Notre Dame Journal of Formal Logic

Infinite Computations with Random Oracles

Merlin Carl and Philipp Schlicht

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Abstract

We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and resetting infinite-time register machines, and α-Turing machines for countable admissible ordinals α.

Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 2 (2017), 249-270.

Dates
Received: 18 March 2014
Accepted: 21 August 2014
First available in Project Euclid: 21 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1487646410

Digital Object Identifier
doi:10.1215/00294527-3832619

Mathematical Reviews number (MathSciNet)
MR3634980

Zentralblatt MATH identifier
06751302

Subjects
Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30] 03D60: Computability and recursion theory on ordinals, admissible sets, etc. 03D65: Higher-type and set recursion theory
Secondary: 68Q05: Models of computation (Turing machines, etc.) [See also 03D10, 68Q12, 81P68] 03E15: Descriptive set theory [See also 28A05, 54H05] 03E35: Consistency and independence results

Keywords
generalized recursion theory randomness constructibility admissible sets infinitary computations

Citation

Carl, Merlin; Schlicht, Philipp. Infinite Computations with Random Oracles. Notre Dame J. Formal Logic 58 (2017), no. 2, 249--270. doi:10.1215/00294527-3832619. https://projecteuclid.org/euclid.ndjfl/1487646410


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