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Exploring Discrete Dynamics - Second Edition

  • July 2016
  • Publisher: Luniver Press
  • ISBN: ISBN-10: 1-905986-47-5 ISBN-13:978-1-905986-47-7
Authors:
Andrew Wuensche at University of the West of England, Bristol
Andrew Wuensche
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Abstract

EXPLORING DISCRETE DYNAMICS (second edition) is a comprehensive guide to studying cellular automata and discrete dynamical networks with the classic software Discrete Dynamics Laboratory (www.DDLab.org), widely used in research and education. These collective networks are at the core of complexity and emergent self-organisation. With interactive graphics, DDLab is able to explore a huge diversity of behaviour, mostly terra incognita -- space-time patterns, and basins of attraction -- mathematical objects representing the convergent flow in state-space. Applications range within physics, mathematics, biology, cognition, society, economics and computation, and more specifically in neural and genetic networks, artificial life, and theories of memory. This second edition covers many new features. Advance Praise by Stuart Kauffman The great John von Neumann invented cellular automata. These discrete state finite automata have become a mainstay in the study of complex systems, exhibiting order, criticality, and chaos. Andy Wuensche's "Exploring Discrete Dynamics" 2016, is by far the most advanced tool for simulating such systems and has become widely important in the field of complexity.
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Supplementary resource (1)

EDD-2ndEd-frontcover
Data
July 2016
Andrew Wuensche
... These rules were found from a short-list [7,8] within an input-entropy scatterplot [17,18] sample of 93000+ isotropic rules, which classify rule-space between order, chaos and complexity. The input-entropy criteria in this sample follow "Life-Like" constraints to the extent that the rules are binary, isotropic, with a 3×3 Moore neighborhood --and with the λ parameter, the density of 1s in the full 512 rule-table, similar to the Game-of-Life where λ = 0.273. ...
... (1) (2) (1) (2) Figure 2: The Ameyalli-rule 2-phase glider moving East at a speed of c/2. 4 time-steps are shown across a fixed background with phases labelled (1) and (2). [16,18] with time running clockwise. Here the glider-gun is confined to its central core by type-B eaters. ...
... An isotropic CA rule based on a 2d binary 3×3 Moore neighborhood --can be defined by a series of methods that become ever simpler, clearer, and more concise, illustrated in figures 10, 11, and 12. In all these methods, a descending order (from left to right) of the neighborhood's decimal equivalent is employed, in line with Wolfram's classic convention [14,15] -rule-tables can then be expressed in decimal or hexadecimal [18]. The decimal equivalent of a 3×3 pattern is taken as a string in the order In these presentations the diagonal symmetry of each 8×8 block is a necessary but insufficient indication of isotropy. ...
The Ameyalli-Rule: Logical Universality in a 2D Cellular Automaton
Preprint
Full-text available
  • Jul 2021
We present a new spontaneously emergent glider-gun in a 2D Cellular Automaton and build the logical gates NOT, AND and OR required for logical universality. The Ameyalli-rule is not based on survival/birth logic but depends on 102 isotropic neighborhood groups making an iso-rule, which can drive an interactive input-frequency histogram for visualising iso-group activity and dependent functions for filtering and mutation. Neutral inputs relative to logical gates are identified which provide an idealized striped-down form of the iso-rule.
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... We present the ideas and methods mainly for 2d square and hexagonal examples as in figures 4 and 5, but also include 1d and 3d. Glider-rules that feature gliders emerging spontaneously are readily found by classifying rule-space by input-entropy variability [28,EDD:33], with examples in [22,23,9]. Glider-guns may also emerge spontaneously [23,25,11] but usually they are elaborately constructed artifacts [7,9,10,12]. ...
... Mutant iso-rule-space can be navigated and explored with the program "Discrete Dynamics Lab" (DDLab) [29] -its many methods for studying space-time patterns [28, and attractor basins [28,] now apply to the new iso-rule paradigm. DDLab is documented in the book "Exploring Discrete Dynamics"(EDD) [28], and we have usually included the relevant sections when citing EDD. ...
... Mutant iso-rule-space can be navigated and explored with the program "Discrete Dynamics Lab" (DDLab) [29] -its many methods for studying space-time patterns [28, and attractor basins [28,] now apply to the new iso-rule paradigm. DDLab is documented in the book "Exploring Discrete Dynamics"(EDD) [28], and we have usually included the relevant sections when citing EDD. ...
Isotropic Cellular Automata: the DDLab iso-rule paradigm
Preprint
Full-text available
  • Aug 2020
To respect physics and nature, cellular automata (CA) models of self-organisation, emergence, computation and logical universality should be isotropic, having equivalent dynamics in all directions. We present a novel paradigm, the iso-rule, a concise expression for isotropic CA by the output table for each isotropic neighborhood group, allowing an efficient method of navigating and exploring iso-rule-space. We describe new functions and tools in DDLab to generate iso-groups and iso-rules, for multi-value as well as binary, in one, two and three dimensions. These methods include filing, filtering, mutating, analysing dynamics by input-frequency and entropy, identifying the critical iso-groups for glider-gun/eater dynamics, and automatically classifying iso-rule-space. We illustrate these ideas and methods for two dimensional CA on square and hexagonal lattices.
