Abstract
For exponentially closed ordinals \(\alpha \), we consider recognizability of constructible subsets of \(\alpha \) for weak and strong \(\alpha \)-register machines (\(\alpha \)-(w)ITRMs) with parameters and their distribution in the constructible hierarchy. In particular, we show that, 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. (We thank our three anonymous referees for their valuable comments.)
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Notes
- 1.
For ITRMs, it is known that decidability, semidecidability and co-semi-decidability, and hence also recognizability, semirecognizability and cosemirecognizability coincide. However, we do not know this for \(\alpha \)-ITRMs.
- 2.
ZF\(^{-}\) denotes Zermelo-Fraenkel set theory without the power set axiom; see [11] for a discussion of the axiomatizations. An ordinal \(\alpha \) is \(\varSigma _2\)-admissible if and only if \(L_{\alpha }\models \varSigma _{2}\)-KP.
- 3.
The condition of exponential closure is a technical convenience; it allows us, for example, to carry out halting algorithms after each other or run nested loops of algorithms without caring for possible register overflows. We conjecture that dropping this condition would not substantially change most of the results, but merely lead to more cumbersome arguments.
- 4.
This case uses ideas similar to those used for proving the lost melody theorem for infinite time Blum-Shub-Smale machines, see [9].
- 5.
Note that \(\alpha \) itself is definable in \(L_{\alpha +1}\) without parameters.
- 6.
The following results are analogues for ITRM-singular \(\alpha \) of results obtained in [7] for \(\alpha =\omega \).
- 7.
This will be further generalized in Lemma 9 below.
- 8.
Note that, for example, the first \(\varPi _{3}\)-reflecting ordinal has this property.
- 9.
Recall that, by Theorem 46 of [3], every \(\varPi _{3}\)-reflecting ordinal is u-weak.
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Carl, M. (2023). All Melodies Are Lost – Recognizability for Weak and Strong \(\alpha \)-Register Machines. In: Della Vedova, G., Dundua, B., Lempp, S., Manea, F. (eds) Unity of Logic and Computation. CiE 2023. Lecture Notes in Computer Science, vol 13967. Springer, Cham. https://doi.org/10.1007/978-3-031-36978-0_7
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