starting with a single black cell and then applying a simple two-dimensional cellular automaton rule 250 times.

But does the existence of this short description mean that picture (c) should not be considered random? From a practical point of view the fact that a short description may exist is presumably not too relevant if we can never find this description by any of the methods of perception and analysis that are available to us. But from a conceptual point of view it may seem unsatisfactory to have a definition of randomness that depends on our methods of perception and analysis, and is not somehow absolute.

So one possibility is to define randomness so that something is considered random only if no short description whatsoever exists of it. And before the discoveries in this book such a definition might have seemed not far from our everyday notion of randomness. For we would probably have assumed that anything generated from a sufficiently short description would necessarily look fairly simple. But what we have discovered in this book is that this is absolutely not the case, and that in fact even from rules with very short descriptions it is easy to generate behavior in which our standard methods of perception and analysis recognize no significant regularities.

So to say that something is random only if no short description whatsoever exists of it turns out to be a highly restrictive definition of randomness. And in fact, as I mentioned in Chapter 7, it essentially implies that no process based on definite rules can ever manage to generate randomness when there is no randomness before. For since the rules themselves have a short description, anything generated by following them will also have a correspondingly short description, and will therefore not be considered random according to this definition.

And even if one is not concerned about where randomness might come from, there is still a further problem: it turns out in general to be impossible to determine in any finite way whether any particular thing can ever be generated from a short description. One might imagine that one could always just try running all programs with progressively longer descriptions, and see whether any of them ever generate what one wants. But the problem is that one can never in general tell in

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From Stephen Wolfram: A New Kind of Science [citation]