dependence on initial conditions, etc. Systems that are nasty in these respects are called chaotic. There are many important, and sometimes deep, facts known about chaotic dynamical systems. The cellular automata are the new style, high performance simulation tool. We hope researchers can bring into play well the modeling power of the CA approach in future in variety complex systems. Cellular automata have been successfully used to model many variety complex systems, such as biology, chemistry, mathematics, physics field. A cellular automaton model allows the formulation of a dynamic complex system application in simple rules. Based on standard CA, there are of course many corrective and extended computational models for different applied objectives. According to simple local transition functions, CA are intuitively regarded as a set of interacting elements are updated during a discrete time interval. References 1.John von Neumann,The general and logical theory of automata, Cerebral Mechanisms in Behavior.[ The Hixon Symposium, John Wiley & Sons Inc., New York, N. Y., 3] 1951, pp K. C., Hurd, L. P., Yu, S.: Computation Theoretic Aspects of Cellular Automata, Physica D, 45, 1990, Wolfram, S. "Universality and Complexity in Cellular Automata." Physica D 10, 1-35,1984. Wolfram 113