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Theoretical Foundations of Applied Discrete Mathematics

On the concept of underdetermined alphabets of equal strength

L. A. Sholomov

Institute of Systems Analysis, Russian Academy of Sciences, Moscow, Russia

Abstract: For underdetermined alphabets, the following two concepts are defined: a) one alphabet is stronger than another, and b) two alphabets have equal strength. To define concepts (a) and (b), several approaches are used. The functional approach is based on expressibility of one alphabet via another; three other approaches – combinatorial, probabilistic, and algorithmic – are terminologically connected with the Kolmogorov's approaches to the notion of the amount of information. It is proved that all these approaches to the comparison of alphabets are equivalent. In case (b), a solution of the optimal compression problem for one of the alphabets, in fact, solves the same problem for the other. It is shown that the concepts (a) and (b) allow polynomial time verification.

Keywords: underdetermined alphabet, alphabets of equal strength, entropy of underdetermined data, Kolmogorov complexity.

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