The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines

TL;DR

This paper extends the lost melody theorem to Infinite Time Blum-Shub-Smale machines, showing the existence of recognizable but non-computable reals and their placement in the constructible hierarchy.

Contribution

It proves the lost melody theorem for ITBMs, characterizes recognizable reals as hyperarithmetic, and analyzes their hierarchy placement.

Findings

Existence of non-computable, recognizable reals for ITBMs.

ITBM-recognizable reals are hyperarithmetic.

Recognizable and unrecognizable reals appear at every level below $L_{\omega_1^{CK}}$.

Abstract

We consider recognizability for Infinite Time Blum-Shub-Smale machines, a model of infinitary computability introduced in Koepke and Seyfferth [KS]. In particular, we show that the lost melody theorem (originally proved for ITTMs in Hamkins and Lewis [HL]), i.e. the existence of non-computable, but recognizable real numbers, holds for ITBMs, that ITBM-recognizable real numbers are hyperarithmetic and that both ITBM-recognizable and ITBM-unrecognizable real numbers appear at every level of the constructible hierarchy below $L_{\omega_{1}^{\text{CK}}}$ at which new real numbers appear at all.