Algorithmic Randomness and Complexity

1. Front Matter

   Pages i-xxviii

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2. Background

   1. Front Matter

      Pages 1-1

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   2. Preliminaries

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 2-6

   3. Computability Theory

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 7-109

   4. Kolmogorov Complexity of Finite Strings

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 110-153

   5. Relating Complexities

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 154-196

   6. Effective Reals

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 197-224

3. Notions of Randomness

   1. Front Matter

      Pages 225-225

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   2. Martin-Löf Randomness

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 226-268

   3. Other Notions of Algorithmic Randomness

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 269-322

   4. Algorithmic Randomness and Turing Reducibility

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 323-401

4. Relative Randomness

   1. Front Matter

      Pages 403-403

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   2. Measures of Relative Randomness

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 404-463

   3. Complexity and Relative Randomness for 1-Random Sets

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 464-499

   4. Randomness-Theoretic Weakness

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 500-553

   5. Lowness and Triviality for Other Randomness Notions

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 554-591

   6. Algorithmic Dimension

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 592-666

5. Further Topics

   1. Front Matter

      Pages 667-667

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   2. Strong Jump Traceability

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 668-704

   3. Ω as an Operator

      • Rodney G. Downey, Denis R. Hirschfeldt

      Pages 705-727

Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we reconcile this intuition with the fact that both are statistically equally likely? What does it mean to say that an individual mathematical object such as a real number is random, or to say that one real is more random than another? And what is the relationship between randomness and computational power. The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these. Much of this theory can be seen as exploring the relationships between three fundamental concepts: relative computability, as measured by notions such as Turing reducibility; information content, as measured by notions such as Kolmogorov complexity; and randomness of individual objects, as first successfully defined by Martin-Löf. Although algorithmic randomness has been studied for several decades, a dramatic upsurge of interest in the area, starting in the late 1990s, has led to significant advances. This is the first comprehensive treatment of this important field, designed to be both a reference tool for experts and a guide for newcomers. It surveys a broad section of work in the area, and presents most of its major results and techniques in depth. Its organization is designed to guide the reader through this large body of work, providing context for its many concepts and theorems, discussing their significance, and highlighting their interactions. It includes a discussion of effective dimension, which allows us to assign concepts like Hausdorff dimension to individual reals, and a focused but detailed introduction to computability theory. It will be of interest to researchers and students in computability theory, algorithmic information theory, and theoretical computer science.

• Algorithms
• algorithm
• complexity
• complexity theory
• computability theory
• computational geometry
• computer
• computer science
• information theory
• logic
• modeling

From the reviews:

“Develops the prerequisites to algorithmic randomness: computability theory and Kolmogorov complexity. … Studying these … one should be able to proceed in the area with confidence. A draft of the book under review has been circulating for years and the reviewer found it to be the best source when attempting to conduct research in the area … . It is advantageous for the future of the area of algorithmic randomness that these two books were published at the cusp of a period of great activity.” (Bjørn Kjos-Hanssen, Mathematical Reviews, Issue 2012 g)

“A thorough and systematic study of algorithmic randomness, this long-awaited work is an irreplaceable source of well-presented classic and new results for advanced undergraduate and graduate students, as well as researchers in the field and related areas. The book joins a select number of books in this category.” (Hector Zenil, ACM Computing Reviews, October, 2011)

• School of Mathematics, Statistics &, Operations Research, Victoria University, Wellington, New Zealand

  Rodney G. Downey

• , Department of Mathematics, University of Chicago, Chicago, USA

  Denis R. Hirschfeldt

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