Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences
Name
Schaeffer-2016-Closed palindromic.pdf
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288.77 KB
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Adobe PDF
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5c29733e46af58fa6da6d0513943a469
Author(s)
Schaeffer, Luke R.
Shallit, Jeffrey
Date Issued
February 2016
Journal
Electronic Journal of Combinatorics
Publisher
European Mathematical Information Service (EMIS)
Citation
Schaeffer, Luke, and Jeffrey Shallit. "Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences." Electronic Journal of Combinatorics, 23:1 (2016), pp. 1-19.
Version
Final published version
Abstract
We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those n for which an automatic sequence x has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length n is itself automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in the prefix of length n of a k-automatic sequence is not k-synchronized.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.37236/5752
