On beta-shifts having arithmetical languages

Authors:
Jakob Grue Simonsen
Affiliations:
Department of Computer Science, University of Copenhagen (DIKU), Copenhagen Ø, Denmark
Venue:
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Year:
2005

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Abstract

Let β be a real number with 1 β β-shift is Δ$^{\rm 0}_{n}$ iff β is a Δn-real. The special case where n is 1 is the independently interesting result that the language of the β-shift is decidable iff β is a computable real. The “if” part of the proof is non-constructive; we show that for Walters' version of the β-shift, no constructive proof exists.