Ordinal machines and admissible recursion theory

Annals of Pure and Applied Logic 160 (3):310-318 (2009)

Abstract

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory

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References found in this work

The Α-Finite Injury Method.G. E. Sacks & S. G. Simpson - 1972 - Annals of Mathematical Logic 4 (4):343-367.
The Fine Structure of the Constructible Hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
Metarecursive Sets.G. Kreisel & Gerald E. Sacks - 1965 - Journal of Symbolic Logic 30 (3):318-338.
Turing Computations on Ordinals.Peter Koepke - 2005 - Bulletin of Symbolic Logic 11 (3):377-397.
Metarecursively Enumerable Sets and Their Metadegrees.Graham C. Driscoll - 1968 - Journal of Symbolic Logic 33 (3):389-411.

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Citations of this work

Supertasks.Jon Pérez Laraudogoitia - 2008 - Stanford Encyclopedia of Philosophy.
Infinite Computations with Random Oracles.Merlin Carl & Philipp Schlicht - 2017 - Notre Dame Journal of Formal Logic 58 (2):249-270.
Taming Koepke's Zoo II: Register Machines.Merlin Carl - 2022 - Annals of Pure and Applied Logic 173 (3):103041.
The Computational Strengths of Α-Tape Infinite Time Turing Machines.Benjamin Rin - 2014 - Annals of Pure and Applied Logic 165 (9):1501-1511.

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