The World of Simple Programs: A New Kind of Science

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height-full"> <div class="heirs-vertical-align-m"> <div class="padding-lr-1"> <h1><a href="/nks/"><img src="/nks/img/a-new-kind-of-science-logo.png" alt="Stephen Wolfram's A New Kind of Science | Online" width="372" height="43" class="fill"></a></h1> </div> <div class="padding-r-1"> <nav id="nav" class="table"> <div class="nav-main heirs-vertical-align-t"> <div class="size-picker"> <i class="icon"></i> <ul> <li class="size-auto checked"><span>auto</span></li> <li class="size-small"><span>small</span></li> <li class="size-medium"><span>medium</span></li> <li class="size-large"><span>large</span></li> <li class="size-x-large"><span>x-large</span></li> </ul> </div> <div class="search"> <div><i class="icon"></i></div> </div> <div class="toc"> <div><a href="/nks/"><i class="icon"></i></a></div> </div> <div class="breadcrumbs"> <div> <span class="chapter"><span>Chapter 3</span></span> </div> </div> </div> <div class="nav-burger"> <div> <div class="hamburger"><i class="icon"></i></div> </div> <div class="burger-nav-l1"> <div id="top-stuff"> <div class="crumb-wrap-supreme"> <div> <span class="toc"><a href="/nks/"><i class="icon"></i></a></span> </div> </div> <div class="size-picker"> <div><i class="icon"></i></div> </div> <ul class="size-pickings"> <li class="size-auto checked"><span>auto</span></li> <li class="size-small"><span>small</span></li> <li class="size-medium"><span>medium</span></li> <li class="size-large"><span>large</span></li> <li class="size-x-large"><span>x-large</span></li> </ul> </div> <ul class="chapters"> <li class="chapter preface"> <a href="/nks/pix--preface/"> <span></span> <span>Preface</span> </a> </li> <li class="chapter chapter-1"> <a href="/nks/chap-1--the-foundations-for-a-new-kind-of-science/"> <span>1</span> <span>The Foundations for a New Kind of Science</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p1--an-outline-of-basic-ideas/"> <span>1</span> <span>An Outline of Basic Ideas</span> </a> </li> <li class="section section-2"> <a href="/nks/p7--relations-to-other-areas/"> <span>2</span> <span>Relations to Other Areas</span> </a> </li> <li class="section section-3"> <a href="/nks/p12--some-past-initiatives/"> <span>3</span> <span>Some Past Initiatives</span> </a> </li> <li class="section section-4"> <a href="/nks/p17--the-personal-story-of-the-science-in-this-book/"> <span>4</span> <span>The Personal Story of the Science in This Book</span> </a> </li> </ul> </li> <li class="chapter chapter-2"> <a href="/nks/chap-2--the-crucial-experiment/"> <span>2</span> <span>The Crucial Experiment</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p23--how-do-simple-programs-behave/"> <span>1</span> <span>How Do Simple Programs Behave?</span> </a> </li> <li class="section section-2"> <a href="/nks/p39--the-need-for-a-new-intuition/"> <span>2</span> <span>The Need for a New Intuition</span> </a> </li> <li class="section section-3"> <a href="/nks/p42--why-these-discoveries-were-not-made-before/"> <span>3</span> <span>Why These Discoveries Were Not Made Before</span> </a> </li> </ul> </li> <li class="chapter chapter-3 selected"> <a href="/nks/chap-3--the-world-of-simple-programs/"> <span>3</span> <span>The World of Simple Programs</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p51--the-search-for-general-features/"> <span>1</span> <span>The Search for General Features</span> </a> </li> <li class="section section-2"> <a href="/nks/p53--more-cellular-automata/"> <span>2</span> <span>More Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p71--mobile-automata/"> <span>3</span> <span>Mobile Automata</span> </a> </li> <li class="section section-4"> <a href="/nks/p78--turing-machines/"> <span>4</span> <span>Turing Machines</span> </a> </li> <li class="section section-5"> <a href="/nks/p82--substitution-systems/"> <span>5</span> <span>Substitution Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p88--sequential-substitution-systems/"> <span>6</span> <span>Sequential Substitution Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p93--tag-systems/"> <span>7</span> <span>Tag Systems</span> </a> </li> <li class="section section-8"> <a href="/nks/p95--cyclic-tag-systems/"> <span>8</span> <span>Cyclic Tag Systems</span> </a> </li> <li class="section section-9"> <a href="/nks/p97--register-machines/"> <span>9</span> <span>Register