Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences
Author(s)
Schaeffer, Luke R.; Shallit, Jeffrey
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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.
Date issued
2016-02Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer ScienceJournal
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
ISSN
1077-8926
1097-1440