Abstract
Parikh-collinear morphisms have recently received a lot of attention. They are defined by the property that the Parikh vectors of the images of letters are collinear. We first show that any fixed point of such a morphism is automatic. Consequently, we get under some mild technical assumption that the abelian complexity of a binary fixed point of a Parikh-collinear morphism is also automatic, and we discuss a generalization to arbitrary alphabets. Then, we consider the abelian complexity function of the fixed point of the Parikh-collinear morphism \(0\mapsto 010011\), \(1\mapsto 1001\). This 5-automatic sequence is shown to be aperiodic, answering a question of Salo and Sportiello.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A long version is available at https://doi.org/10.48550/arXiv.2201.04603.
References
Allouche, J.P., Dekking, M., Queffélec, M.: Hidden automatic sequences. Comb. Theory 1(#20) (2021). https://doi.org/10.5070/C61055386
Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511546563
Allouche, J.-P., Shallit, J.: Automatic sequences are also non-uniformly morphic. In: Raigorodskii, A.M., Rassias, M.T. (eds.) Discrete Mathematics and Applications. SOIA, vol. 165, pp. 1–6. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-55857-4_1
Bruyère, V., Hansel, G., Michaux, C., Villemaire, R.: Logic and \(p\)-recognizable sets of integers. Bull. Belg. Math. Soc. Simon Stevin 1(2), 191–238 (1994). http://projecteuclid.org/euclid.bbms/1103408547, journées Montoises (Mons, 1992)
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960). https://doi.org/10.1002/malq.19600060105
Béal, M.P., Perrin, D., Restivo, A.: Recognizability of morphisms. Ergodic Theory Dyn. Syst. 1–25 (2023). https://doi.org/10.1017/etds.2022.109
Cassaigne, J., Nicolas, F.: Quelques propriétés des mots substitutifs. Bull. Belg. Math. Soc. Simon Stevin 10(suppl.), 661–676 (2003). http://projecteuclid.org/euclid.bbms/1074791324
Cassaigne, J., Richomme, G., Saari, K., Zamboni, L.Q.: Avoiding abelian powers in binary words with bounded abelian complexity. Int. J. Found. Comput. S. 22(4), 905–920 (2011). https://doi.org/10.1142/S0129054111008489
Charlier, É., Leroy, J., Rigo, M.: Asymptotic properties of free monoid morphisms. Linear Algebra Appl. 500, 119–148 (2016). https://doi.org/10.1016/j.laa.2016.02.030
Charlier, É., Rampersad, N., Shallit, J.: Enumeration and decidable properties of automatic sequences. Internat. J. Found. Comput. Sci. 23(5), 1035–1066 (2012). https://doi.org/10.1142/S0129054112400448
Cobham, A.: Uniform tag sequences. Math. Syst. Theory 6, 164–192 (1972). https://doi.org/10.1007/BF01706087
Dekking, F.M.: The spectrum of dynamical systems arising from substitutions of constant length. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 41(3), 221–239 (1978). https://doi.org/10.1007/BF00534241
Durand, F., Leroy, J.: The constant of recognizability is computable for primitive morphisms. Journal of Integer Sequences 20(#17.4.5) (2017). https://cs.uwaterloo.ca/journals/JIS/VOL20/Leroy/leroy4.html
Erdős, P.: Some unsolved problems. Michigan Math. J. 4, 291–300 (1958). https://doi.org/10.1307/mmj/1028997963
Fici, G., Puzynina, S.: Abelian combinatorics on words: a survey. Comput. Sci. Rev. 47, 100532 (2023). https://doi.org/10.1016/j.cosrev.2022.100532
Goč, D., Rampersad, N., Rigo, M., Salimov, P.: On the number of abelian bordered words (with an example of automatic theorem-proving). Internat. J. Found. Comput. Sci. 25(8), 1097–1110 (2014). https://doi.org/10.1142/S0129054114400267