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... Under this last approach, Sapin found a universal CA in 2004 [11], and Gómez-Soto/Wuensche published three more: the the X-Rule in 2015 [5], the Precursor-Rule in 2017 [6], and the Sayab-Rule in 2018 [7]. [13,16] incorporates the sub-glider-gun GGc. The period is 22 time-steps showing all phases/patterns of the GGa. ...
... The P15 oscillator, named for its 15 time-step frequency, is detailed in figure 15, and is used to build G2a and Ga glider-guns in figure 16. Figure 15: The P15 oscillator [22] showing all 15 phases (time-steps) as an attractor cycle [13,16] where the direction of time is clockwise. Inset: An oscillator phase shown at a larger scale alongside the same phase on the attractor cycle. ...
... The P22 oscillator showing all 22 phases (time-steps) as an attractor cycle[13,16] where the direction of time is clockwise. Inset: An oscillator phase shown at a larger scale alongside the same phase on the attractor cycle. ...
The Variant-Rule, Another Logically Universal Rule
Preprint
Full-text available
  • Sep 2019
The Variant-rule derives from the Precursor-rule by interchanging two classes of its 28 isotropic mappings. Although this small mutation conserves most glider types and stable blocks, glider-gun engines are changed, as are most large scale pattern behaviors, illustrating both the robustness and fragility of evolution. We demonstrate these newly discovered structures and dynamics, and utilising two different glider types, build the logical gates required for universality in the logical sence.
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... • The rules can be reinterpreted as reaction-diffusion systems with inhibitoractivator reagents in a chemical medium [6,7,2], where the three CA values are seen as: Activator, Inhibitor, and Substrate. • The availability of short-lists of glider rules, extracted from large samples of complex rules that are found (and sorted automatically) by the variability of input-entropy [8,9,10]. • The CAP model can be appled to bio-oscillations in excitable tissue according to classical 3-state neuronal dynamics: Firing, Refractory, and Ready to Fire. ...
... The entropy of the input-histogram can be calculated 2 and plotted with its characteristic wavelength (wl), wave-height (wh, twice amplitude), and waveform (its shape or phase), which in turn can generate an entropy-density scatter plot [8] (fig 5). From space-time patterns, the density or proportion of each value, (0, 1, 2) if v=3, can be plotted, and this can generate a density return-map scatter plot [10] (fig 6). The scatter plots have the characteristics of chaotic strange attractors, and successive dots can be connected to create a linked history -this option is much faster to produce the characteristic plot because just a few time-step are needed. ...
... We summarise below the steps in DDLab to run the experiments, referring to chapters and sections (denoted by #x.x) from the book "Exploring Discrete Dynamics -Second Edition" [10], -its pdf is kept updated online [11]. Having installed DDLab 4 , from a terminal in the directory ddlab/ddfiles, enter ../ddlabz07 -w & to start in a white screen. ...
The cellular automaton pulsing model, experiments with DDLab
Preprint
Full-text available
  • Nov 2018
The cellular automaton (CA) pulsing model (arXiv:1806.06416) described the surprising phenomenon of spontaneous, sustained and robust rhythmic oscillations, pulsing dynamics, when random wiring is applied to a 2D `glider' rule running in a 3-value totalistic CA. Case studies, pulsing measures, possible mechanisms, and implications for oscillatory networks in biology were presented. In this paper we summarise the results, extend the entropy-density and density-return map plots to include a linked history, look at totalistic glider rules with neighborhoods of 3, 4 and 5, as well as 6 and 7 studied previously, introduce methods to automatically recognise the wavelength, and extend results for randomly asynchronous updating. We show how the model is implemented in DDLab to validate results, output data, and allow experiments and research by others.
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... In 1993, a different reverse algorithm was invented [17] for the pre-images and basins of attraction of random Boolean networks (RBN) (figure 15) just in time to make the cover of Kauffman's seminal book [10] "The Origins of Order" (figure 3). The RBN algorithm was later generalised for "discrete dynamical networks" (DDN) described in "Exploring Discrete Dynamics" [26]. The algorithms, implemented in the software DDLab [27], compute pre-images directly, see detail Figure 1: Top: The basin of attraction field of a 1D binary CA, k=7, n=16 [21]. ...
... Below: A detail of the second basin, where states are shown as 4×4 bit patterns. Kauffman's (1993) "The Origins of Order" [10], and Wuensche's(2016) "Exploring Discrete Dynamics" 2nd Ed [26]. and basins of attraction are drawn automatically following flexible graphic conventions. ...
... There are countless variations, intermediate architectures, and hybrid systems, between CA and DDN. These systems can also be seen as instances of "random maps with out-degree one" (MAP) [19,26], a list of "exhausive pairs" where each state in state-space is assigned a random sucessor, possibly with some bias. All these systems reorganise state-space into basins of attraction. ...
