The distribution of ITRM-recognizable reals

TL;DR

This paper studies the distribution of ITRM-recognizable reals within the constructible hierarchy, revealing gaps and limitations in recognizability without a universal recognizer.

Contribution

It provides a detailed analysis of how ITRM-recognizable reals are distributed along the constructible hierarchy, including the existence of gaps and the absence of a universal recognizer.

Findings

Recognizable reals have gaps in the well-ordering of L.

No universal ITRM recognizer exists.

Relativized recognizability is also considered.

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. Koepke (2009), Koepke and Welch (2011) and Koepke and Miller (2008). We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in Hamkins and Lewis (200) and applied to $ITRM$'s in Carl et al. (2010). A real $x$ is $ITRM$-recognizable iff there is an $ITRM$-program $P$ such that $P^{y}$ stops with output 1 iff $y=x$, and otherwise stops with output 0. In Carl et al. (2010), it is shown that the recognizable reals are not contained in the computable reals. Here, we investigate in detail how the $ITRM$-recognizable reals are distributed along the canonical well-ordering $<_{L}$ of G\"odel'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.