Degrees of randomized computability

Bulletin of Symbolic Logic 28 (1):27-70 (2022)
  Copy   BIBTEX

Abstract

In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$, $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated with this ordering, the Levin–V’yugin degrees, can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 93,745

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Analytics

Added to PP
2022-04-07

Downloads
12 (#317,170)

6 months
5 (#1,552,255)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Christopher Porter
Drake University

Citations of this work

No citations found.

Add more citations

References found in this work

Notions of weak genericity.Stuart A. Kurtz - 1983 - Journal of Symbolic Logic 48 (3):764-770.
Cone avoidance and randomness preservation.Stephen G. Simpson & Frank Stephan - 2015 - Annals of Pure and Applied Logic 166 (6):713-728.
Randomness and Semimeasures.Laurent Bienvenu, Rupert Hölzl, Christopher P. Porter & Paul Shafer - 2017 - Notre Dame Journal of Formal Logic 58 (3):301-328.

Add more references