Shannon entropy vs. kolmogorov complexity
CSR'06 Proceedings of the First international computer science conference on Theory and Applications
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In this paper we construct a structure R that is a "finite Version" of the semilattice of Turing degrees. Its elements are strings (technically, sequences of strings) and x = y means that K(x|y)=(conditional Kolmogorov complexity of x relative to y) is small.We construct two elements in R that do not have greatest lower bound. We give a series of examples that show how natural algebraic constructions give two elements that have lower bound 0 (minimal element) but significant mutual information. (A first example of that kind was constructed by Gács--Körner using completely different technique.)We define a notion of "complexity profile" of the pair of elements of R and give (exact) upper and lower bounds for it in a particular case.