History
Algorithmic information theory was founded by Ray Solomonoff, who published the basic ideas on which the field is based as part of his invention of algorithmic probability - a way to overcome serious problems associated with the application of Bayes rules in statistics. He first described his results at a Conference at Caltech in 1960, and in a report, Feb. 1960, "A Preliminary Report on a General Theory of Inductive Inference." Algorithmic information theory was later developed independently by Andrey Kolmogorov, in 1965 and Gregory Chaitin, around 1966.
There are several variants of Kolmogorov complexity or algorithmic information; the most widely used one is based on self-delimiting programs and is mainly due to Leonid Levin (1974). Per Martin-Löf also contributed significantly to the information theory of infinite sequences. An axiomatic approach to algorithmic information theory based on Blum axioms (Blum 1967) was introduced by Mark Burgin in a paper presented for publication by Andrey Kolmogorov (Burgin 1982). The axiomatic approach encompasses other approaches in the algorithmic information theory. It is possible to treat different measures of algorithmic information as particular cases of axiomatically defined measures of algorithmic information. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of the axiomatic approach in mathematics. The axiomatic approach to algorithmic information theory was further developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003).
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