Machines</span> </a> </li> <li class="section section-10"> <a href="/nks/p102--symbolic-systems/"> <span>10</span> <span>Symbolic Systems</span> </a> </li> <li class="section section-11"> <a href="/nks/p105--some-conclusions/"> <span>11</span> <span>Some Conclusions</span> </a> </li> <li class="section section-12"> <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/"> <span>12</span> <span>How the Discoveries in This Chapter Were Made</span> </a> </li> </ul> </li> <li class="chapter chapter-4"> <a href="/nks/chap-4--systems-based-on-numbers/"> <span>4</span> <span>Systems Based on Numbers</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p115--the-notion-of-numbers/"> <span>1</span> <span>The Notion of Numbers</span> </a> </li> <li class="section section-2"> <a href="/nks/p117--elementary-arithmetic/"> <span>2</span> <span>Elementary Arithmetic</span> </a> </li> <li class="section section-3"> <a href="/nks/p128--recursive-sequences/"> <span>3</span> <span>Recursive Sequences</span> </a> </li> <li class="section section-4"> <a href="/nks/p132--the-sequence-of-primes/"> <span>4</span> <span>The Sequence of Primes</span> </a> </li> <li class="section section-5"> <a href="/nks/p136--mathematical-constants/"> <span>5</span> <span>Mathematical Constants</span> </a> </li> <li class="section section-6"> <a href="/nks/p145--mathematical-functions/"> <span>6</span> <span>Mathematical Functions</span> </a> </li> <li class="section section-7"> <a href="/nks/p149--iterated-maps-and-the-chaos-phenomenon/"> <span>7</span> <span>Iterated Maps and the Chaos Phenomenon</span> </a> </li> <li class="section section-8"> <a href="/nks/p155--continuous-cellular-automata/"> <span>8</span> <span>Continuous Cellular Automata</span> </a> </li> <li class="section section-9"> <a href="/nks/p161--partial-differential-equations/"> <span>9</span> <span>Partial Differential Equations</span> </a> </li> <li class="section section-10"> <a href="/nks/p167--continuous-versus-discrete-systems/"> <span>10</span> <span>Continuous Versus Discrete Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-5"> <a href="/nks/chap-5--two-dimensions-and-beyond/"> <span>5</span> <span>Two Dimensions and Beyond</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p169--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p170--cellular-automata/"> <span>2</span> <span>Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p184--turing-machines/"> <span>3</span> <span>Turing Machines</span> </a> </li> <li class="section section-4"> <a href="/nks/p187--substitution-systems-and-fractals/"> <span>4</span> <span>Substitution Systems and Fractals</span> </a> </li> <li class="section section-5"> <a href="/nks/p193--network-systems/"> <span>5</span> <span>Network Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p204--multiway-systems/"> <span>6</span> <span>Multiway Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p210--systems-based-on-constraints/"> <span>7</span> <span>Systems Based on Constraints</span> </a> </li> </ul> </li> <li class="chapter chapter-6"> <a href="/nks/chap-6--starting-from-randomness/"> <span>6</span> <span>Starting from Randomness</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p223--the-emergence-of-order/"> <span>1</span> <span>The Emergence of Order</span> </a> </li> <li class="section section-2"> <a href="/nks/p231--four-classes-of-behavior/"> <span>2</span> <span>Four Classes of Behavior</span> </a> </li> <li class="section section-3"> <a href="/nks/p250--sensitivity-to-initial-conditions/"> <span>3</span> <span>Sensitivity to Initial Conditions</span> </a> </li> <li class="section section-4"> <a href="/nks/p255--systems-of-limited-size-and-class-2-behavior/"> <span>4</span> <span>Systems of Limited Size and Class 2 Behavior</span> </a> </li> <li class="section section-5"> <a href="/nks/p261--randomness-in-class-3-systems/"> <span>5</span> <span>Randomness in Class 3 Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p266--special-initial-conditions/"> <span>6</span> <span>Special Initial Conditions</span> </a> </li> <li class="section section-7"> <a href="/nks/p275--the-notion-of-attractors/"> <span>7</span> <span>The Notion of Attractors</span> </a> </li> <li class="section section-8"> <a href="/nks/p281--structures-in-class-4-systems/"> <span>8</span> <span>Structures in Class 4 Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-7"> <a