By B. Goertzel
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Extra info for The Structure of Intelligence
But is there any other way to define a random number generator? One could define a random number generator as a machine which generates a normal sequence of numbers, but it is easy to see that there is no way to prove a finite sequence is part of a normal sequence. This is because the definition of normality involves going "far enough out" in a sequence. Once you go far enough out in a normal sequence, all subsequences of length n must occur equally often. But as n increases, so does the precise meaning of "far enough".
This denotes the complexity of obtaining x from (y,z). If y is a Turing machine program and z is a finite binary sequence, we shall let CT (y,z) denote the number of time steps which the Turing machine takes to stop when equipped with program y and given z as initial input. For many purposes, the numbers a, b and c are not important. Often they can all be taken to equal 1, and then they don't appear in the formula at all. But in some cases it may be useful to set a=b=1 and c=0, for instance. Then the formula reads %y% + %z% < %x%.
So, what is the smallest value that %y%% + %z%% assumes for any (y,z) such that y*z=x? Clearly, this smallest value must be either equal to %x%%, or very close to it. For, what if %y%% + %z%% is bigger than %x%%? Then it cannot be the smallest %y%% + %z%%, because if one took z to be the "empty sequence" (the sequence consisting of no characters) and then took y to be the shortest program for computing x, one would have %z%%=0 and %y%%=%x%%. And, on the other hand, is it possible for %y%%+%z%% to be smaller than %x%%?