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A Statistical Mechanical Interpretation of Algorithmic Information Theory

Part of the SpringerBriefs in Mathematical Physics Book Series (volume 36)

DOI: 10.1007/978-981-15-0739-7

ISBN: 978-981-15-0738-0

Publisher: Springer Singapore - Springer Journals

Nov 12, 2019

1 - 9 of 9 Chapters

LitStream

Chapter 1Statistical Mechanical Interpretation of Noiseless Source Coding

Tadaki, Kohtaro

Pages: 1 –12

[In this chapter we develop a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the...
LitStream

Chapter 2Algorithmic Information Theory

Tadaki, Kohtaro

Pages: 13 –21

[In this chapter, we review the basic framework of algorithmic information theory to the extent necessary to read the rest of the book.]
LitStream

Chapter 3Partial Randomness

Tadaki, Kohtaro

Pages: 23 –34

[At the turn of the 21st century, Tadaki [40, 41] generalized Chaitin’s halting probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}...
LitStream

Chapter 4Temperature Equals to Partial Randomness

Tadaki, Kohtaro

Pages: 35 –52

[In 2006 Calude and Stay [6] pointed out a superficial similarity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Chapter 5Fixed Point Theorems on Partial Randomness

Tadaki, Kohtaro

Pages: 53 –73

[In the preceding chapter, we saw that the partial randomness (and therefore the compression rate) of the values of all the thermodynamic quantities equals to the temperature T in the statistical mechanical interpretation of AIT. However, in statistical mechanics or thermodynamics, among all...
LitStream

Chapter 6Statistical Mechanical Meaning of the Thermodynamic Quantities of AIT

Tadaki, Kohtaro

Pages: 75 –82

[Generally speaking, in order to give a statistical mechanical interpretation to a framework which looks unrelated to statistical mechanics at first glance, it is important to identify a microcanonical ensemble in the framework. Once we can do so, we can easily develop an equilibrium statistical...
LitStream

Chapter 7The Partial Randomness of Recursively Enumerable Reals

Tadaki, Kohtaro

Pages: 83 –101

[In this chapter, we investigate the partial randomness of a recursively enumerable real, also known as a left-computable real, and gives a theorem which plays an important role in the analysis of the thermodynamic quantities of AIT.]
LitStream

Chapter 8Computation-Theoretic Clarification of the Phase Transition at Temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=1$$\end{document}

Tadaki, Kohtaro

Pages: 103 –125

[The notion of weak truth-table reducibility plays an important role in recursion theory (see e.g. Nies [27] and Downey and Hirschfeldt [14]). For any sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}...
LitStream

Chapter 9Other Related Results and Future Development

Tadaki, Kohtaro

Pages: 127 –130

[In addition to the works described in the preceding chapters, we did various works about the statistical mechanical interpretation of AIT. This section summarizes some of them. At present, we are trying various approaches to reveal the overall picture of the statistical mechanical interpretation...

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