href="/nks/chap-7--mechanisms-in-programs-and-nature/"> <span>7</span> <span>Mechanisms in Programs and Nature</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p297--universality-of-behavior/"> <span>1</span> <span>Universality of Behavior</span> </a> </li> <li class="section section-2"> <a href="/nks/p299--three-mechanisms-for-randomness/"> <span>2</span> <span>Three Mechanisms for Randomness</span> </a> </li> <li class="section section-3"> <a href="/nks/p301--randomness-from-the-environment/"> <span>3</span> <span>Randomness from the Environment</span> </a> </li> <li class="section section-4"> <a href="/nks/p304--chaos-theory-and-randomness-from-initial-conditions/"> <span>4</span> <span>Chaos Theory and Randomness from Initial Conditions</span> </a> </li> <li class="section section-5"> <a href="/nks/p315--the-intrinsic-generation-of-randomness/"> <span>5</span> <span>The Intrinsic Generation of Randomness</span> </a> </li> <li class="section section-6"> <a href="/nks/p327--the-phenomenon-of-continuity/"> <span>6</span> <span>The Phenomenon of Continuity</span> </a> </li> <li class="section section-7"> <a href="/nks/p337--origins-of-discreteness/"> <span>7</span> <span>Origins of Discreteness</span> </a> </li> <li class="section section-8"> <a href="/nks/p342--the-problem-of-satisfying-constraints/"> <span>8</span> <span>The Problem of Satisfying Constraints</span> </a> </li> <li class="section section-9"> <a href="/nks/p351--origins-of-simple-behavior/"> <span>9</span> <span>Origins of Simple Behavior</span> </a> </li> </ul> </li> <li class="chapter chapter-8"> <a href="/nks/chap-8--implications-for-everyday-systems/"> <span>8</span> <span>Implications for Everyday Systems</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p363--issues-of-modelling/"> <span>1</span> <span>Issues of Modelling</span> </a> </li> <li class="section section-2"> <a href="/nks/p369--the-growth-of-crystals/"> <span>2</span> <span>The Growth of Crystals</span> </a> </li> <li class="section section-3"> <a href="/nks/p374--the-breaking-of-materials/"> <span>3</span> <span>The Breaking of Materials</span> </a> </li> <li class="section section-4"> <a href="/nks/p376--fluid-flow/"> <span>4</span> <span>Fluid Flow</span> </a> </li> <li class="section section-5"> <a href="/nks/p383--fundamental-issues-in-biology/"> <span>5</span> <span>Fundamental Issues in Biology</span> </a> </li> <li class="section section-6"> <a href="/nks/p400--growth-of-plants-and-animals/"> <span>6</span> <span>Growth of Plants and Animals</span> </a> </li> <li class="section section-7"> <a href="/nks/p422--biological-pigmentation-patterns/"> <span>7</span> <span>Biological Pigmentation Patterns</span> </a> </li> <li class="section section-8"> <a href="/nks/p429--financial-systems/"> <span>8</span> <span>Financial Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-9"> <a href="/nks/chap-9--fundamental-physics/"> <span>9</span> <span>Fundamental Physics</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p433--the-problems-of-physics/"> <span>1</span> <span>The Problems of Physics</span> </a> </li> <li class="section section-2"> <a href="/nks/p435--the-notion-of-reversibility/"> <span>2</span> <span>The Notion of Reversibility</span> </a> </li> <li class="section section-3"> <a href="/nks/p441--irreversibility-and-the-second-law-of-thermodynamics/"> <span>3</span> <span>Irreversibility and the Second Law of Thermodynamics</span> </a> </li> <li class="section section-4"> <a href="/nks/p458--conserved-quantities-and-continuum-phenomena/"> <span>4</span> <span>Conserved Quantities and Continuum Phenomena</span> </a> </li> <li class="section section-5"> <a href="/nks/p465--ultimate-models-for-the-universe/"> <span>5</span> <span>Ultimate Models for the Universe</span> </a> </li> <li class="section section-6"> <a href="/nks/p472--the-nature-of-space/"> <span>6</span> <span>The Nature of Space</span> </a> </li> <li class="section section-7"> <a href="/nks/p475--space-as-a-network/"> <span>7</span> <span>Space as a Network</span> </a> </li> <li class="section section-8"> <a href="/nks/p481--the-relationship-of-space-and-time/"> <span>8</span> <span>The Relationship of Space and Time</span> </a> </li> <li class="section section-9"> <a href="/nks/p486--time-and-causal-networks/"> <span>9</span> <span>Time and Causal Networks</span> </a> </li> <li class="section section-10"> <a href="/nks/p497--the-sequencing-of-events-in-the-universe/"> <span>10</span> <span>The