Rainbow of Computer Science pp 90-101

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6570) | Cite as

Local Squares, Periodicity and Finite Automata

  • Mari Huova
  • Juhani Karhumäki
  • Aleksi Saarela
  • Kalle Saari

Abstract

We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on whether the square is that of words, that of abelian words or, as an in between case, that of so called k-abelian words. The above nine variants of the problem are completely solved, and some open problems are addressed in the k-abelian case. Finally, an amazing unavoidability result for 2-abelian squares is obtained.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adian, S.I., Novikov, P.S.: Infinite periodic groups I, II, III. Izv. Akad. Nauk SSSR. Ser. Mat. 32(1,2,3), 212–244, 251–524, 709–731 (1968)Google Scholar
  2. 2.
    Avgustinovich, S., Karhumäki, J., Puzynina, S.: On abelian versions of Critical Factorization Theorem. In: Proceedings of the 13th Mons. Theoretical Computer Science Days (2010)Google Scholar
  3. 3.
    Burnside, W.: On an unsettled question in the theory of discontinuous groups. Quart. J. Pure and Appl. Math. 33, 230–238 (1902)MATHGoogle Scholar
  4. 4.
    Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 329–438. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  5. 5.
    Dekking, F.M.: Strongly nonrepetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27(2), 181–185 (1979)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Harmaala, E.: Private communicationGoogle Scholar
  7. 7.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  8. 8.
    Karhumäki, J., Lepistö, A., Plandowski, W.: Locally periodic versus globally periodic infinite words. J. Combin. Theory Ser. A 100(2), 250–264 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Keränen, V.: Abelian squares are avoidable on 4 letters. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 41–52. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  10. 10.
    Lepistö, A.: On Relations between Local and Global Periodicity. PhD thesis, University of Turku (2002)Google Scholar
  11. 11.
    Lothaire, M.: Combinatorics on Words. Addison-Wesley, Reading (1983)MATHGoogle Scholar
  12. 12.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  13. 13.
    Mignosi, F., Restivo, A., Salemi, S.: Periodicity and the golden ratio. Theoret. Comput. Sci. 204(1-2), 153–167 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Morse, M., Hedlund, G.A.: Unending chess, symbolic dynamics and a problem in semigroups. Duke Math. J. 11, 1–7 (1944)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mari Huova
    • 1
  • Juhani Karhumäki
    • 1
  • Aleksi Saarela
    • 1
  • Kalle Saari
    • 1
  1. 1.Department of Mathematics and Turku Centre for Computer Science TUCSUniversity of TurkuTurkuFinland

Personalised recommendations