# A Statistical Mechanical Interpretation of Algorithmic Information TheoryComputation-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}

A Statistical Mechanical Interpretation of Algorithmic Information Theory: Computation-Theoretic... [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} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A, B\subset \mathbb {N}$$\end{document}, we say that Ais weak truth-table reducible toBWeak truth-table reducibility, denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\le _{wtt} B$$\end{document}, if there exist an oracle Turing machine M and a total recursive function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:\mathbb {N}\rightarrow \mathbb {N}$$\end{document} such that (i) A is Turing reducible to B via M and (ii) on every input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document}, the machine M only queries natural numbers at most g(n).] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Statistical Mechanical Interpretation of Algorithmic Information TheoryComputation-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}

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

/lp/springer-journals/a-statistical-mechanical-interpretation-of-algorithmic-information-i9uM5OoNps
Publisher
Springer Singapore
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019
ISBN
978-981-15-0738-0
Pages
103 –125
DOI
10.1007/978-981-15-0739-7_8
Publisher site
See Chapter on Publisher Site

### Abstract

[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} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A, B\subset \mathbb {N}$$\end{document}, we say that Ais weak truth-table reducible toBWeak truth-table reducibility, denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\le _{wtt} B$$\end{document}, if there exist an oracle Turing machine M and a total recursive function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:\mathbb {N}\rightarrow \mathbb {N}$$\end{document} such that (i) A is Turing reducible to B via M and (ii) on every input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document}, the machine M only queries natural numbers at most g(n).]

Published: Nov 12, 2019