Conditional complexity and codes
Theoretical Computer Science
Non-reducible descriptions for conditional kolmogorov complexity
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
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Assume that a program p produces an output string b for an input string a: p(a) = b. We look for a "reduction" (simplification) of p, i.e., a program q such that q(a) = b but q has Kolmogorov complexity smaller than p and contains no additional information as compared to p (this means that the conditional complexity K(q|p) is negligible). We show that, for any two strings a and b (except for some degenerate cases), one can find a nonreducible program p of any arbitrarily large complexity (any value larger than K(a) + K(b|a) is possible).