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The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines

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Connecting with Computability (CiE 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12813))

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Abstract

We consider recognizability for Infinite Time Blum-Shub-Smale machines, a model of infinitary computability introduced by Koepke and Seyfferth. In particular, we show that the lost melody theorem (originally proved for ITTMs by Hamkins and Lewis), i.e. the existence of non-computable, but recognizable real numbers, holds for ITBMs, that ITBM-recognizable real numbers are hyperarithmetic and that both ITBM-recognizable and ITBM-unrecognizable real numbers appear at every level of the constructible hierarchy below \(L_{\omega _{1}^{\text {CK}}}\) above \(\omega ^{\omega }\).

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Notes

  1. 1.

    “Weak ITRMs”, also known as “unresetting ITRM”, differ from ITRMs in that a computation in which the inferior limit of the sequence of contents of some register is infinite at some limit time, the computation is undefined, while for ITRMs, the content of such a register is just reset to 0; they were defined by Koepke in [11].

  2. 2.

    For ITRMs, this is proved in [4]; for ITTMs, it is known that there are, e.g., no recognizable real numbers in \(L_{\varSigma }\setminus L_{\lambda }\), see, e.g., [8], Theorem 4.2.6.

  3. 3.

    For example, we can take \(\psi _{0}(x)\) to be \(\forall {y\in x}(y\ne y)\) and then let \(\psi _{k+1}(x)\) be \(\forall {y}(y\in x\leftrightarrow (\exists {z}(\psi _{k}(z)\wedge y\in z)\vee \psi _{k}(y)))\).

  4. 4.

    Here is a sketch for the construction: Use p to split \(\omega \) into \(\omega \) many disjoint portions of the form \(\{p(k,i):i\in \omega \}\). For \(i\in \omega \), let \(f_{0}\) be the \(<_{L}\)-minimal bijection \(f_{0}:\omega \rightarrow L_{\omega }\) and, for \(i>0\), let \(f_{i}\) be the \(<_{L}\)-minimal bijection \(f_{i}:\omega \rightarrow L_{\omega ^{i+1}}\setminus L_{\omega ^{i}}\); for \(k\in \omega \), let \(F_{k}:=\bigcup _{i<k}f_{i}\). Then let \(c_{k}:=\{p(i,j):F_{k}(i)\in F_{k}(j)\}\) and \(c:=\bigcup _{k\in \omega }c_{k}\). Thus, c is a code for \(L_{\omega ^{\omega }}\). To compute c, it suffices to compute \(f_{k}\) and \(c_{k}\), uniformly in k. For this, run through the ITBM-programs and use H to identify the first program Q that computes a code d for \(L_{\omega ^{k+1}}\). Using H, we can actually obtain d by considering, for each \(i\in \omega \), the program \(Q^{\prime }_{i}\) which, for \(i\in \omega \), halts when Q(i) halts with output 0 and loops otherwise and using H to check whether \(Q^{\prime }_{i}\) halts (note that \(Q^{\prime }_{i}\) is recursive in Q and i). From d, one can compute \(f_{0}\), ..., \(f_{k-1}\), and hence also \(F_{k}\), using truth predicate evaluation (since natural numbers are definable without parameters); again using truth predicate evaluation, one obtains \(c_{k}\).

  5. 5.

    The same approach was used in [3] to obtain non-recognizables for ITRMs.

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Acknowledgements

We thank our three anonymous referees for various helpful suggestions for improving the presentation of this paper.

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  1. Europa-Universität Flensburg, Auf dem Campus 1b, 24943, Flensburg, Germany

    Merlin Carl

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  1. Merlin Carl
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Correspondence to Merlin Carl .

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  1. University of Lille, Lille, France

    Liesbeth De Mol

  2. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

  3. University of Göttingen, Göttingen, Germany

    Florin Manea

  4. Ghent University, Ghent, Belgium

    David Fernández-Duque

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Carl, M. (2021). The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_7

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  • DOI: https://doi.org/10.1007/978-3-030-80049-9_7

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