Sequencing of Events in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p504--uniqueness-and-branching-in-time/"> <span>11</span> <span>Uniqueness and Branching in Time</span> </a> </li> <li class="section section-12"> <a href="/nks/p508--evolution-of-networks/"> <span>12</span> <span>Evolution of Networks</span> </a> </li> <li class="section section-13"> <a href="/nks/p516--space-time-and-relativity/"> <span>13</span> <span>Space, Time and Relativity</span> </a> </li> <li class="section section-14"> <a href="/nks/p525--elementary-particles/"> <span>14</span> <span>Elementary Particles</span> </a> </li> <li class="section section-15"> <a href="/nks/p530--the-phenomenon-of-gravity/"> <span>15</span> <span>The Phenomenon of Gravity</span> </a> </li> <li class="section section-16"> <a href="/nks/p537--quantum-phenomena/"> <span>16</span> <span>Quantum Phenomena</span> </a> </li> </ul> </li> <li class="chapter chapter-10"> <a href="/nks/chap-10--processes-of-perception-and-analysis/"> <span>10</span> <span>Processes of Perception and Analysis</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p547--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p548--what-perception-and-analysis-do/"> <span>2</span> <span>What Perception and Analysis Do</span> </a> </li> <li class="section section-3"> <a href="/nks/p552--defining-the-notion-of-randomness/"> <span>3</span> <span>Defining the Notion of Randomness</span> </a> </li> <li class="section section-4"> <a href="/nks/p557--defining-complexity/"> <span>4</span> <span>Defining Complexity</span> </a> </li> <li class="section section-5"> <a href="/nks/p560--data-compression/"> <span>5</span> <span>Data Compression</span> </a> </li> <li class="section section-6"> <a href="/nks/p572--irreversible-data-compression/"> <span>6</span> <span>Irreversible Data Compression</span> </a> </li> <li class="section section-7"> <a href="/nks/p577--visual-perception/"> <span>7</span> <span>Visual Perception</span> </a> </li> <li class="section section-8"> <a href="/nks/p585--auditory-perception/"> <span>8</span> <span>Auditory Perception</span> </a> </li> <li class="section section-9"> <a href="/nks/p588--statistical-analysis/"> <span>9</span> <span>Statistical Analysis</span> </a> </li> <li class="section section-10"> <a href="/nks/p598--cryptography-and-cryptanalysis/"> <span>10</span> <span>Cryptography and Cryptanalysis</span> </a> </li> <li class="section section-11"> <a href="/nks/p606--traditional-mathematics-and-mathematical-formulas/"> <span>11</span> <span>Traditional Mathematics and Mathematical Formulas</span> </a> </li> <li class="section section-12"> <a href="/nks/p620--human-thinking/"> <span>12</span> <span>Human Thinking</span> </a> </li> <li class="section section-13"> <a href="/nks/p632--higher-forms-of-perception-and-analysis/"> <span>13</span> <span>Higher Forms of Perception and Analysis</span> </a> </li> </ul> </li> <li class="chapter chapter-11"> <a href="/nks/chap-11--the-notion-of-computation/"> <span>11</span> <span>The Notion of Computation</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p637--computation-as-a-framework/"> <span>1</span> <span>Computation as a Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p638--computations-in-cellular-automata/"> <span>2</span> <span>Computations in Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p642--the-phenomenon-of-universality/"> <span>3</span> <span>The Phenomenon of Universality</span> </a> </li> <li class="section section-4"> <a href="/nks/p644--a-universal-cellular-automaton/"> <span>4</span> <span>A Universal Cellular Automaton</span> </a> </li> <li class="section section-5"> <a href="/nks/p656--emulating-other-systems-with-cellular-automata/"> <span>5</span> <span>Emulating Other Systems with Cellular Automata</span> </a> </li> <li class="section section-6"> <a href="/nks/p664--emulating-cellular-automata-with-other-systems/"> <span>6</span> <span>Emulating Cellular Automata with Other Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p674--implications-of-universality/"> <span>7</span> <span>Implications of Universality</span> </a> </li> <li class="section section-8"> <a href="/nks/p675--the-rule-110-cellular-automaton/"> <span>8</span> <span>The Rule 110 Cellular Automaton</span> </a> </li> <li class="section section-9"> <a href="/nks/p690--the-significance-of-universality-in-rule-110/"> <span>9</span> <span>The Significance of