Wednesday, August 23, 2017

Langton’s Ant

If you knew everything in the present, exactly, and you also knew the “laws of nature” could you predict the future?  We live in a dynamic system that we can measure, and we know all of the rules we can measure, so like predicting the eclipse of the sun, can’t we predict more complicated things as well?    

A conventional dynamic system (CDS) is defined as a system where all the parts (every part) are described precisely as to position, and motion, (and by implication; the rules that govern their interaction - what they can do).  The rules are that we don’t make up outside influences; they are all defined in our complete and precisely defined system.

The Rule takes the system one step from its current state to the next (possible, ergo defined state) step.  If we’ve done everything right, the result will be predictable.
If we work with a planetary solar system, (this means all the planets, their sizes and densities, distances, velocities, moons, gravitational constant, etc), it’s pretty complicated.  But we can, and do, predict eclipses and the position of our planets and moons millions of years into the future with excellent accuracy.  Even the Mayans did this 1,400 years ago for eclipses, (hold that thought*).  So this means we can predict at least some parts of the future; we live among predetermined systems, right?
Consider a far simpler CDS, it has only four or five defined parts and only three rules.  The parts are a graph paper like grid (two parts?) and a locator of the  starting point, the initial state.  The rules are:

   Rule 1). Each step in this CDS takes the locator one square forward.
  Rule 2). At a white square, turn 90° right, flip the color of the square, move forward one unit
 Rule 3). At a black square, turn 90° left, flip the color of the square, move forward one unit

This called “Langton’s Ant**.  The ant is the locator that moves one step at a time, following the “rules”.  This can be run on a very simple computer program (OK, the computer is complicated but it is doing a very simple step by step process).
This is much simpler than planets, moons, velocities, orbits, distances and all their interactions as they whizz around our sun.
So when you run it, it should be very predictable, right?  Info in, info out; always the same result…  Nope.  When run, it is unpredictable for the first few hundred moves then a strange locked in pattern emerges to infinity. 
This simple CDA leads to complex behavior. Three distinct modes of behavior are apparent.
Simplicity.  During the first few hundred moves it creates very simple patterns which are often symmetrical.

Chaos.  After a few hundred moves, a big, irregular pattern of black and white squares appears. The ant traces a pseudo-random path until around 10,000 steps.

Emergent order.  Finally the ant starts building a recurrent "highway" pattern of 104 steps that repeats indefinitely.

All finite initial configurations tested eventually converge to the same repetitive pattern.  No one has been able to prove why this is true for all such initial configurations. It is only known that the ant's trajectory is always unbounded regardless of the initial configuration.
Here is a super simple CDS that results in a pattern that no one could predict, and even now that we know what happens, we can’t explain why, we had to run it to see it happen.
 So the debate about the future being determinable from the present is over (not to mention quantum mechanics which suggests that we can even accurately observe the exact initial conditions).  Now do a CDS with 200,000 atoms in three dimensional space…. You know, like a virus.  Oh, and no one has been able to suggest what spot in the universe is holding still or not accelerating.  Even simple Newtonian physics only works in an inertial frame.

          *The Mayans could predict the eclipses with great accuracy (at the same time that most Europeans believed the earth to be flat and the center of the Universe).  But before you get too laudatory of the Mayans’ scientific abilities, at the same time that they could do this, they still had religious mythology about jaguars eating the sun.  Their CDS still had outside influences, Gods.

         **If you’d like to see this in action, go to:

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