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Computability and randomness. (English) Zbl 1169.03034
Oxford Logic Guides 51. Oxford: Oxford University Press (ISBN 978-0-19-923076-1/hbk). xv, 433 p. £ 55.00 (2009).

The book presents recent results in algorithmic information theory. It contains the following chapters: 1. The complexity of sets; 2. The descriptive complexity of strings; 3. Martin-Löf randomness and invariants; 4. Diagonally noncomputable functions; 5. Lowness properties and $K$-triviality; 6. Some advanced computability theory; 7. Randomness and betting strategies; 8. Classes of computational complexity; 9. Higher computability and randomness. The book includes many exercises (some with solutions).

Written by a researcher who contributed with significant results to the field, the treatment is mathematical rigourous, with many open problems, and an up-dated list of references. The book is suited for advanced courses and research.

##### MSC:
 03D32 Algorithmic randomness and dimension 68Q30 Algorithmic information theory 03-02 Research monographs (mathematical logic) 68-02 Research monographs (computer science)