# Algorithmic Information Theory

Finding community structure in networks using the eigenvectors of matrices.

## Search space topology, algorithmic information theory and biological evolution

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## Algorithmic information theory | mathematics | sukarasri.cf

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Reprints and Permissions. Entropy Nucleic Acids Research The mathematical formalization of the concept of probability or chance has a long intertwined history. The now standard axioms of probability, learned by all students, are due to Kolmogorov While mathematically convincing, the semantics is far from clear.

Frequentists interpret probabilities as limits of observed relative frequencies, objectivists think of them as real aspects of the world, subjectivists regard them as one's degree of belief often elicited from betting ratios , while Cournot only assigns meaning to events of high probability, namely as happening for sure in our world.

None of these approaches answers the question of whether some specific individual object or observation, like the binary strings above, is random. Kolmogorov's axioms do not allow one to ask such questions.

### A new version of algorithmic information theory

Von Mises , with refinements to his approach by Wald , and Church with various degrees of success attempted to formalize the intuitive notion of one string looking more random than another see the example in the introduction. Unfortunately no sequence can satisfy all randomness tests. The Mises-Wald-Church approach seemed satisfactory until Ville showed that some sequences are random according to their definition and yet lack certain properties that are universally agreed to be satisfied by random sequences. Martin-Loef , rather than give a definition and check whether it satisfied all requirements, took the approach to formalize the notion of all effectively testable requirements in the form of tests for randomness.

The tests are constructive namely all and only lower semi-computable ones, which are typically all one ever cares about. Since the tests are constructed from Turing machines, they can be effectively enumerated according to the effective enumeration of the Turing machines they derive from. Since the set of sequences satisfying a test having the randomness property the test verifies has measure one, and there are only countably many tests, the set of sequences satisfying all such tests also has measure one.