Designing complex dynamics in cellular automata with memory

Since their inception at Macy conferences in later 1940s, complex systems have remained the most controversial topic of interdisciplinary sciences. The term "complex system" is the most vague and liberally used scientific term. Using elementary cellular automata (ECA), and exploiting the CA classification, we demonstrate elusiveness of "complexity" by shifting space-time dynamics of the automata from simple to complex by enriching cells with memory. This way, we can transform any ECA class to another ECA class - without changing skeleton of cell-state transition function - and vice versa by just selecting a right kind of memory. A systematic analysis displays that memory helps "discover" hidden information and behavior on trivial - uniform, periodic, and nontrivial - chaotic, complex - dynamical systems. © World Scientific Publishing Company

Compression-based investigation of the dynamical properties of cellular automata and other systems

A method for studying the qualitative dynamical properties of abstract computing machines based on the approximation of their program-size complexity using a general lossless compression algorithm is presented. It is shown that the compression-based approach classifies cellular automata (CA) into clusters according to their heuristic behavior, with these clusters showing a correspondence with Wolfram's main classes of CA behavior. A compression based method to estimate a characteristic exponent to detect phase transitions and measure the resiliency or sensitivity of a system to its initial conditions is also proposed. A conjecture regarding the capability of a system to reach computational universality related to the values of this phase transition coefficient is formulated. These ideas constitute a compression-based framework for investigating the dynamical properties of cellular automata and other systems.Comment: 28 pages. This version includes the conjecture relating the transition coefficient to computational universality. Camera ready versio

Layered Cellular Automata

Layered Cellular Automata (LCA) extends the concept of traditional cellular automata (CA) to model complex systems and phenomena. In LCA, each cell's next state is determined by the interaction of two layers of computation, allowing for more dynamic and realistic simulations. This thesis explores the design, dynamics, and applications of LCA, with a focus on its potential in pattern recognition and classification. The research begins by introducing the limitations of traditional CA in capturing the complexity of real-world systems. It then presents the concept of LCA, where layer 0 corresponds to a predefined model, and layer 1 represents the proposed model with additional influence. The interlayer rules, denoted as f and g, enable interactions not only from adjacent neighboring cells but also from some far-away neighboring cells, capturing long-range dependencies. The thesis explores various LCA models, including those based on averaging, maximization, minimization, and modified ECA neighborhoods. Additionally, the implementation of LCA on the 2-D cellular automaton Game of Life is discussed, showcasing intriguing patterns and behaviors. Through extensive experiments, the dynamics of different LCA models are analyzed, revealing their sensitivity to rule changes and block size variations. Convergent LCAs, which converge to fixed points from any initial configuration, are identified and used to design a two-class pattern classifier. Comparative evaluations demonstrate the competitive performance of the LCA-based classifier against existing algorithms. Theoretical analysis of LCA properties contributes to a deeper understanding of its computational capabilities and behaviors. The research also suggests potential future directions, such as exploring advanced LCA models, higher-dimensional simulations, and hybrid approaches integrating LCA with other computational models.Comment: This thesis represents the culmination of my M.Tech research, conducted under the guidance of Dr. Sukanta Das, Associate Professor at the Department of Information Technology, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India. arXiv admin note: substantial text overlap with arXiv:2210.13971 by other author

Exponential convergence to equilibrium in cellular automata asymptotically emulating identity

We consider the problem of finding the density of 1's in a configuration obtained by $n$ iterations of a given cellular automaton (CA) rule, starting from disordered initial condition. While this problems is intractable in full generality for a general CA rule, we argue that for some sufficiently simple classes of rules it is possible to express the density in terms of elementary functions. Rules asymptotically emulating identity are one example of such a class, and density formulae have been previously obtained for several of them. We show how to obtain formulae for density for two further rules in this class, 160 and 168, and postulate likely expression for density for eight other rules. Our results are valid for arbitrary initial density. Finally, we conjecture that the density of 1's for CA rules asymptotically emulating identity always approaches the equilibrium point exponentially fast.Comment: 20 pages, 4 figures, 2 table

A framework for the local information dynamics of distributed computation in complex systems

The nature of distributed computation has often been described in terms of the component operations of universal computation: information storage, transfer and modification. We review the first complete framework that quantifies each of these individual information dynamics on a local scale within a system, and describes the manner in which they interact to create non-trivial computation where "the whole is greater than the sum of the parts". We describe the application of the framework to cellular automata, a simple yet powerful model of distributed computation. This is an important application, because the framework is the first to provide quantitative evidence for several important conjectures about distributed computation in cellular automata: that blinkers embody information storage, particles are information transfer agents, and particle collisions are information modification events. The framework is also shown to contrast the computations conducted by several well-known cellular automata, highlighting the importance of information coherence in complex computation. The results reviewed here provide important quantitative insights into the fundamental nature of distributed computation and the dynamics of complex systems, as well as impetus for the framework to be applied to the analysis and design of other systems.Comment: 44 pages, 8 figure

An Experimental Study of Robustness to Asynchronism for Elementary Cellular Automata

Cellular Automata (CA) are a class of discrete dynamical systems that have been widely used to model complex systems in which the dynamics is specified at local cell-scale. Classically, CA are run on a regular lattice and with perfect synchronicity. However, these two assumptions have little chance to truthfully represent what happens at the microscopic scale for physical, biological or social systems. One may thus wonder whether CA do keep their behavior when submitted to small perturbations of synchronicity. This work focuses on the study of one-dimensional (1D) asynchronous CA with two states and nearest-neighbors. We define what we mean by ``the behavior of CA is robust to asynchronism'' using a statistical approach with macroscopic parameters. and we present an experimental protocol aimed at finding which are the robust 1D elementary CA. To conclude, we examine how the results exposed can be used as a guideline for the research of suitable models according to robustness criteria.Comment: Version : Feb 13th, 2004, submitted to Complex System