The strange world of transfinite Melodies -- Recognizability for weak and strong infinite time $\alpha$-register machines

TL;DR

This paper investigates recognizability of constructible subsets of exponential closed ordinals for infinite time register machines, revealing complex behaviors and phenomena such as lost melodies and variability in recognizability across different ordinals.

Contribution

It provides a comprehensive analysis of recognizability for $ ext{(w)}$ITRMs at exponential closed ordinals, including new phenomena and complete characterizations of when lost melodies occur.

Findings

Existence of lost melodies recognizable without parameters for all $ ext{(w)}$ITRMs.

Most $ ext{(w)}$ITRMs exhibit absolute recognizability between $L$ and $V$.

The recognizability sets for $ ext{(w)}$ITRMs vary widely depending on $eta$, including empty, disjoint, or overlapping with computable sets.

Abstract

For exponentially closed ordinals $\alpha$, we consider recognizability of constructible subsets of $\alpha$ for $\alpha$-(w)ITRMs and their distribution in the constructible hierarchy. In particular, for $\alpha$-ITRMs, we show that, there are lost melodies that are recognizable without parameters for all $\alpha$, that the iterated recognizability is absolute between $L$ and $V$ for most values of $\alpha$ and generalize "all or nothing"-phenomenon known from ITRMs occurs for a proper class of $\alpha$. For $\alpha$-wITRMs, we offer a complete characterization of those $\alpha$ for which lost melodies exist and that the relation between the sets of computable and recognizable subsets of $\alpha$ varies wildly, depending on $\alpha$: The computable sets may be included among the recognizable sets (which is usually the case in ordinal computability), but there are also class many values of $\alpha$ for which the set of recognizable sets is empty and such for which the set of recognizable sets is non-empty, but disjoint from the set of computable sets. %for class many values of $\alpha$, the sets of $\alpha$-wITRM-computable and $\alpha$-wITRM-recognizable subsets of $\alpha$ are both non-empty, but disjoint, and, also for class many values of $\alpha$, the set of $\alpha$-wITRM-recognizable subsets of $\alpha$ is empty. This paper is an extension of our paper in the CiE 2023 proceedings.