Universality in Rule 110</span> </a> </li> <li class="section section-10"> <a href="/nks/p691--class-4-behavior-and-universality/"> <span>10</span> <span>Class 4 Behavior and Universality</span> </a> </li> <li class="section section-11"> <a href="/nks/p694--the-threshold-of-universality-in-cellular-automata/"> <span>11</span> <span>The Threshold of Universality in Cellular Automata</span> </a> </li> <li class="section section-12"> <a href="/nks/p706--universality-in-turing-machines-and-other-systems/"> <span>12</span> <span>Universality in Turing Machines and Other Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-12"> <a href="/nks/chap-12--the-principle-of-computational-equivalence/"> <span>12</span> <span>The Principle of Computational Equivalence</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p715--basic-framework/"> <span>1</span> <span>Basic Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p716--outline-of-the-principle/"> <span>2</span> <span>Outline of the Principle</span> </a> </li> <li class="section section-3"> <a href="/nks/p719--the-content-of-the-principle/"> <span>3</span> <span>The Content of the Principle</span> </a> </li> <li class="section section-4"> <a href="/nks/p726--the-validity-of-the-principle/"> <span>4</span> <span>The Validity of the Principle</span> </a> </li> <li class="section section-5"> <a href="/nks/p735--explaining-the-phenomenon-of-complexity/"> <span>5</span> <span>Explaining the Phenomenon of Complexity</span> </a> </li> <li class="section section-6"> <a href="/nks/p737--computational-irreducibility/"> <span>6</span> <span>Computational Irreducibility</span> </a> </li> <li class="section section-7"> <a href="/nks/p750--the-phenomenon-of-free-will/"> <span>7</span> <span>The Phenomenon of Free Will</span> </a> </li> <li class="section section-8"> <a href="/nks/p753--undecidability-and-intractability/"> <span>8</span> <span>Undecidability and Intractability</span> </a> </li> <li class="section section-9"> <a href="/nks/p772--implications-for-mathematics-and-its-foundations/"> <span>9</span> <span>Implications for Mathematics and Its Foundations</span> </a> </li> <li class="section section-10"> <a href="/nks/p822--intelligence-in-the-universe/"> <span>10</span> <span>Intelligence in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p840--implications-for-technology/"> <span>11</span> <span>Implications for Technology</span> </a> </li> <li class="section section-12"> <a href="/nks/p844--historical-perspectives/"> <span>12</span> <span>Historical Perspectives</span> </a> </li> </ul> </li> <li class="chapter notes"> <a href="/nks/notes-for-chapters/"> <span></span> <span>Notes</span> </a> </li> </ul> <div class="search"> </div> </div> </div> </nav> </div> </div> </div> <div id="search"> <div class="text-align-c__600"> <div class="inline-grid cols-3 cols-1__600"> <form id="search-index-form" action="/nks/index/"><div><label for="search-index-input">Look up in index</label><input type="search" id="search-index-input" name="search-index-input"></div></form> <form id="search-all-form" action="/nks/search/"><div><label for="search-all-input">Text search</label><input type="search" id="search-all-input" name="search-all-input"></div></form> <form id="search-jump-form"><div><label for="search-jump-input">Jump to page</label><input type="search" id="search-jump-input" name="search-jump-input"></div></form> </div> </div> </div> </header>  <main id="main"> <div class="thin-wrapper"> <div id="previous"><a href="/nks/chap-2--the-crucial-experiment/"><span>‹</span></a></div> <div id="next"><a href="/nks/chap-4--systems-based-on-numbers/"><span>›</span></a></div> <div id="content" class="thin-padding"> <div class="ribbon-wrapper top"> </div> <section class="chapter-title"> <h1 class="chapter-heading"><a href="/nks/p51--the-search-for-general-features/">Chapter 3: <span>The World of Simple Programs</span></a></h1> <hr> </section> <section class="chapter-content"> <div> <h2 id="sect-3-1--the-search-for-general-features" class="section-heading"><a href="/nks/p51--the-search-for-general-features/">Section 1: <span>The Search for General Features</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-1--the-search-for-general-features</span></span></h2> <div class="pages"> <a href="/nks/p51--the-search-for-general-features/"><img src="/nks/pages/p51--the-search-for-general-features/image-tiny.png" alt="Page 51" width="70" height="86"></a> <a