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published_or_final_version / Physics / Master / Master of Philosophy

New tools are available for reconstructing the attractor basins of discrete dynamical networks where state-space is linked according the network's dynamics. In this thesis the computer software "Discrete Dynamics Lab" is applied to examine simple networks ranging from cellular automata (CA) to random Boolean networks (RBN), that have been widely applied as idealised models of physical and biological systems, to search for general principles underlying their dynamics. The algorithms and methods for generating pre-images for both CA and RBN, and reconstructing and representing attractor basins are described, and also considered in the mathematical context of random directed graphs. RBN and CA provide contrasting notions of self-organisation. RBN provide models of hierarchical categorisation in biology, for example memory in neural and genomic networks. CA provide models at the lower level of emergent complex pattern. New measures and results are presented on CA attractor basins and how they relate to measures on local dynamics and the Z parameter, characterising ordered to "complex" to chaotic behaviour. A method is described for classifying CA rules by an entropy-variance measure which allows glider rules and related complex rules to be found automatically giving a virtually unlimited sample for further study. The dynamics of RBN and intermediate network architectures are examined in the context of memory, where categorisation occurs at the roots of subtrees as well as at attractors. Learning algorithms are proposed for "sculpting" the basin of attraction field. RBN are proposed as a possible neural network model, and also discussed as a model of genomic regulatory networks, where cell types have been explained as attractors

The creation and use of models is critical not only to the scientific process, but also to life in general. Selected features of a system are abstracted into a model that can then be used to gain knowledge of the workings of the observed system and even anticipate its future behaviour. A key feature of the modelling process is the identification of commonality. This allows previous experience of one model to be used in a new or unfamiliar situation. This recognition of commonality between models allows standards to be formed, especially in areas such as measurement. How everyday physical objects are measured is built on an ingrained acceptance of their underlying commonality. Complex systems, often with their layers of interwoven interactions, are harder to model and, therefore, to measure and predict. Indeed, the inability to compute and model a complex system, except at a localised and temporal level, can be seen as one of its defining attributes. The establishing of commonality between complex systems provides the opportunity to find common metrics. This work looks at two dimensional cellular automata, which are widely used as a simple modelling tool for a variety of systems. This has led to a very diverse range of systems using a common modelling environment based on a lattice of cells. This provides a possible common link between systems using cellular automata that could be exploited to find a common metric that provided information on a diverse range of systems. An enhancement of a categorisation of cellular automata model types used for biological studies is proposed and expanded to include other disciplines. The thesis outlines a new metric, the C-Value, created by the author. This metric, based on the connectedness of the active elements on the cellular automata grid, is then tested with three models built to represent three of the four categories of cellular automata model types. The results show that the new C-Value provides a good indicator of the gathering of active cells on a grid into a single, compact cluster and of indicating, when correlated with the mean density of active cells on the lattice, that their distribution is random. This provides a range to define the disordered and ordered state of a grid. The use of the C-Value in a localised context shows potential for identifying patterns of clusters on the grid.

by Siu Chi Sang Obadiah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 66-68). / Abstracts in English and Chinese. / Chapter 1 --- Pseudorandom Number Generator --- p.5 / Chapter 1.1 --- Introduction --- p.5 / Chapter 1.2 --- Statistical Indistingushible and Entropy --- p.7 / Chapter 1.3 --- Example of PNG --- p.9 / Chapter 2 --- Basic Knowledge of Cellular Automata --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Elementary and Totalistic Cellular Automata --- p.14 / Chapter 2.3 --- Four classes of Cellular Automata --- p.17 / Chapter 2.4 --- Entropy --- p.20 / Chapter 3 --- Theoretical analysis of the CA PNG --- p.26 / Chapter 3.1 --- The Generator --- p.26 / Chapter 3.2 --- Global Properties --- p.27 / Chapter 3.3 --- Stability Properties --- p.31 / Chapter 3.4 --- Particular Initial States --- p.33 / Chapter 3.5 --- Functional Properties --- p.38 / Chapter 3.6 --- Computational Theoretical Properties --- p.42 / Chapter 3.7 --- Finite Size Behaviour --- p.44 / Chapter 3.8 --- Statistical Properties --- p.51 / Chapter 3.8.1 --- statistical test used --- p.54 / Chapter 4 --- Practical Implementation of the CA PNG --- p.56 / Chapter 4.1 --- The implementation of the CA PNG --- p.56 / Chapter 4.2 --- Applied to the set of integers --- p.58 / Chapter 5 --- Application to Cryptography --- p.61 / Chapter 5.1 --- Stream Cipher --- p.61 / Chapter 5.2 --- One Time Pad --- p.62 / Chapter 5.3 --- Probabilistic Encryption --- p.63 / Chapter 5.4 --- Probabilistic Encryption with RSA --- p.64 / Chapter 5.5 --- Prove yourself --- p.65 / Bibliography

