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Randomness conservation inequalities; information and independence in mathematical theories
Levin L. Information and Control61 (1):15-37,1984.Type:Article
Date Reviewed: Sep 1 1985

The need for a firm foundation for probability was recognized by Hibert in the early 1900s [1]. While various foundations have been proposed which serve adequately in many practical situations, there are still nagging questions about the meaning of randomness which are usually swept under the carpet. Kolmogorov’s work (see, for example, [2]) is recognized as settling some of these questions, but his work is more alluded to than studied.

The present article proposes a new definition of randomness which satisfies more invariance properties than Kolmogorov’s definition. The author gives examples of how his definition many be used to simplify probability theory, intuitionistic mathematics, and the theory of Turing degrees.

This paper is succinct and tightly written. This reviewer had to constantly refer to the definitions and is still not certain that he followed all the arguments. While the paper suggests the usefulness of the new definition, more expository articles will be necessary before the author’s ideas have any impact.

Reviewer:  Paul Cull Review #: CR109136
1) Hibert, D.Mathematical problems, Bull. Amer. Math. Soc. 2, 8 (1902), 437–479.
2) Komogorov, A. N.Grundbegriffe der wahrscheinlichkeitre-chung, Berlin, 1933; 2nd Russian edition published by Nauka, Moscow, 1974. Translated as Foundations of the theory of probability (2nd ed.), Chelsea Publ. Co., New York.
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