Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-rbzxz Total loading time: 0.313 Render date: 2022-05-17T19:05:38.162Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

8 - Higher randomness

Published online by Cambridge University Press:  07 May 2020

Johanna N. Y. Franklin
Affiliation:
Hofstra University, New York
Christopher P. Porter
Affiliation:
Drake University, Iowa
Get access

Summary

We present an overview of higher randomness and its recent developments. After an introduction, we provide in the second section some background on higher computability, presenting in particular $\Pi^1_1$ and $\Sigma^1_1$ sets from the viewpoint of the computability theorist. In the third section we give an overview of the different higher randomness classes: $\Delta^1_1$-randomness, $\Pi^1_1$-Martin-Löf randomness, higher weak-2 randomness, higher difference randomness, and $\Pi^1_1$-randomness. We then move on to study each of these classes, separating them and inspecting their respective lowness classes. We put more attention on $\Pi^1_1$-Martin-Löf randomness and $\Pi^1_1$-randomness: The former is the higher analogue of the most well-known and studied class in classical algorithmic randomness. We show in particular how to lift the main classical randomness theorems to the higher settings by putting continuity in higher reductions and relativisations. The latter presents, as we will see, many remarkable properties and does not have any analogue in classical randomness. Finally in the eighth section we study randomness along with a higher hierarchy of complexity of sets, motivated by the notion of higher weak-2 randomness. We show that this hierarchy collapses eventually.

Type
Chapter
Information
Algorithmic Randomness
Progress and Prospects
, pp. 232 - 300
Publisher: Cambridge University Press
Print publication year: 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×