Strong Medvedev Reducibilities and the KL-Randomness Problem

While it is not known whether each real that is Kolmogorov-Loveland random is Martin-Löf random, i.e., whether KLR⊆MLR $$\mathrm {KLR}\subseteq \mathrm {MLR}$$ , Kjos-Hanssen and Webb (2021) showed that MLR is truth-table Medvedev reducible (≤s,tt $$\le _{s,tt}$$ ) to KLR. They did this by studying...

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Bibliographic Details
Published inRevolutions and Revelations in Computability Vol. 13359; pp. 151 - 161
Main Authors Kjos-Hanssen, Bjørn, Webb, David J.
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2022
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Martin-Löf Randomness
Medvedev reducibility
truth-table reducibility
Online AccessGet full text
ISBN3031087399
9783031087394
ISSN0302-9743
1611-3349
DOI10.1007/978-3-031-08740-0_13

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Summary:While it is not known whether each real that is Kolmogorov-Loveland random is Martin-Löf random, i.e., whether KLR⊆MLR $$\mathrm {KLR}\subseteq \mathrm {MLR}$$ , Kjos-Hanssen and Webb (2021) showed that MLR is truth-table Medvedev reducible (≤s,tt $$\le _{s,tt}$$ ) to KLR. They did this by studying a natural class Either (MLR) and showing that MLR≤s,ttEither(MLR)⊇KLR $$\mathrm {MLR}\le _{s,tt}\mathrm {Either}\,(\mathrm {MLR})\supseteq \mathrm {KLR}$$ . We show that Degtev’s stronger reducibilities (positive and linear) do not suffice for the reduction of MLR to Either (MLR), and some related results.
Bibliography:Original Abstract: While it is not known whether each real that is Kolmogorov-Loveland random is Martin-Löf random, i.e., whether KLR⊆MLR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KLR}\subseteq \mathrm {MLR}$$\end{document}, Kjos-Hanssen and Webb (2021) showed that MLR is truth-table Medvedev reducible (≤s,tt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le _{s,tt}$$\end{document}) to KLR. They did this by studying a natural class Either (MLR) and showing that MLR≤s,ttEither(MLR)⊇KLR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {MLR}\le _{s,tt}\mathrm {Either}\,(\mathrm {MLR})\supseteq \mathrm {KLR}$$\end{document}. We show that Degtev’s stronger reducibilities (positive and linear) do not suffice for the reduction of MLR to Either (MLR), and some related results.
Bjørn Kjos-Hanssen—This work was partially supported by a grant from the Simons Foundation (#704836 to Bjørn Kjos-Hanssen). The authors would like to thank Reviewer 2 for Theorem 5. The second author would also like to thank Carl Eadler for pointing him towards results about Karnaugh maps, which were helpful in thinking about Theorem 4.
ISBN:3031087399
9783031087394
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-031-08740-0_13