Some Applications of Algebra to Automatic Sequences

  • Jason Bell
Part of the Trends in Mathematics book series (TM)


We give an overview of the theory of rings satisfying a polynomial identity and use this to give a proof of a characterization due to Berstel and Reutenauer of automatic and regular sequences in terms of two properties, which we call the shuffle property and the power property. These properties show that if one views an automatic sequence f as a map on a free monoid on k-letters to a finite subset of a ring, then the values of f are closely related to values of f on related words obtained by permuting letters of the word. We use this characterization to give answers to three questions from Allouche and Shallit, two of which have not appeared in the literature. The final part of the chapter deals more closely with the shuffle property, and we view this as giving a generalization of regular sequences. We show that sequences with the shuffle property are closed under the process of taking sums and taking products; in addition we show that there is closure under a noncommutative product, which turns the collection of shuffled sequences into a noncommutative algebra. We show that this algebra is very large, in the sense that it contains a copy of a free associative algebra on countably many generators. We conclude by giving some open questions, which we hope will begin a more careful study of shuffled sequences.



I thank Jean-Paul Allouche and Jeffrey Shallit for many helpful comments. I also thank Jean-Paul Allouche for raising Question 4.9.4.


  1. 13.
    Allouche, J.-P., Scheicher, K., Tichy, R.F.: Regular maps in generalized number systems. Math. Slovaca 50, 41–58 (2000)MathSciNetzbMATHGoogle Scholar
  2. 14.
    Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  3. 16.
    Allouche, J.-P., Shallit, J.O.: The ring of k-regular sequences. In: Choffrut, C., Lengauer, T. (eds.) STACS 90, Proceedings of the 7th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 415, pp. 12–23. Springer, Berlin (1990)Google Scholar
  4. 20.
    Amitsur, A.S., Levitzki, J.: Minimal identities for algebras. Proc. Am. Math. Soc. 1, 449–463 (1950)MathSciNetCrossRefGoogle Scholar
  5. 21.
    Amitsur, S.A., Small, L.W.: Affine algebras with polynomial identities. Rend. Circ. Mat. Palermo (2) Suppl. 31, 9–43 (1993). Recent developments in the theory of algebras with polynomial identities (Palermo, 1992)Google Scholar
  6. 77.
    Berstel, J., Reutenauer, C.: Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications, vol. 137. Cambridge University Press, Cambridge (2011)Google Scholar
  7. 105.
    Braun, A.: The nilpotency of the radical in a finitely generated PI ring. J. Algebra 89(2), 375–396 (1984)MathSciNetCrossRefGoogle Scholar
  8. 162.
    Conway, J.H.: On Numbers and Games. Academic Press, New York (1976)zbMATHGoogle Scholar
  9. 189.
    Dekking, F.M., Mendès France, M., Poorten, A.J.v.d.: Folds! Math. Intelligencer 4, 130–138, 173–181, 190–195 (1982). Erratum, 5 (1983), 5Google Scholar
  10. 282.
    Hansel, G.: A simple proof of the Skolem-Mahler-Lech theorem. In: Brauer, W. (ed.) Proceedings of the 12th International Conference on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 194, pp. 244–249. Springer, Berlin (1985)Google Scholar
  11. 406.
    Mahler, K.: An unsolved problem on the powers of 3/2. J. Aust. Math. Soc. 8, 313–321 (1968)MathSciNetCrossRefGoogle Scholar
  12. 425.
    Moshe, Y.: On some questions regarding k-regular and k-context-free sequences. Theor. Comput. Sci. 400(1-3), 62–69 (2008)MathSciNetCrossRefGoogle Scholar
  13. 484.
    Pirillo, G.: A proof of Shirshov’s theorem. Adv. Math. 124(1), 94–99 (1996)MathSciNetCrossRefGoogle Scholar
  14. 498.
    Regev, A.: Existence of identities in A ⊗ B. Isr. J. Math. 11, 131–152 (1972)Google Scholar
  15. 512.
    Rosset, S.: A new proof of the Amitsur-Levitski identity. Isr. J. Math. 23(2), 187–188 (1976)MathSciNetCrossRefGoogle Scholar
  16. 542.
    Shallit, J.O.: A generalization of automatic sequences. Theor. Comput. Sci. 61, 1–16 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations