Conway's Game of Life
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Think Complexity: Exploring Complexity Science in Python
Think Complexity Version 2.6.2 Think Complexity Version 2.6.2 Allen B. Downey Green Tea Press Needham, Massachusetts Copyright © 2016 Allen B. Downey. Green Tea Press 9 Washburn Ave Needham MA 02492 Permission is granted to copy, distribute, transmit and adapt this work under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License: https://thinkcomplex.com/license. If you are interested in distributing a commercial version of this work, please contact the author. The LATEX source for this book is available from https://github.com/AllenDowney/ThinkComplexity2 iv Contents Preface xi 0.1 Who is this book for?...................... xii 0.2 Changes from the first edition................. xiii 0.3 Using the code.......................... xiii 1 Complexity Science1 1.1 The changing criteria of science................3 1.2 The axes of scientific models..................4 1.3 Different models for different purposes............6 1.4 Complexity engineering.....................7 1.5 Complexity thinking......................8 2 Graphs 11 2.1 What is a graph?........................ 11 2.2 NetworkX............................ 13 2.3 Random graphs......................... 16 2.4 Generating graphs........................ 17 2.5 Connected graphs........................ 18 2.6 Generating ER graphs..................... 20 2.7 Probability of connectivity................... 22 vi CONTENTS 2.8 Analysis of graph algorithms.................. 24 2.9 Exercises............................. 25 3 Small World Graphs 27 3.1 Stanley Milgram......................... 27 3.2 Watts and Strogatz....................... 28 3.3 Ring lattice........................... 30 3.4 WS graphs............................ 32 3.5 Clustering............................ 33 3.6 Shortest path lengths...................... 35 3.7 The WS experiment....................... 36 3.8 What kind of explanation is that?............... 38 3.9 Breadth-First Search..................... -
Automatic Detection of Interesting Cellular Automata
Automatic Detection of Interesting Cellular Automata Qitian Liao Electrical Engineering and Computer Sciences University of California, Berkeley Technical Report No. UCB/EECS-2021-150 http://www2.eecs.berkeley.edu/Pubs/TechRpts/2021/EECS-2021-150.html May 21, 2021 Copyright © 2021, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement First I would like to thank my faculty advisor, professor Dan Garcia, the best mentor I could ask for, who graciously accepted me to his research team and constantly motivated me to be the best scholar I could. I am also grateful to my technical advisor and mentor in the field of machine learning, professor Gerald Friedland, for the opportunities he has given me. I also want to thank my friend, Randy Fan, who gave me the inspiration to write about the topic. This report would not have been possible without his contributions. I am further grateful to my girlfriend, Yanran Chen, who cared for me deeply. Lastly, I am forever grateful to my parents, Faqiang Liao and Lei Qu: their love, support, and encouragement are the foundation upon which all my past and future achievements are built. Automatic Detection of Interesting Cellular Automata by Qitian Liao Research Project Submitted to the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, in partial satisfaction of the requirements for the degree of Master of Science, Plan II. -
Interview with John Horton Conway
Interview with John Horton Conway Dierk Schleicher his is an edited version of an interview with John Horton Conway conducted in July 2011 at the first International Math- ematical Summer School for Students at Jacobs University, Bremen, Germany, Tand slightly extended afterwards. The interviewer, Dierk Schleicher, professor of mathematics at Jacobs University, served on the organizing com- mittee and the scientific committee for the summer school. The second summer school took place in August 2012 at the École Normale Supérieure de Lyon, France, and the next one is planned for July 2013, again at Jacobs University. Further information about the summer school is available at http://www.math.jacobs-university.de/ summerschool. John Horton Conway in August 2012 lecturing John H. Conway is one of the preeminent the- on FRACTRAN at Jacobs University Bremen. orists in the study of finite groups and one of the world’s foremost knot theorists. He has written or co-written more than ten books and more than one- received the Pólya Prize of the London Mathemati- hundred thirty journal articles on a wide variety cal Society and the Frederic Esser Nemmers Prize of mathematical subjects. He has done important in Mathematics of Northwestern University. work in number theory, game theory, coding the- Schleicher: John Conway, welcome to the Interna- ory, tiling, and the creation of new number systems, tional Mathematical Summer School for Students including the “surreal numbers”. He is also widely here at Jacobs University in Bremen. Why did you known as the inventor of the “Game of Life”, a com- accept the invitation to participate? puter simulation of simple cellular “life” governed Conway: I like teaching, and I like talking to young by simple rules that give rise to complex behavior. -
