Abstract
Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the \(\limsup \) of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the \(\limsup \) rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content of that cell has stabilized before that limit step and is then equal to this constant value. We call these machines weak infinite time Turing machines (wITTMs). We study different variants of wITTMs adding multiple tapes, heads, or bidimensional tapes. We show that some of these models are equivalent to each other concerning their computational strength. We show that wITTMs decide exactly the arithmetic relations on natural numbers.
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Notes
After I gave a talk on wITTMs at the Computability in Europe 2015 meeting, Merlin Carl pointed out to me that he and Joel D. Hamkins briefly discussed machines of this type in 2012 on MathOverflow (see https://mathoverflow.net/questions/111902/transfinitely-iterated-limit-computability). In that discussion, Hamkins stated the results that we here present as Theorems 2 and 26 and hinted briefly at their proofs. Apparently, Hamkins did not publish anything on wITTMs. Carl discusses wITTMs in chapter 2 of his forthcoming [3]. There Carl also presents a complete proof of Theorem 2, which he communicated to me.
See [4] p. 298.
We will later show that one could also imagine to have two input tapes at disposal. Adding finitely many input tapes does not alter the computational power of wITTMs. See, in particular, Equivalence Theorem 15.
In his forthcoming book Ordinal computability. An introduction to infinitary machines, Merlin Carl proves the following statements:
- 1.
If \(S\subseteq \omega \) is wITTM-writable, then, for some n, \(S\le _T\emptyset ^{(n)}\).
- 2.
For every \(S\subseteq \omega \), for some wITTM M, M on input S outputs \(S'\).
Carl’s proofs of these statements essentially work also as proofs of, respectively, Theorems 24 and 26. We slightly adapt Carl’s ideas to provide the proofs in this article.
- 1.
In a personal conversation with me, Dan Turetsky communicated the following observation together with its proof.
References
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Bianchetti, M.: Infinite time computation: strong and weak infinite time turing machines. Master’s thesis, University of Notre Dame (2017)
Carl, M.: Ordinal computability. An introduction to infinitary machines (forthcoming)
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Hamkins, J.D., Lewis, A.: Infinite time turing machines. J. Symb. Log. 65(2), 567–604 (2000)
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Acknowledgements
This work is largely a part of my master thesis [2] under the supervision of Julia F. Knight. I am very grateful to Julia F. Knight for her skillful guidance and insightful comments at every stage of the project. I would like to warmly thank Quinn Culver for numerous, extremely helpful discussions. I am also grateful to Merlin Carl, Dan Turetsky, and Greg Igusa for useful discussions and to an anonymous reviewer for valuable suggestions. I am also grateful to everyone that attended to presentations of earlier versions of this work either at the Notre Dame Computability seminar or at the CiE 2015 meeting.
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Bianchetti, M. Weaker variants of infinite time Turing machines. Arch. Math. Logic 59, 335–365 (2020). https://doi.org/10.1007/s00153-019-00692-9
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DOI: https://doi.org/10.1007/s00153-019-00692-9
Keywords
- Ordinal computability
- Infinite time Turing machine
- Transfinite computation
- Supertask
- Arithmetic hierarchy
- Real arithmetic