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International Conference on Combinatorics on Words

WORDS 2015: Combinatorics on Words pp 170–190Cite as

Mechanical Proofs of Properties of the Tribonacci Word

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9304)

Abstract

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) “Tribonacci-automatic”. This class includes, for example, the famous Tribonacci word \(\mathbf{T} = 0102010010201\cdots \), the fixed point of the morphism \(0 \rightarrow 01\), \(1 \rightarrow 02\), \(2 \rightarrow 0\). We use our decision procedure to reprove some old results about the Tribonacci word from the literature, such as assertions about the occurrences in \(\mathbf T\) of squares, cubes, palindromes, and so forth. We also obtain some new results, including on enumeration.

Keywords

  • Linear Representation
  • Critical Exponent
  • Decision Procedure
  • Canonical Representation
  • Linear Recurrence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

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    There is also a version where “prefixes” is replaced by “suffixes”.

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Acknowledgments

We are very grateful to Amy Glen for her recommendations and advice. We thank Ondrej Turek and the referees for pointing out errors.

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Hamoon Mousavi & Jeffrey Shallit

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  1. Hamoon Mousavi
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  2. Jeffrey Shallit
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Correspondence to Jeffrey Shallit .

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Editors and Affiliations

  1. Universität Kiel, Kiel, Germany

    Florin Manea

  2. Universität Kiel, Kiel, Germany

    Dirk Nowotka

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Mousavi, H., Shallit, J. (2015). Mechanical Proofs of Properties of the Tribonacci Word. In: Manea, F., Nowotka, D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science(), vol 9304. Springer, Cham. https://doi.org/10.1007/978-3-319-23660-5_15

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  • DOI: https://doi.org/10.1007/978-3-319-23660-5_15

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