Randomness conservation inequalities; information and independence in mathematical theories*

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The article further develops Kolmogorov's algorithmic complexity theory. The definition of randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as mutual information in individual infinite sequences. Applications to several areas, like probability theory, theory of algorithms, intuitionistic logic are considered. These theories are simplified substantially with the postulate that the objects they consider are independent of (have small mutual information with) any sequence specified by a mathematical property.

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Supported in 1978–83 by NSF grants MCS 77-19754, 81-04211, and 83-04498. Correspondence should be addressed to the author, 150-3 Kenrick Street, Boston, Mass. 02135.