Cellular Automata
Cellular Automata Dr. Michael Lones Room EM.G31 [email protected] ⟐ Genetic programming ▹ Is about evolving computer programs ▹ Mostly conventional GP: tree, graph, linear ▹ Mostly conventional issues: memory, syntax ⟐ Developmental GP encodings ▹ Programs that build other things ▹ e.g. programs, structures ▹ Biologically-motivated process ▹ The developed programs are still “conventional” forwhile + 9 WRITE ✗ 2 3 1 i G0 G1 G2 G3 G4 ✔ ⟐ Conventional ⟐ Biological ▹ Centralised ▹ Distributed ▹ Top-down ▹ Bottom-up (emergent) ▹ Halting ▹ Ongoing ▹ Static ▹ Dynamical ▹ Exact ▹ Inexact ▹ Fragile ▹ Robust ▹ Synchronous ▹ Asynchronous ⟐ See Mitchell, “Biological Computation,” 2010 http://www.santafe.edu/media/workingpapers/10-09-021.pdf ⟐ What is a cellular automaton? ▹ A model of distributed computation • Of the sort seen in biology ▹ A demonstration of “emergence” • complex behaviour emerges from Stanislaw Ulam interactions between simple rules ▹ Developed by Ulam and von Neumann in the 1940s/50s ▹ Popularised by John Conway’s work on the ‘Game of Life’ in the 1970s ▹ Significant later work by Stephen Wolfram from the 1980s onwards John von Neumann ▹ Stephen Wolfram, A New Kind of Science, 2002 ▹ https://www.wolframscience.com/nksonline/toc.html ⟐ Computation takes place on a grid, which may have 1, 2 or more dimensions, e.g. a 2D CA: ⟐ At each grid location is a cell ▹ Which has a state ▹ In many cases this is binary: State = Off State = On ⟐ Each cell contains an automaton ▹ Which observes a neighbourhood around the cell ⟐ Each cell contains -
Complexity Properties of the Cellular Automaton Game of Life
Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 11-14-1995 Complexity Properties of the Cellular Automaton Game of Life Andreas Rechtsteiner Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Physics Commons Let us know how access to this document benefits ou.y Recommended Citation Rechtsteiner, Andreas, "Complexity Properties of the Cellular Automaton Game of Life" (1995). Dissertations and Theses. Paper 4928. https://doi.org/10.15760/etd.6804 This Thesis is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. THESIS APPROVAL The abstract and thesis of Andreas Rechtsteiner for the Master of Science in Physics were presented November 14th, 1995, and accepted by the thesis committee and the department. COMMITTEE APPROVALS: 1i' I ) Erik Boaegom ec Representativi' of the Office of Graduate Studies DEPARTMENT APPROVAL: **************************************************************** ACCEPTED FOR PORTLAND STATE UNNERSITY BY THE LIBRARY by on /..-?~Lf!c-t:t-?~~ /99.s- Abstract An abstract of the thesis of Andreas Rechtsteiner for the Master of Science in Physics presented November 14, 1995. Title: Complexity Properties of the Cellular Automaton Game of Life The Game of life is probably the most famous cellular automaton. Life shows all the characteristics of Wolfram's complex Class N cellular automata: long-lived transients, static and propagating local structures, and the ability to support universal computation. We examine in this thesis questions about the geometry and criticality of Life. -
Cellular Automata Analysis of Life As a Complex Physical System
Cellular automata analysis of life as a complex physical system Simon Tropp Thesis submitted for the degree of Bachelor of Science Project duration: 3 Months Supervised by Andrea Idini and Alex Arash Sand Kalaee Department of Physics Division of Mathematical Physics May 2020 Abstract This thesis regards the study of cellular automata, with an outlook on biological systems. Cellular automata are non-linear discrete mathematical models that are based on simple rules defining the evolution of a cell, de- pending on its neighborhood. Cellular automata show surprisingly complex behavior, hence these models are used to simulate complex systems where spatial extension and non-linear relations are important, such as in a system of living organisms. In this thesis, the scale invariance of cellular automata is studied in relation to the physical concept of self-organized criticality. The obtained power laws are then used to calculate the entropy of such systems and thereby demonstrate their tendency to increase the entropy, with the intention to increase the number of possible future available states. This finding is in agreement with a new definition of entropic relations called causal entropy. 1 List of Abbreviations CA cellular automata. GoL Game of Life. SOC self-organized criticality. 2 Contents 1 Introduction 4 1.1 Cellular Automata and The Game of Life . .4 1.1.1 Fractals in Nature and CA . .7 1.2 Self-organized Criticality . 10 1.3 The Role of Entropy . 11 1.3.1 Causal Entropic Forces . 13 1.3.2 Entropy Maximization and Intelligence . 13 2 Method 15 2.1 Self-organized Criticality in CA . -