Basins of Attraction of Cellular Automata and Discrete Dynamical Networks: A Volume in the Encyclopedia of Complexity and Systems Science, Second Edition
Chapter
Full-text available
  • Jan 2018
Basins of attraction of cellular automata and discrete dynamical networks link state-space according to deterministic transitions, giving a topology of trees rooted on attractor cycles. Applying reverse algorithms, basins of attraction can be computed and drawn automatically. They provide insights and applications beyond single trajectories, including notions of order, complexity, chaos, self-organisation, mutation, the genotype-phenotype, encryption, content addressable memory, learning, and gene regulation. Attractor basins are interesting as mathematical objects in their own right.
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... Complex glider dynamics arise from rare "complex" rules. Otherwise the dynamics and rule types, broadly speaking, are either "ordered" or "disordered" judged by subjective impressions of space-time patterns, but also by objective measures such as input-entropy and its variability [33], basin of attraction topology [32], and Derrida plots [11,39]. By any evaluation, disorder comprises the vast majority of rule-space. ...
... We pose the question: while preserving a homogeneous rule, what kind of dynamics would result if the regular local neighborhood connections (the wiring) of classical CA are randomised? -an experiment readily implemented in DDLab [39,40], with its functionality for toggling between regular and random wiring on-the-fly, and where random wiring can be fully random or confined in a local zone, or even re-randomised at each time-step. ...
... The pulsing is obvious to the subjective eye when observing spacetime patterns, but is also characterised by objective measures: the density of each value across the network, and the collective input-entropy. Time-plots of pulsing measures for each complex rule maintain a particular wavelength (wl), wave height (twice amplitude), and waveform (shape of the wave), and scatter plots of entropy/density [33] and the density return-map [39] show unique signatures, which have the characteristics of chaotic strange attractors. We will use the term "waveform" to sum up these pulsing measures. ...
Pulsing dynamics in randomly wired complex cellular automata
Preprint
Full-text available
  • Jun 2018
Sustained rhythmic oscillations, pulsing dynamics, directly emerge when the local connection scheme is randomised in 3-value complex cellular automata --- those characterised by emergent "glider" dynamics. With random wiring, time-plots of pulsing measures maintain a particular waveform for each complex rule, and scatter plots of entropy/density and the density return-map, show unique signatures, which have the characteristics of chaotic strange attractors. We present case studies, possible mechanisms, and implications for oscillatory networks in biology.
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... We note that properties of CA phase-spaces were examined among others by Wuensche et al. (2001). Precisely for this purpose, a software was designed by Wuensche (2016). ...
Classification of Discrete Dynamical Systems Based on Transients
Preprint
Full-text available
  • Aug 2021
In order to develop systems capable of artificial evolution, we need to identify which systems can produce complex behavior. We present a novel classification method applicable to any class of deterministic discrete space and time dynamical systems. The method is based on classifying the asymptotic behavior of the average computation time in a given system before entering a loop. We were able to identify a critical region of behavior that corresponds to a phase transition from ordered behavior to chaos across various classes of dynamical systems. To show that our approach can be applied to many different computational systems, we demonstrate the results of classifying cellular automata, Turing machines, and random Boolean networks. Further, we use this method to classify 2D cellular automata to automatically find those with interesting, complex dynamics. We believe that our work can be used to design systems in which complex structures emerge. Also, it can be used to compare various versions of existing attempts to model open-ended evolution (Ray (1991), Ofria et al. (2004), Channon (2006)).
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... Experiments were done with Discrete Dynamics Lab [24,25], Mathematica and Golly. The Sayab-Rule was found during a collaboration at June workshops in 2017 at the DDLab Complex Systems Institute in Ariege, France, and also at the Universidad Autónoma de Zacatecas, México, and in London, UK. ...
Logical Universality from a Minimal Two-Dimensional Glider Gun
Article
Full-text available
  • Mar 2018
To understand the underlying principles of self-organization and computation in cellular automata, it would be helpful to find the simplest form of the essential ingredients, glider guns and eaters, because then the dynamics would be easier to interpret. Such minimal components emerge spontaneously in the newly discovered Sayab rule, a binary two-dimensional cellular automaton with a Moore neighborhood and isotropic dynamics. The Sayab rule’s glider gun, which has just four live cells at its minimal phases, can implement complex dynamical interactions and the gates required for logical universality. © 2018, Complex Systems Publications, Inc. All rights reserved.
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Classification of Complex Systems Based on Transients
Preprint
  • Aug 2020
In order to develop systems capable of modeling artificial life, we need to identify, which systems can produce complex behavior. We present a novel classification method applicable to any class of deterministic discrete space and time dynamical systems. The method distinguishes between different asymptotic behaviors of a system's average computation time before entering a loop. When applied to elementary cellular automata, we obtain classification results, which correlate very well with Wolfram's manual classification. Further, we use it to classify 2D cellular automata to show that our technique can easily be applied to more complex models of computation. We believe this classification method can help to develop systems, in which complex structures emerge.
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