The Structure of Intelligence by B. Goertzel

By B. Goertzel

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Suppose that we are not dealing with Turing machines, but rather with "substitution machines" -- machines which are capable of running only programs of the form P(A,B,C)="Wherever sequence B occurs in sequence C, replace it with sequence A". Instead of writing P(A,B,C) each time, we shall denote such a program with the symbol (A,B,C). For instance, (1,10001,1000110001100011000110001) = 11111. (A,B,C) should be read "substitute A for B in C". e. %x%=%x%T , and the complexity %y% of a substitution program y as the number of symbols required to express y in the form (A,B,C).

G. Spencer-Brown has discovered a particularly poignant way of illustrating the implications of this logical peculiarity. Suppose, he says, that you have built a random number generator -- a machine which is intended to generate numbers in such a way that each number it generates has absolutely nothing to do with the others. Then how can you test it to see if it works? com 35 THE STRUCTURE OF INTELLIGENCE Suppose you tested it and it gave out a hundred zeros in a row. You would probably assume that it was broken.

Then you'd probably be confident its answers were random, since you're probably confident the results of a coin toss are random. But this is nothing more or less than intuition: you're assuming something is random because you haven't seen any structure to it in the past. An intuition is not the same as a theoretical guarantee. Essentially, this paradox arises from the assumption that a random number generator must give out every sequence of length n with equal frequency. But is there any other way to define a random number generator?

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