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Compression of a bit string x of length n involves creating a program shorter than n...

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Advance Math

Compression of a bit string x of length n involves creating aprogram shorter than n bits that returns

x. The Kolmogorov complexity of a string K(x) is the length ofshortest program that returns x (i.e.

the length of a maximally compressed version of x).

(a) Explain why "the smallest positive integer not definable inunder 100 characters" is paradoxical.

(b) Prove that for any length n, there must be at least one bitstring that cannot be compressed to

fewer than n bits.

(c) Imagine you had the program K, which outputs the Kolmogorovcomplexity of string. Design

a program P that when given integer n outputs the bit string oflength n with the highest

Kolmogorov complexity. If there are multiple strings with thehighest complexity, output the

lexicographically first (i.e. the one that would come first in adictionary).

(d) Suppose the program P you just wrote can be written in mbits. Show that P and by extension,

K, cannot exist, for a sufficiently large input n.

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