The distribution of ITRM-recognizable reals

Abstract

Infinite Time Register Machines (ITRM's) are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. [12,13,11]. We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in [6] and applied to ITRM's in [3]. A real x is ITRM-recognizable iff there is an ITRM-program P such that Py stops with output 1 iff y=x, and otherwise stops with output 0. In [3], it is shown that the recognizable reals are not contained in the ITRM-computable reals. Here, we investigate in detail how the ITRM-recognizable reals are distributed along the canonical well-ordering <L of Gödel's constructible hierarchy L. In particular, we prove that the recognizable reals have gaps in <L, that there is no universal ITRM in terms of recognizability and consider a relativized notion of recognizability.