Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317) (1999)

Atlanta, Georgia

May 4, 1999 to May 6, 1999

ISSN: 1093-0159

ISBN: 0-7695-0075-7

pp: 114

Andrei Muchnik , Institute of New Technologies

Andrei Romashchenko , Moscow State University

Nikolai Vereshchagin , Moscow State University

Alexander Shen , Institute of Problems of Information Transmission

ABSTRACT

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 G?cs--K?rner 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.

INDEX TERMS

Kolmogorov complexity, mutual information, Turing degrees

CITATION

A. Muchnik, N. Vereshchagin, A. Shen and A. Romashchenko, "Upper Semilattice of Binary Strings with the Relation "x is Simple Conditional to y","

*Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)(CCC)*, Atlanta, Georgia, 1999, pp. 114.

doi:10.1109/CCC.1999.766270

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