Infinite time Turing machines with only one tape

  • J. D. Hamkins and D. E. Seabold, “Infinite Time Turing Machines With Only One Tape,” Mathematical Logic Quarterly, vol. 47, iss. 2, pp. 271-287, 2001.  
    @article{HamkinsSeabold2001:OneTape,
    author = {Hamkins, Joel David and Seabold, Daniel Evan},
    title = {Infinite Time Turing Machines With Only One Tape},
    journal = {Mathematical Logic Quarterly},
    volume = {47},
    number = {2},
    publisher = {WILEY-VCH Verlag Berlin GmbH},
    issn = {1521-3870},
    MRNUMBER = {1829946 (2002f:03074)},
    url = {http://dx.doi.org/10.1002/1521-3870(200105)47:2<271::AID-MALQ271>3.0.CO;2-6},
    doi = {10.1002/1521-3870(200105)47:2<271::AID-MALQ271>3.0.CO;2-6},
    pages = {271--287},
    keywords = {One-tape infinite Turing machine, Supertask computation},
    year = {2001},
    eprint = {math/9907044},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for functions $f:\mathbb{R}\to\mathbb{N}$, the same class of computable functions. Nevertheless, there are infinite time computable functions $f:\mathbb{R}\to\mathbb{R}$ that are not one-tape computable, and so the two models of supertask computation are not equivalent. Surprisingly, the class of one-tape computable functions is not closed under composition; but closing it under composition yields the full class of all infinite time computable functions. Finally, every ordinal which is clockable by an infinite time Turing machine is clockable by a one-tape machine, except certain isolated ordinals that end gaps in the clockable ordinals.

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