Day 003 - Complexity from simplicity

Submitted by Sam on 23 May, 2011 - 21:24

The height of excitement in a deterministic universe is the truth that hiding somewhere in the space of all possible equations is an expression that describes everything – a rule-set that perfectly recreates the whole of reality, from beginning to end.

An equation that produces general relativity, the miracle of life, and the works of Pelham Grenville Wodehouse sounds as though it must be impenetrably complex. Counter-intuitively, this is not necessarily the case. Very simple programs (so simple that they can be completely described by a graphical illustration) can be made to generate extremely complex patterns from just a few basic rules. Sometimes the results that these well-defined rules produce are so complex that they can be used as random number generators, as is the case with Stephen Wolfram's Rule 30.

Rule 30 is an arresting example of how rapidly complexity can arise even in a fully determined system. Cellular automata that exhibit the complexity of Rule 30 give a practical insight into chaos theory, visually illustrating how the slightest change in initial conditions can produce highly unpredictable changes in a system's behaviour. This is the idea behind the butterfly effect, which shows how great differences can arise from the smallest changes in past states of a system. The butterfly effect explains neatly why the weather forecast is such a compelling work of fiction.

In a deterministic universe, it is possible to short-circuit chaos theory altogether and find out with utter certainty whether it will rain or not tomorrow evening1 . Stephen Wolfram is obviously very partial to this idea, and believes that there might just be a chance that we ourselves are lucky enough to live in a deterministic universe. The very possibility of holding a rule for the universe in our hands, as he puts it, is enough motivation to commit serious research and resources into finding it. Watch him say so himself in the TED talk below.

  • 1. Some poetic license is operational here. There is the small matter of computational irreducibility that might get in the way of predictions about tomorrow's weather, as a full simulation of the entire universe might be rather slow to run.
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