What the hack is Rule 30? Cellular Automata Explained

Introduction

If you ask the same question, what is this Rule 30, then you are not alone. This rule is related to one-dimensional Cellular Automata introduced by Stephen Wolfram in 1983 and later described in his A New Kind Of Science book. Even though, I skimmed through this book previously, I was never able to understand what was it all about. It had some neat diagrams, but I didn’t try to understand how the diagrams where generated. This changed a day ago when I watched an interesting interview that Lex Fridman had with Stephen Wolfram, where they discussed among other things Rule 30, which was the one that Wolfram described first. This rule is capable of generating a complex behavior even though the rule itself is very simple to define.

Elementary Cellular Automaton

The Elementary Cellular Automaton consists of a one-dimensional array of cells that can be in just two sates, namely, black or white color. The cellular automaton starts from initial state, and then transition to the next state based on a certain function. The next state of the automaton is calculated based on the color of current cell and its left and right neighbors’ colors. By the way, that transition function, gives its name to the cellular automaton rule, just like Rule 30.

Rule 30 transition function

Current Sate111110101100011010001000
Next state00011110

Where, for example, the binary digits ‘111’ in the first column indicate the black color of the left, current and right cells, and the values, ‘0’ or ‘1’ indicate what will be the color of the current cell in the next state (iteration) of the automaton. If we write down all the the binary values of the next state together as a single 8 digit binary number, which is 00011110 and convert it to a decimal number we get 30, and hence the name of the Rule 30. In a Boolean form this transition function is

(left, current, right) -> left XOR (current OR right)

We’ll use this function later in the C++ implementation of the Rule 30.

This is how the output from Rule 30 looks like after 20 and 100 steps respectively.

source: WolframAlpha

How to generate Rule 30?

It is easy to implement the Rule 30 using a 64 bit integer in C++. The only drawback is that we can get only 32 steps with the implementation below taken from Wikipedia.

Note: If this is a little bit to much for you now, then skip to the next part where we’ll use WolframAlpha to program it for us.

In the code below a 64 bit integer is used to generate an initial state using a bit mask in line 6. Then the outer for loop generates 32 steps of the Rule 30 by applying the transition function in line 19. The inner loop generates the black (using character ‘1’) and white (using character ‘-‘) colors using state bit mask.

#include <stdint.h>
#include <iostream>

int main() {
 // This is our bit mask with the 32 bit set to '1' for initial state
  uint64_t state = 1u << 31;

  for (int i = 0 ; i < 32 ; i++) {

    for (int j = sizeof(uint64_t) * 8 - 1 ; j  >=  0 ; j--) {
      // Here we decide what should be the color of the current cell based on the current state bit mask.
     // Bitwise operator is used to accomplish this efficiently
      std::cout << char(state  >>  j & 1 ? '1' : '-');
    }
    std::cout << '\n';
    
    // This is just the (left, current, right) -> left XOR (current OR right) functioned mentioned previously
   // Bitwise operators  are used to accomplish this efficiently
    state = (state >> 1) ^ (state | state << 1);
  }
}

It is possible to run this code in the online C++ compile. Just click on the link and then click on the green Run button at the top of the screen. The result looks similar to below.

The full output for 32 steps looks like this

--------------------------------1-------------------------------
-------------------------------111------------------------------
------------------------------11--1-----------------------------
-----------------------------11-1111----------------------------
----------------------------11--1---1---------------------------
---------------------------11-1111-111--------------------------
--------------------------11--1----1--1-------------------------
-------------------------11-1111--111111------------------------
------------------------11--1---111-----1-----------------------
-----------------------11-1111-11--1---111----------------------
----------------------11--1----1-1111-11--1---------------------
---------------------11-1111--11-1----1-1111--------------------
--------------------11--1---111--11--11-1---1-------------------
-------------------11-1111-11--111-111--11-111------------------
------------------11--1----1-111---1--111--1--1-----------------
-----------------11-1111--11-1--1-11111--1111111----------------
----------------11--1---111--1111-1----111------1---------------
---------------11-1111-11--111----11--11--1----111--------------
--------------11--1----1-111--1--11-111-1111--11--1-------------
-------------11-1111--11-1--111111--1---1---111-1111------------
------------11--1---111--1111-----1111-111-11---1---1-----------
-----------11-1111-11--111---1---11----1---1-1-111-111----------
----------11--1----1-111--1-111-11-1--111-11-1-1---1--1---------
---------11-1111--11-1--111-1---1--1111---1--1-11-111111--------
--------11--1---111--1111---11-11111---1-11111-1--1-----1-------
-------11-1111-11--111---1-11--1----1-11-1-----11111---111------
------11--1----1-111--1-11-1-1111--11-1--11---11----1-11--1-----
-----11-1111--11-1--111-1--1-1---111--1111-1-11-1--11-1-1111----
----11--1---111--1111---1111-11-11--111----1-1--1111--1-1---1---
---11-1111-11--111---1-11----1--1-111--1--11-1111---111-11-111--
--11--1----1-111--1-11-1-1--11111-1--111111--1---1-11---1--1--1-
-11-1111--11-1--111-1--1-1111-----1111-----1111-11-1-1-111111111

Compare the image above to the one generated using WolframAlpha symbolic language in the Wolfram Notebook.

Using WolframAlpha to generate Rule 30

WolframAlpha is a computational knowledge engine developed by Wolfram Research. It is embedded in Mathematica symbolic language and it has a declarative programming style. Which means that you specify what you want to be done instead of how it should be done.

For example, to generate Rule 30 and visualize it one simple writes

ArrayPlot[CellularAutomaton[30, {{1},0}, 64]]

where CellularAutomaton function generates 64 states of the automaton, starting with a one black cell in accordance with the Rule 30 generation function, and ArrayPlot function prints the result.

And the output is

Please follow the this link to the Wolfram Notebook where the various Elementary Cellular Automata Rules are generated.

Summary

Playing with cellular automat rules seems like an interesting game to play, plus doing it using WolframAlpha language is a piece of cake.

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