The Fantastic Combinations of John Conway's New Solitaire Game "Life" - M
Mathematical Games - The fantastic combinations of John Conway's new solitaire game "life" - M. Gardner - 1970 Suche Home Einstellungen Anmelden Hilfe MATHEMATICAL GAMES The fantastic combinations of John Conway's new solitaire game "life" by Martin Gardner Scientific American 223 (October 1970): 120-123. Most of the work of John Horton Conway, a mathematician at Gonville and Caius College of the University of Cambridge, has been in pure mathematics. For instance, in 1967 he discovered a new group-- some call it "Conway's constellation"--that includes all but two of the then known sporadic groups. (They are called "sporadic" because they fail to fit any classification scheme.) Is is a breakthrough that has had exciting repercussions in both group theory and number theory. It ties in closely with an earlier discovery by John Conway of an extremely dense packing of unit spheres in a space of 24 dimensions where each sphere touches 196,560 others. As Conway has remarked, "There is a lot of room up there." In addition to such serious work Conway also enjoys recreational mathematics. Although he is highly productive in this field, he seldom publishes his discoveries. One exception was his paper on "Mrs. Perkin's Quilt," a dissection problem discussed in "Mathematical Games" for September, 1966. My topic for July, 1967, was sprouts, a topological pencil-and-paper game invented by Conway and M. S. Paterson. Conway has been mentioned here several other times. This month we consider Conway's latest brainchild, a fantastic solitaire pastime he calls "life". Because of its analogies with the rise, fall and alternations of a society of living organisms, it belongs to a growing class of what are called "simulation games"--games that resemble real-life processes. -
From Conway's Soldier to Gosper's Glider Gun in Memory of J. H
From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Zhangchi Chen Université Paris-Sud June 17, 2020 Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Math in games Figure: Conway’s Life Game Figure: Peg Solitaire Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: The rule Figure: A valide move Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier Question How many pegs (soldiers) do you need to reach the height of 1, 2, 3, 4 or 5? Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier Baby case To reach the height of 1-3, one needs 2,4,8 soldiers Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier A bit harder case To reach the height of 4, one needs 20 (not 16) soldiers. One solution (not unique). Further? (Conway 1961) It is impossible to reach the height of 5. Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier 1 ' '2 '4 '3 '4 '6 '5 '4 '5 '6 '10 '9 '8 '7 '6 '5 '6 '7 '8 '9 '10 '11 '10 '9 '8 '7 '6 '7 '8 '9 '10 '11 .. -
Logical Universality from a Minimal Two-Dimensional Glider Gun 3
Logical Universality from a Minimal Two- Dimensional Glider Gun José Manuel Gómez Soto Department of Mathematics Autonomous University of Zacatecas Zacatecas, Zac. Mexico [email protected] http://matematicas.reduaz.mx/~jmgomez Andrew Wuensche Discrete Dynamics Lab [email protected] http://www.ddlab.org To understand the underlying principles of self-organization and com- putation 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 interac- tions and the gates required for logical universality. Keywords: universality; cellular automata; glider gun; logical gates 1. Introduction The study of two-dimensional (2D) cellular automata (CAs) with com- plex properties has progressed over time in a kind of regression from the complicated to the simple. Just to mention a few key moments in cellular automaton (CA) history, the original CA was von Neumann’s with 29 states designed to model self-reproduction, and by exten- sion—universality [1]. Codd simplified von Neumann’s CA to eight states [2], and Banks simplified it further to three and four states [3,�4]. In modeling self-reproduction it is also worth mentioning Lang- ton’s “Loops” [5] with eight states, which was simplified by Byl to six states [6]. These 2D CAs all featured the five-cell von Neumann neigh- borhood. -
Minimal Glider-Gun in a 2D Cellular Automaton