href="/nks/p52--the-search-for-general-features/"><img src="/nks/pages/p52--the-search-for-general-features/image-tiny.png" alt="Page 52" width="70" height="86"></a><a href="/nks/sect-3-1--the-search-for-general-features--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-2--more-cellular-automata" class="section-heading"><a href="/nks/p53--more-cellular-automata/">Section 2: <span>More Cellular Automata</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-2--more-cellular-automata</span></span></h2> <div class="pages"> <a href="/nks/p53--more-cellular-automata/"><img src="/nks/pages/p53--more-cellular-automata/image-tiny.png" alt="Page 53" width="70" height="86"></a> <a href="/nks/p54--more-cellular-automata/"><img src="/nks/pages/p54--more-cellular-automata/image-tiny.png" alt="Page 54" width="70" height="86"></a> <a href="/nks/p55--more-cellular-automata/"><img src="/nks/pages/p55--more-cellular-automata/image-tiny.png" alt="Page 55" width="70" height="86"></a> <a href="/nks/p56--more-cellular-automata/"><img src="/nks/pages/p56--more-cellular-automata/image-tiny.png" alt="Page 56" width="70" height="86"></a> <a href="/nks/p57--more-cellular-automata/"><img src="/nks/pages/p57--more-cellular-automata/image-tiny.png" alt="Page 57" width="70" height="86"></a> <a href="/nks/p58--more-cellular-automata/"><img src="/nks/pages/p58--more-cellular-automata/image-tiny.png" alt="Page 58" width="70" height="86"></a> <a href="/nks/p59--more-cellular-automata/"><img src="/nks/pages/p59--more-cellular-automata/image-tiny.png" alt="Page 59" width="70" height="86"></a> <a href="/nks/p60--more-cellular-automata/"><img src="/nks/pages/p60--more-cellular-automata/image-tiny.png" alt="Page 60" width="70" height="86"></a> <a href="/nks/p61--more-cellular-automata/"><img src="/nks/pages/p61--more-cellular-automata/image-tiny.png" alt="Page 61" width="70" height="86"></a> <a href="/nks/p62--more-cellular-automata/"><img src="/nks/pages/p62--more-cellular-automata/image-tiny.png" alt="Page 62" width="70" height="86"></a> <a href="/nks/p63--more-cellular-automata/"><img src="/nks/pages/p63--more-cellular-automata/image-tiny.png" alt="Page 63" width="70" height="86"></a> <a href="/nks/p64--more-cellular-automata/"><img src="/nks/pages/p64--more-cellular-automata/image-tiny.png" alt="Page 64" width="70" height="86"></a> <a href="/nks/p65--more-cellular-automata/"><img src="/nks/pages/p65--more-cellular-automata/image-tiny.png" alt="Page 65" width="70" height="86"></a> <a href="/nks/p66--more-cellular-automata/"><img src="/nks/pages/p66--more-cellular-automata/image-tiny.png" alt="Page 66" width="70" height="86"></a> <a href="/nks/p67--more-cellular-automata/"><img src="/nks/pages/p67--more-cellular-automata/image-tiny.png" alt="Page 67" width="70" height="86"></a> <a href="/nks/p68--more-cellular-automata/"><img src="/nks/pages/p68--more-cellular-automata/image-tiny.png" alt="Page 68" width="70" height="86"></a> <a href="/nks/p69--more-cellular-automata/"><img src="/nks/pages/p69--more-cellular-automata/image-tiny.png" alt="Page 69" width="70" height="86"></a> <a href="/nks/p70--more-cellular-automata/"><img src="/nks/pages/p70--more-cellular-automata/image-tiny.png" alt="Page 70" width="70" height="86"></a><a href="/nks/sect-3-2--more-cellular-automata--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-3--mobile-automata" class="section-heading"><a href="/nks/p71--mobile-automata/">Section 3: <span>Mobile Automata</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-3--mobile-automata</span></span></h2> <div class="pages"> <a href="/nks/p71--mobile-automata/"><img src="/nks/pages/p71--mobile-automata/image-tiny.png" alt="Page 71" width="70" height="86"></a> <a href="/nks/p72--mobile-automata/"><img src="/nks/pages/p72--mobile-automata/image-tiny.png" alt="Page 72" width="70" height="86"></a> <a href="/nks/p73--mobile-automata/"><img src="/nks/pages/p73--mobile-automata/image-tiny.png" alt="Page 73" width="70" height="86"></a> <a href="/nks/p74--mobile-automata/"><img src="/nks/pages/p74--mobile-automata/image-tiny.png" alt="Page 74" width="70" height="86"></a> <a href="/nks/p75--mobile-automata/"><img src="/nks/pages/p75--mobile-automata/image-tiny.png" alt="Page 75" width="70" height="86"></a> <a href="/nks/p76--mobile-automata/"><img src="/nks/pages/p76--mobile-automata/image-tiny.png" alt="Page 76" width="70" height="86"></a> <a href="/nks/p77--mobile-automata/"><img