The strange world of transfinite melodies—Recognizability for weak and strong infinite time α-register machines

For exponentially closed ordinals α , we consider recognizability of constructible subsets of α for α -(w)ITRMs and their distribution in the constructible hierarchy. In particular, for α -ITRMs, we show that there are lost melodies that are recognizable without parameters for all α and that iterated recognizability is absolute between L and V for most values of α ; moreover, we generalize the “all or nothing”-phenomenon that was hitherto known only for the case α = ω to a proper class of α . For α -wITRMs, we offer a complete characterization of those α for which lost melodies exist and show that the relation between the sets of computable and recognizable subsets of α varies wildly, depending on α : 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 α for which the set of recognizable sets is empty and others for which the set of recognizable sets is non-empty, but disjoint from the set of computable sets. This paper is an extension of our paper in the CiE 2023 proceedings.