On Beta-Shifts Having Arithmetical Languages

Let β be a real number with 1 < β < 2. We prove that the language of the β-shift is ∆0 n iff β is a ∆n-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.

• Type: Conference
• Year: 2005
• Where: MFCS
• Authors: Jakob Grue Simonsen
• Keywords: Computable Real, Constructive Proof Exists, MFCS 2005, Real Number