Minimal Glider-Gun in a 2D Cellular Automaton Jos´eManuel G´omezSoto∗ Universidad Aut´onomade Zacatecas. Unidad Acad´emica de Matem´aticas. Zacatecas, Zac. M´exico. Andrew Wuenschey Discrete Dynamics Lab. September 11, 2017 Abstract To understand the underlying principles of self-organisation and compu- tation in cellular automata, it would be helpful to find the simplest form of the essential ingredients, glider-guns and eaters, because then the dy- namics would be easier to interpret. Such minimal components emerge spontaneously in the newly discovered Sayab-rule, a binary 2D cellular au- tomaton with a Moore neighborhood and isotropic dynamics. The Sayab- rule has the smallest glider-gun reported to date, consisting of just four live cells at its minimal phases. We show that the Sayab-rule can imple- ment complex dynamical interactions and the gates required for logical universality. keywords: universality, cellular automata, glider-gun, logical gates. 1 Introduction The study of 2D cellular automata (CA) with complex properties has progressed over time in a kind of regression from the complicated to the simple. Just to mention a few key moments in CA history, the original CA was von Neu- mann's with 29 states designed to model self-reproduction, and by extension { universality[18]. Codd simplified von Neumann's CA to 8 states[5], and Banks simplified it further to 3 and 4 states[2, 1]. In modelling self-reproduction its also worth mentioning Langton's \Loops"[12] with 8 states, which was simpli- fied by Byl to 6 states[4]. These 2D CA all featured the 5-cell \von Neumann" arXiv:1709.02655v1 [nlin.CG] 8 Sep 2017 neighborhood. -
Behaviors and Patterns in Low Period Oscillators in Life Abbie Gibson Computer Science Tufts University Appalachian State University [email protected]
Behaviors and Patterns in Low Period Oscillators in Life Abbie Gibson Computer Science Tufts University Appalachian State University [email protected] I. Abstract There are natural occurring cellular automatons seen everywhere in the world, yet there is little known about how these patterns behave. This summer for ten weeks I participated in a research experience that will increase the knowledge known about cellular automata. This paper introduces common patterns and behaviors among minimal period oscillators. Through the program Golly we have been able to analyze a small set of oscillators using the game of life rule set. For each oscillator in our set we looked as it cycled through its different patterns and noted what kind of patterns and behaviors we saw. Along with common patterns and behaviors we looked at the overall density of the oscillator and the density of each pattern the oscillator contained. In this paper we will show some of the common behaviors and patterns we found in minimal period oscillators. II. Introduction A Cellular Automaton is a infinite grid of cells used to represent artificial life. Each cell has a finite number of states and a set of neighbors. Cellular automatons behaviors and properties are determined by the rule set that is applied to the cellular automata. The most common rule set is Conway’s Game of Life. Life uses a two dimensional with cells that have two states, alive or dead. The neighborhood of a cell in game of life consists of the cell and the eight closest cells to it, including the top, left, right, bottom, and four. -
The Game of Life and Other Cellular Automata
The Game of Life and Other Cellular Automata Mi Yu/Adam Reevesman January 28, 2015 The Game Of Life Last class we talked about Cellular Automata, Elementary Cellular Automata, Rule 110, and the classifications of Cellular Automata. This week, we will look at a special form of Cellular Automata| The Game of Life. This game was devised by the British mathemati- cian, John Horton Conway in 1970. This game is a zero-player game, which means that we only need one initial input, and then we can observe how the configuration evolves. There are many interest patterns about The Game of Life. This week, we will take a glance of the beauty of The Game of Life. First of all, as a quick recap of the first week's notes, let me define the rules of The Game of Life. People play Life1 on a two-dimensional grid of square cells, each of which is in one of two possible states, alive or dead. The square will interact with its eight neighbors2. The rules for each update are the followings: 1. Any live cell with fewer than two live neighbors dies 2. Any live cell with two or three live neighbors lives 3. Any live cell with more than three live neighbors dies 4. Any dead cell with exactly three live neighbors becomes alive Now, since we have the rules, let us play the game. If we have the following initial state: Trying to update this state, we find out that we will always get the initial state as the output.