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2-State 2-Symbol Turing Machines with Periodic Support Produce Regular Sets

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Descriptional Complexity of Formal Systems (DCFS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10316))

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Abstract

We say that a Turing machine has periodic support if there is an infinitely repeated word to the left of the input and another infinitely repeated word to the right. In the search for the smallest universal Turing machines, machines that use periodic support have been significantly smaller than those for the standard model (i.e. machines with the usual blank tape on either side of the input). While generalising the model allows us to construct smaller universal machines it makes proving decidability results for the various state-symbol products that restrict program size more difficult. Here we show that given an arbitrary 2-state 2-symbol Turing machine and a configuration with periodic support the set of reachable configurations is regular. Unlike previous decidability results for 2-state 2-symbol machines, here we include in our consideration machines that do not reserve a transition rule for halting, which further adds to the difficulty of giving decidability results.

This work is supported by Swiss National Science Foundation grant numbers 200021-153295 and 200021-166231. The author thanks the anonymous reviewers for their careful reading of the paper and their helpful comments.

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Notes

  1. 1.

    Note that applying mapping (9) to machines of the form given by Eq. (7) also flips the read symbols and so applying it to \(\varSigma =(\sigma _1,\sigma _2,\sigma _3,\sigma _4)\) gives \((\overline{\sigma _2},\overline{\sigma _1},\overline{\sigma _4},\overline{\sigma _3})\) instead of \((\overline{\sigma _1},\overline{\sigma _2},\overline{\sigma _3},\overline{\sigma _4})\).

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Author information

Authors and Affiliations

  1. Institute of Neuroinformatics, University of Zürich and ETH Zürich, Zürich, Switzerland

    Turlough Neary

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  1. Turlough Neary
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Correspondence to Turlough Neary .

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Editors and Affiliations

  1. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  2. University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada

    Cezar Câmpeanu

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Neary, T. (2017). 2-State 2-Symbol Turing Machines with Periodic Support Produce Regular Sets. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_22

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  • DOI: https://doi.org/10.1007/978-3-319-60252-3_22

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  • Online ISBN: 978-3-319-60252-3

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