advance how many steps of evolution one will need to look at in order to be sure that any particular piece of behavior will not occur. And as a result, no finite process can in general be used to guarantee that there is no short description that exists of a particular thing.

By setting up various restrictions, say on the number of steps of evolution that will be allowed, it is possible to obtain slightly more tractable definitions of randomness. But even in such cases the amount of computational work required to determine whether something should be considered random is typically astronomically large. And more important, while such definitions may perhaps be of some conceptual interest, they correspond very poorly with our intuitive notion of randomness. In fact, if one followed such a definition most of the pictures in this book that I have said look random—including for example picture (c) on page 553—would be considered not random. And following the discussion of Chapter 7, so would at least many of the phenomena in nature that we normally think of as random.

Indeed, what I suspect is that ultimately no useful definition of randomness can be based solely on the issue of what short descriptions of something may in principle exist. Rather, any useful definition must, I believe, make at least some reference to how such short descriptions are supposed to be found.

Over the years, a variety of definitions of randomness have been proposed that are based on the absence of certain specific regularities. Often these definitions are presented as somehow being fundamental. But in fact they typically correspond just to seeing whether some particular process—and usually a rather simple one—succeeds in recognizing regularities and thus in generating a shorter description.

A common example—to be discussed further two sections from now—involves taking, say, a sequence of black and white cells, and then counting the frequency with which each color and each block of colors occurs. Any deviation from equality among these frequencies represents a regularity in the sequence and reveals nonrandomness. But despite some confusion in the past it is certainly not true that just checking equality of frequencies of blocks of colors—even arbitrarily long ones—is sufficient to ensure that no regularities at all exist. This

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