By Gregory J. Chaitin
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Extra info for THINKING ABOUT GГ–DEL AND TURING: Essays on Complexity, 1970-2007
Consequently, the great majority of strings of length n are random, that is, need programs of approximately length n, that is to say, are of complexity approximately n. What happens if one wishes to show that a particular string is random? What if one wishes to prove that the complexity of a certain string is almost equal to its length? What if one wishes to exhibit a specific example of a string of length n and complexity close to n, and assure oneself by means of a proof that there is no shorter program for calculating this string?
A program for U consists of two parts: the left-hand part indicates which computer is to be simulated, and the right-hand part gives the program to be simulated. We now suppose that some particular universal computer U has been chosen as the standard one for measuring complexities, and shall henceforth write I(s) instead of IU (s). Definition 4. The rules of inference of a class of formal axiom systems is a recursive function F (a, h) (a a binary string, h a natural number) with the property that F (a, h) ⊂ F (a, h + 1).
Recall that we consider the nth binary string to be the natural number n. Definition 6. , B(n) = max k = max U (p). I(k)≤n lg(p)≤n Theorem 4. Let f be a partial recursive function that carries natural numbers into natural numbers. Then B(n) ≥ f (n) for all sufficiently great values of n. Proof. Consider the computer C such that C(p) = f (p) for all p. I(f (n)) ≤ IC (f (n)) + c ≤ lg(n) + c = [log2 (n + 1)] + c < n for all sufficiently great values of n. Thus B(n) ≥ f (n) for all sufficiently great values of n.
THINKING ABOUT GГ–DEL AND TURING: Essays on Complexity, 1970-2007 by Gregory J. Chaitin