src="/nks/pages/p77--mobile-automata/image-tiny.png" alt="Page 77" width="70" height="86"></a><a href="/nks/sect-3-3--mobile-automata--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-4--turing-machines" class="section-heading"><a href="/nks/p78--turing-machines/">Section 4: <span>Turing Machines</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-4--turing-machines</span></span></h2> <div class="pages"> <a href="/nks/p78--turing-machines/"><img src="/nks/pages/p78--turing-machines/image-tiny.png" alt="Page 78" width="70" height="86"></a> <a href="/nks/p79--turing-machines/"><img src="/nks/pages/p79--turing-machines/image-tiny.png" alt="Page 79" width="70" height="86"></a> <a href="/nks/p80--turing-machines/"><img src="/nks/pages/p80--turing-machines/image-tiny.png" alt="Page 80" width="70" height="86"></a> <a href="/nks/p81--turing-machines/"><img src="/nks/pages/p81--turing-machines/image-tiny.png" alt="Page 81" width="70" height="86"></a><a href="/nks/sect-3-4--turing-machines--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-5--substitution-systems" class="section-heading"><a href="/nks/p82--substitution-systems/">Section 5: <span>Substitution Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-5--substitution-systems</span></span></h2> <div class="pages"> <a href="/nks/p82--substitution-systems/"><img src="/nks/pages/p82--substitution-systems/image-tiny.png" alt="Page 82" width="70" height="86"></a> <a href="/nks/p83--substitution-systems/"><img src="/nks/pages/p83--substitution-systems/image-tiny.png" alt="Page 83" width="70" height="86"></a> <a href="/nks/p84--substitution-systems/"><img src="/nks/pages/p84--substitution-systems/image-tiny.png" alt="Page 84" width="70" height="86"></a> <a href="/nks/p85--substitution-systems/"><img src="/nks/pages/p85--substitution-systems/image-tiny.png" alt="Page 85" width="70" height="86"></a> <a href="/nks/p86--substitution-systems/"><img src="/nks/pages/p86--substitution-systems/image-tiny.png" alt="Page 86" width="70" height="86"></a> <a href="/nks/p87--substitution-systems/"><img src="/nks/pages/p87--substitution-systems/image-tiny.png" alt="Page 87" width="70" height="86"></a><a href="/nks/sect-3-5--substitution-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-6--sequential-substitution-systems" class="section-heading"><a href="/nks/p88--sequential-substitution-systems/">Section 6: <span>Sequential Substitution Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-6--sequential-substitution-systems</span></span></h2> <div class="pages"> <a href="/nks/p88--sequential-substitution-systems/"><img src="/nks/pages/p88--sequential-substitution-systems/image-tiny.png" alt="Page 88" width="70" height="86"></a> <a href="/nks/p89--sequential-substitution-systems/"><img src="/nks/pages/p89--sequential-substitution-systems/image-tiny.png" alt="Page 89" width="70" height="86"></a> <a href="/nks/p90--sequential-substitution-systems/"><img src="/nks/pages/p90--sequential-substitution-systems/image-tiny.png" alt="Page 90" width="70" height="86"></a> <a href="/nks/p91--sequential-substitution-systems/"><img src="/nks/pages/p91--sequential-substitution-systems/image-tiny.png" alt="Page 91" width="70" height="86"></a> <a href="/nks/p92--sequential-substitution-systems/"><img src="/nks/pages/p92--sequential-substitution-systems/image-tiny.png" alt="Page 92" width="70" height="86"></a><a href="/nks/sect-3-6--sequential-substitution-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-7--tag-systems" class="section-heading"><a href="/nks/p93--tag-systems/">Section 7: <span>Tag Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-7--tag-systems</span></span></h2> <div class="pages"> <a href="/nks/p93--tag-systems/"><img src="/nks/pages/p93--tag-systems/image-tiny.png" alt="Page 93" width="70" height="86"></a> <a href="/nks/p94--tag-systems/"><img src="/nks/pages/p94--tag-systems/image-tiny.png" alt="Page 94" width="70" height="86"></a><a href="/nks/sect-3-7--tag-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-8--cyclic-tag-systems" class="section-heading"><a href="/nks/p95--cyclic-tag-systems/">Section 8: <span>Cyclic Tag Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-8--cyclic-tag-systems</span></span></h2> <div class="pages"> <a href="/nks/p95--cyclic-tag-systems/"><img src="/nks/pages/p95--cyclic-tag-systems/image-tiny.png" alt="Page 95" width="70" height="86"></a> <a href="/nks/p96--cyclic-tag-systems/"><img