procedure can indeed be used to check that no purely repetitive pattern exists, but as we will see later in this chapter, it does not successfully detect the presence of even certain highly regular nested patterns.

So how then can we develop a useful yet precise definition of randomness? What we need is essentially just a precise version of the statement at the beginning of this section: that something should be considered random if none of our standard methods of perception and analysis succeed in detecting any regularities in it. But how can we ever expect to find any kind of precise general characterization of what all our various standard methods of perception and analysis do?

The key point that will emerge in this chapter is that in the end essentially all these methods can be viewed as being based on rather simple programs. So this suggests a definition that can be given of randomness: something should be considered to be random whenever there is essentially no simple program that can succeed in detecting regularities in it.

Usually if what one is studying was itself created by a simple program then there will be a few closely related programs that always succeed in detecting regularities. But if something can reasonably be considered random, then the point is that the vast majority of simple programs should not be able to detect any regularities in it.

So does one really need to try essentially all sufficiently simple programs in order to determine this? In my experience, the answer tends to be no. For once a few simple programs corresponding to a few standard methods of perception and analysis have failed to detect regularities, it is extremely rare for any other simple program to succeed in detecting them.

So this means that the everyday definition of randomness that we discussed at the very beginning of this section is in the end already quite unambiguous. For it typically will not matter much which of the standard methods of perception and analysis we use: after trying a few of them we will almost always be in a position to come to a quite definite conclusion about whether or not something should be considered random.

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