Abstract
We show that the subword complexity function ρ x (n), which counts the number of distinct factors of length n of a sequence x, is k-synchronized in the sense of Carpi if x is k-automatic. As an application, we generalize recent results of Goldstein. We give analogous results for the number of distinct factors of length n that are primitive words or powers. In contrast, we show that the function that counts the number of unbordered factors of length n is not necessarily k-synchronized for k-automatic sequences.
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References
Allouche, J.-P., Shallit, J.: The ring of k-regular sequences. Theoret. Comput. Sci. 98, 163–197 (1992)
Allouche, J.-P., Shallit, J.: The ring of k-regular sequences, II. Theoret. Comput. Sci. 307, 3–29 (2003)
Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalization. Cambridge University Press (2003)
Carpi, A., D’Alonzo, V.: On the repetitivity index of infinite words. Internat. J. Algebra Comput. 19, 145–158 (2009)
Carpi, A., D’Alonzo, V.: On factors of synchronized sequences. Theoret. Comput. Sci. 411, 3932–3937 (2010)
Carpi, A., Maggi, C.: On synchronized sequences and their separators. RAIRO Inform. Théor. App. 35, 513–524 (2001)
Cassaigne, J.: Special factors of sequences with linear subword complexity. In: Dassow, J., Rozenberg, G., Salomaa, A. (eds.) Developments in Language Theory II, pp. 25–34. World Scientific (1996)
Charlier, É., Rampersad, N., Shallit, J.: Enumeration and decidable properties of automatic sequences. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 165–179. Springer, Heidelberg (2011)
Cobham, A.: Uniform tag sequences. Math. Systems Theory 6, 164–192 (1972)
Damanik, D.: Local symmetries in the period-doubling sequence. Disc. Appl. Math. 100, 115–121 (2000)
Dekking, F.M., Mendès France, M., van der Poorten, A.J.: Folds! Math. Intelligencer 4, 130–138, 173–181, 190–195 (1982); Erratum 5, 5 (1983)
Fine, N.J., Wilf, H.S.: Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16, 109–114 (1965)
Goč, D., Mousavi, H., Shallit, J.: On the number of unbordered factors. In: Dediu, A.-H., Martín-Vide, C., Truthe, B. (eds.) LATA 2013. LNCS, vol. 7810, pp. 299–310. Springer, Heidelberg (2013)
Goldstein, I.: Asymptotic subword complexity of fixed points of group substitutions. Theoret. Comput. Sci. 410, 2084–2098 (2009)
Goldstein, I.: Subword complexity of uniform D0L words over finite groups. Theoret. Comput. Sci. 412, 5728–5743 (2011)
Lyndon, R.C., Schützenberger, M.P.: The equation a M = b N c P in a free group. Michigan Math. J. 9, 289–298 (1962)
Schaeffer, L., Shallit, J.: The critical exponent is computable for automatic sequences. Int. J. Found. Comput. Sci. 23, 1611–1626 (2012)
Shallit, J.: The critical exponent is computable for automatic sequences. In: Ambroz, P., Holub, S., Másaková, Z. (eds.) Proceedings 8th International Conference Words 2011. Elect. Proc. Theor. Comput. Sci., vol. 63, pp. 231–239 (2011)
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Goč, D., Schaeffer, L., Shallit, J. (2013). Subword Complexity and k-Synchronization. In: Béal, MP., Carton, O. (eds) Developments in Language Theory. DLT 2013. Lecture Notes in Computer Science, vol 7907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38771-5_23
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DOI: https://doi.org/10.1007/978-3-642-38771-5_23
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