Partial Sub-Elementary Cellular Automaton

Chaos Found in 5-rules

After doing plenty of experiments with arbitrarily defined rulesets, I made it my mission to find an even simpler ruleset than the partial permutable subcellular automaton with 6-rules that generates complexity. The image below was created using a Partial Sub-elementary Cellular Automaton. Notice I didn't say 'Partial Permutable.' This ruleset contains no permutable rules of any n-space. Instead, it is generated by amalgamation of rules. It is the Frankenstein's Monster of CA in n-spaces.

Figure: Shows the ruleset for a Cellular Automaton with only 5 rules

In the image above, you can see Solitons on the right side of the triangle. This SCA may be an even better candidate for Turing Completeness and if a NAND gate can be programmed in the initial array, this will be the simplest Cellular Automaton capable of universal computation.

In section 6 we will look at a ECA with 4-rules that generates complexity. It seems with the right combination of rules, complexity can still be generated. Find the link to the next section at the bottom of this page.

package com.flotow;import java.util.Scanner;import java.lang.Math.*;public class Main { public static void main(String[] args) { Scanner scan = new Scanner(System.in); String shape; int r000; int r001; int r010; int r011; int r1; int runs = 0; int[] in = new int[1002]; for(int i = 0; i < in.length; i++){ if(i == 500){ in[i] = 1; }else{ in[i] = 0; } } System.out.println("Enter number of iterations."); System.out.println("----------------------------------"); int cycles = scan.nextInt(); int nums = 0; System.out.println("Choose the rule set. For the following cases\nEnter 1 if the rule should output a 1\nor Enter 0 if it should output 0."); System.out.println("001 -> "); r000 = scan.nextInt(); System.out.println("101 -> "); r001 = scan.nextInt(); System.out.println("10 -> "); r010 = scan.nextInt(); System.out.println("11 -> "); r011 = scan.nextInt(); System.out.println("00 -> "); r1 = scan.nextInt();
System.out.println(""); System.out.println("What shape would you like to draw the fractal with?"); System.out.println(""); String wait = scan.nextLine(); shape = wait; String wait2 = scan.nextLine(); shape = wait2; System.out.println("Starting..."); System.out.println("----------------------------------"); while (true) { while (nums < cycles) { if(runs == 0) { Automata a = new Automata(in,r000,r001,r010,r011,r1,shape); nums++; in = a.arrayOut(); } if(runs>0){ if(runs == 1){ RandomizeArray r = new RandomizeArray(); in = r.out(); } Automata b = new Automata(in,r000,r001,r010,r011,r1,shape); in = b.arrayOut(); nums++; runs = 2; } } runs = 1; System.out.println("Go Again? ENTER 1 for yes or 0 for no."); int again = scan.nextInt(); if(again != 1){ break; } System.out.println("Enter number of iterations:"); System.out.println("----------------------------------"); cycles = scan.nextInt(); nums = 0; System.out.println("Choose the rule set. For the following cases\nEnter 1 if the rule should output a 1\nor Enter 0 if it should output 0."); System.out.println("001 -> "); r000 = scan.nextInt(); System.out.println("101 -> "); r001 = scan.nextInt(); System.out.println("10 -> "); r010 = scan.nextInt(); System.out.println("11 -> "); r011 = scan.nextInt(); System.out.println("00 -> "); r1 = scan.nextInt(); System.out.println(""); System.out.println("What shape would you like to draw the fractal with?"); System.out.println(""); wait = scan.nextLine(); wait2 = scan.nextLine(); shape = wait2; System.out.println("Starting..."); System.out.println("----------------------------------"); } }}class Automata{ int[] X = new int[1002]; int[] newX = new int[1002]; int r000; int r001; int r010; int r011; int r1; public Automata(int[] X, int r000, int r001, int r010, int r011,int r1, String shape) { this.r000 = r000; this.r001 = r001; this.r010 = r010; this.r011 = r011; this.r1 = r1;
this.X = this.newX; CountCharacter c = new CountCharacter(shape); int spaces = c.stringCountOut(); String totalSpaces = ""; for(int s = 0; s < spaces; s++){ totalSpaces = totalSpaces+" "; } for(int j = 0; j < X.length-1; j++){ if(X[j]==0){ System.out.print(totalSpaces); }else{ System.out.print(shape); } } for (int i = 0; i < X.length-1; i++) { if (i == 0) { if ((X[i] == 0 && X[i + 1] == 0 && X[1000] ==1)) { this.newX[i] = this.r000; } else if ((X[i] == 1 && X[i + 1] == 0 && X[1000] == 1)) { this.newX[i] = this.r001;; } else if ((X[i] == 1 && X[i + 1] == 0)) { this.newX[i] = this.r010; } else if ((X[i] == 1 && X[i + 1] == 1)) { this.newX[i] = this.r011; }else if ((X[i] == 0 && X[i+1]==0)) { this.newX[i] = this.r1; } } else if (i >= 1 && i <= 998) { if ((X[i] == 0 && X[i + 1] == 0 && X[i-1] ==1)) { this.newX[i] = this.r000; } else if ((X[i] == 1 && X[i + 1] == 0 && X[i-1] == 1)) { this.newX[i] = this.r001; } else if ((X[i] == 1 && X[i + 1] == 0)) { this.newX[i] = this.r010; } else if ((X[i] == 1 && X[i + 1] == 1)) { this.newX[i] = this.r011; }else if ((X[i] == 0 && X[i+1]==0)) { this.newX[i] = this.r1; } } else if (i == 999) { if ((X[i] == 0 && X[i + 1] == 0 && X[i-1] ==1)) { this.newX[i] = this.r000; }else if ((X[i] == 1 && X[i + 1] == 0 && X[i-1] == 1)) { this.newX[i] = this.r001; }else if ((X[i] == 1 && X[0] == 0)) { this.newX[i] = this.r010+1; }else if ((X[i] == 1 && X[0] == 1)) { this.newX[i] = this.r011; }else if ((X[i] == 0 && X[i+1]==0)) { this.newX[i] = this.r1; } } } for(int k = 0; k < X.length-1; k++){ this.X[k] = this.newX[k]; } System.out.println(""); } public int[] arrayOut(){ return this.X; }}class RandomizeArray{ int[] rando = new int[1002]; double randomNum = 0; public RandomizeArray(){ for(int i = 0; i < 1002; i++){ randomNum = Math.random(); if(i==600){ rando[i] = 1; }else{ rando[i] = 0; } } } public int[] out(){ return this.rando; }}class CountCharacter { String in; int count; public CountCharacter(String in) { this.count = 0; for (int i = 0; i < in.length(); i++) { this.count++; } } public int stringCountOut(){ return this.count; }}

Section 6

- 4-ruled automaton that generates complexity