Abstract
We consider clockable ordinals for Infinite Time Turing Machines (ITTMs), i.e., halting times of ITTMs on the empty input. It is well-known that, in contrast to the writable ordinals, the set of clockable ordinals has ‘gaps’. In this paper, we show several results on gaps, mainly related to the admissible ordinals they may properly contain. We prove that any writable ordinal can occur as the order type of the sequence of admissible ordinals in such a gap. We give precise information on their ending points. We also investigate higher rank ordinals (recursively inaccessible, etc.). Moreover, we show that those gaps can have any reasonably effective length (in the sense of ITTMs) compared to their starting point.
The authors would like to express their thanks to the anonymous referees, who made numerous suggestions and interesting remarks.
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Notes
- 1.
Here, by ‘parallel’, we mean that we alternately perform one step of the first and one of the second algorithm. After \(\omega \) many steps of the parallel execution, we will thus have performed \(\omega \) many steps of both algorithms.
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Carl, M., Durand, B., Lafitte, G., Ouazzani, S. (2017). Admissibles in Gaps. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_18
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DOI: https://doi.org/10.1007/978-3-319-58741-7_18
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