Guo, Y.J., Lü, X.T., Wen, Z.X.: On the boundary sequence of an automatic sequence. Discrete Math. 345(1) (2022). https://doi.org/10.1016/j.disc.2021.112632
Krawczyk, E., Müllner, C.: Automaticity of uniformly recurrent substitutive sequences (2023). https://doi.org/10.48550/arXiv.2111.13134
Mossé, B.: Reconnaissabilité des substitutions et complexité des suites automatiques. Bulletin de la Société Mathématique de France 124(2), 329–346 (1996). https://doi.org/10.24033/bsmf.2283
Mossé, B.: Puissance de mots et reconnaissabilité des points fixes d’une substitution. Theor. Comput. Sci. 99(2), 327–334 (1992). https://doi.org/10.1016/0304-3975(92)90357-L
Mousavi, H.: Automatic theorem proving in Walnut (2016). https://doi.org/10.48550/ARXIV.1603.06017
Parreau, A., Rigo, M., Rowland, E., Vandomme, É.: A new approach to the 2-regularity of the \(\ell \)-abelian complexity of 2-automatic sequences. Electron. J. Comb. 22(#P1.27) (2022). https://doi.org/10.37236/4478
Puzynina, S., Whiteland, M.A.: Abelian closures of infinite binary words. J. Comb. Theory Ser. A 185, 105524 (2022). https://doi.org/10.1016/j.jcta.2021.105524
Rigo, M.: Formal languages, automata and numeration systems. Vol. 1. ISTE, London; John Wiley & Sons Inc, Hoboken, NJ (2014), introduction to combinatorics on words, With a foreword by Valérie Bethé. https://doi.org/10.1002/9781119008200
Rigo, M.: Formal languages, automata and numeration systems. Vol. 2. Networks and Telecommunications Series, ISTE, London; John Wiley & Sons Inc, Hoboken, NJ (2014), applications to recognizability and decidability, With a foreword by Valérie Berthé. https://doi.org/10.1002/9781119042853
Rigo, M., Salimov, P.: Another generalization of abelian equivalence: binomial complexity of infinite words. Theor. Comput. Sci. 601, 47–57 (2015). https://doi.org/10.1016/j.tcs.2015.07.025
Rigo, M., Stipulanti, M., Whiteland, M.A.: Binomial complexities and Parikh-collinear morphisms. In: Diekert, V., Volkov, M.V. (eds.) Developments in Language Theory - 26th International Conference, DLT 2022, Tampa, FL, USA, May 9–13, 2022, Proceedings. Lecture Notes in Computer Science, vol. 13257, pp. 251–262. Springer (2022). https://doi.org/10.1007/978-3-031-05578-2_20
Schaeffer, L.: Deciding Properties of Automatic Sequences. Master’s thesis, Univ. of Waterloo (2013). https://uwspace.uwaterloo.ca/handle/10012/7899
Shallit, J.: Abelian complexity and synchronization. INTEGERS: Electron. J. Comb. Number Theory (#A.36) (2021). http://math.colgate.edu/integers/v36/v36.pdf
Shallit, J.: The Logical Approach to Automatic Sequences: Exploring Combinatorics on Words with Walnut. London Mathematical Society Lecture Note Series, Cambridge University Press (2022). https://doi.org/10.1017/9781108775267
Acknowledgments
M. Stipulanti and M. Whiteland are supported by the FNRS Research grants 1.B.397.20F and 1.B.466.21F, respectively. We thank Julien Leroy for fruitful discussions on morphic words and pointing out useful references. We thank A. Sportiello and V. Salo for asking the question leading to this paper. We also thank the reviewers for their suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Rigo, M., Stipulanti, M., Whiteland, M.A. (2023). Automaticity and Parikh-Collinear Morphisms. In: Frid, A., Mercaş, R. (eds) Combinatorics on Words. WORDS 2023. Lecture Notes in Computer Science, vol 13899. Springer, Cham. https://doi.org/10.1007/978-3-031-33180-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-031-33180-0_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-33179-4
Online ISBN: 978-3-031-33180-0
eBook Packages: Computer ScienceComputer Science (R0)