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This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.

This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.

There has been a move recently in academia, industry, and the consumer space towards the use of unsupervised parallel computation and distributed networks (i.e., networks of computing elements working together to achieve a global outcome with only local knowledge). To fully understand the types of problems that these systems are applied to regularly, a representative member of this group of unsupervised parallel and distributed systems is needed to allow the development of generalizable results. Although not the only potential candidate, the field of cellular automata is an excellent choice for analyzing how these systems work as it is one of the simplest members of this group in terms of design specification. The current ability of the field of cellular automata to represent the realm of unsupervised parallel and distributed systems is limited to only a subset of the possible systems, which leads to the main goal of this work of finding a method of allowing cellular automata to represent a much larger range of systems. To achieve this goal, a conceptual framework has been developed that allows the definition of interconnected systems of cellular automata that can represent most, if not all, unsupervised parallel and distributed systems. The framework introduces the concept of allowing the boundary conditions of a cellular automaton to be defined by a separately specified system, which can be any system that is capable of producing the information needed, including another cellular automaton. Using this interconnection concept, two forms of computational simplification are enabled: the deconstruction of a large system into smaller, modular pieces; and the construction of a large system built from a heterogeneous set of smaller pieces. This framework is formally defined using an interconnection graph, where edges signify the flow of information from one node to the next and the nodes are the various systems involved. A library has been designed which implements the interconnection graphs defined by the framework for a subset of the possible nodes, primarily to allow an exploration of the field of cellular automata as a potential representational member of unsupervised parallel and distributed systems. This library has been developed with a number of criteria in mind that will allow it to be instantiated on both hardware and software using an open and extendable architecture to enable interaction with external systems and future expansion to take into account novel research. This extendability is discussed in terms of combining the library with genetic algorithms to find an interconnected system that will satisfy a specific computational goal. There are also a number of novel components of the library that further enhance the capabilities of potential research, including methods for automatically building interconnection graphs from sets of cellular automata and the ability to skip over static regions of a given cellular automaton in an intelligent way to reduce computation time. With a particular set of cellular automaton parameters, the use of this feature reduced the computation time by 75%. As a demonstration of the usefulness of both the library and the framework that it implements, a hardware application has been developed which makes use of many of the novel aspects that have been introduced to produce an interactive art installation named 'Aurora'. This application has a number of design requirements that are directly achieved through the use of library components and framework definitions. These design requirements included a lack of centralized control or data storage, a need for visibly dynamic behaviour in the installation, and the desire for the visitors to the installation to be able to affect the visible movement of patterns across the surface of the piece. The success of the library in this application was heavily dependent on its instantiation on a mixture of hardware and software, as well as the ability to extend the library to suit particular needs and aspects of the specific application requirements. The main goal of this thesis research, finding a method that allows cellular automata to represent a much larger range of unsupervised parallel and distributed systems, has been partially achieved in the creation of a novel framework which defines the concept of interconnection, and the design of an interconnection graph using this concept. This allows the field of cellular automata, in combination with the framework, to be an excellent representational member of an extended set of unsupervised parallel and distributed systems when compared to the field alone. A library has been developed that satisfies a broad set of design criteria that allow it to be used in any future research built on the use of cellular automata as this representational member. A hardware application was successfully created that makes use of a number of novel aspects of both the framework and the library to demonstrate their applicability in a real world situation.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 65-66).