src="/nks/pages/p96--cyclic-tag-systems/image-tiny.png" alt="Page 96" width="70" height="86"></a><a href="/nks/sect-3-8--cyclic-tag-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-9--register-machines" class="section-heading"><a href="/nks/p97--register-machines/">Section 9: <span>Register Machines</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-9--register-machines</span></span></h2> <div class="pages"> <a href="/nks/p97--register-machines/"><img src="/nks/pages/p97--register-machines/image-tiny.png" alt="Page 97" width="70" height="86"></a> <a href="/nks/p98--register-machines/"><img src="/nks/pages/p98--register-machines/image-tiny.png" alt="Page 98" width="70" height="86"></a> <a href="/nks/p99--register-machines/"><img src="/nks/pages/p99--register-machines/image-tiny.png" alt="Page 99" width="70" height="86"></a> <a href="/nks/p100--register-machines/"><img src="/nks/pages/p100--register-machines/image-tiny.png" alt="Page 100" width="70" height="86"></a> <a href="/nks/p101--register-machines/"><img src="/nks/pages/p101--register-machines/image-tiny.png" alt="Page 101" width="70" height="86"></a><a href="/nks/sect-3-9--register-machines--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-10--symbolic-systems" class="section-heading"><a href="/nks/p102--symbolic-systems/">Section 10: <span>Symbolic Systems</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-10--symbolic-systems</span></span></h2> <div class="pages"> <a href="/nks/p102--symbolic-systems/"><img src="/nks/pages/p102--symbolic-systems/image-tiny.png" alt="Page 102" width="70" height="86"></a> <a href="/nks/p103--symbolic-systems/"><img src="/nks/pages/p103--symbolic-systems/image-tiny.png" alt="Page 103" width="70" height="86"></a> <a href="/nks/p104--symbolic-systems/"><img src="/nks/pages/p104--symbolic-systems/image-tiny.png" alt="Page 104" width="70" height="86"></a><a href="/nks/sect-3-10--symbolic-systems--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-11--some-conclusions" class="section-heading"><a href="/nks/p105--some-conclusions/">Section 11: <span>Some Conclusions</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-11--some-conclusions</span></span></h2> <div class="pages"> <a href="/nks/p105--some-conclusions/"><img src="/nks/pages/p105--some-conclusions/image-tiny.png" alt="Page 105" width="70" height="86"></a> <a href="/nks/p106--some-conclusions/"><img src="/nks/pages/p106--some-conclusions/image-tiny.png" alt="Page 106" width="70" height="86"></a> <a href="/nks/p107--some-conclusions/"><img src="/nks/pages/p107--some-conclusions/image-tiny.png" alt="Page 107" width="70" height="86"></a><a href="/nks/sect-3-11--some-conclusions--notes/" class="notes-thumb"><img src="/nks/img/notes-thumb.png" alt="Notes" width="70" height="86"></a> </div> </div> <hr> <div> <h2 id="sect-3-12--how-the-discoveries-in-this-chapter-were-made" class="section-heading"><a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/">Section 12: <span>How the Discoveries in This Chapter Were Made</span></a><span class="section-anchor"><img src="/nks/img/icon-hyperlink.svg" alt="" width="20" height="20"><span class="clipboard-inline clipboard-sectionlink">https://www.wolframscience.com/nks/chap-3--the-world-of-simple-programs#sect-3-12--how-the-discoveries-in-this-chapter-were-made</span></span></h2> <div class="pages"> <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p108--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 108" width="70" height="86"></a> <a href="/nks/p109--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p109--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 109" width="70" height="86"></a> <a href="/nks/p110--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p110--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 110" width="70" height="86"></a> <a href="/nks/p111--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p111--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 111" width="70" height="86"></a> <a href="/nks/p112--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p112--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 112" width="70" height="86"></a> <a href="/nks/p113--how-the-discoveries-in-this-chapter-were-made/"><img src="/nks/pages/p113--how-the-discoveries-in-this-chapter-were-made/image-tiny.png" alt="Page 113" width="70" height="86"></a><a 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