A Characterization of Infinite LSP Words

  • Gwenaël RichommeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)


G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of S-adicity. More precisely we provide a finite set of morphisms S and an automaton \(\mathcal{A}\) such that an infinite word is LSP if and only if it is S-adic and all its directive words are recognizable by \(\mathcal{A}\).


Generalizations of Sturmian words Morphisms S-adicity 



Many thanks to referees for their careful readings and their interesting suggestions and questions.


  1. 1.
    Berstel, J., Séébold, P.: Sturmian words. In: Lothaire, M. (ed.) Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90, pp. 45–110. Cambridge University Press, Cambridge (2002)Google Scholar
  2. 2.
    Berthé, V.: S-adic expansions related to continued fractions. In: Akiyama, S. (ed.) Natural Extension of Arithmetic Algorithms and S-adic System. RIMS Kôkyûroku Bessatsu, vol. B58, pp. 61–84 (2016)Google Scholar
  3. 3.
    Berthé, V., Delecroix, V.: Beyond substitutive dynamical systems: S-adic expansions. In: Akiyama, S. (ed.) Numeration and Substitution 2012. RIMS Kôkyûroku Bessatsu, vol. B46, pp. 81–123 (2014)Google Scholar
  4. 4.
    Berthé, V., Holton, C., Zamboni, L.Q.: Initial powers of Sturmian sequences. Acta Arith. 122, 315–347 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berthé, V., Labbé, S.: Factor complexity of S-adic words generated by the Arnoux-Rauzy-Poincaré algorithm. Adv. App. Math. 63, 90–130 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Berthé, V., Rigo, M. (eds.): Combinatorics, Automata and Number Theory, Encyclopedia of Mathematics and its Applications, vol. 135. Cambridge University Press, Cambridge (2010)Google Scholar
  7. 7.
    Ferenczi, S.: Rank and symbolic complexity. Ergod. Theor. Dyn. Syst. 16, 663–682 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fici, G.: Special factors and the combinatorics of suffix and factor automata. Theor. Comput. Sci. 412, 3604–3615 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Leroy, J.: Contribution à la résolution de la conjecture \(S\)-adique. Doctoral thesis, Université de Picardie Jules Verne (2012)Google Scholar
  10. 10.
    Leroy, J.: An \(S\)-adic characterization of minimal subshifts with first difference of complexity \(p(n+1)-p(n)\le 2\). Discrete Math. Theor. Comput. Sci. 16(1), 233–286 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Leroy, J., Richomme, G.: A combinatorial proof of S-adicity for sequences with linear complexity. Integers 13, 19 (2013). Article #A5MathSciNetzbMATHGoogle Scholar
  12. 12.
    Levé, F., Richomme, G.: Quasiperiodic Sturmian words and morphisms. Theor. Comput. Sci. 372(1), 15–25 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lothaire, M.: Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 17. Addison-Wesley, Reading (1983). Reprinted in the Cambridge Mathematical Library. Cambridge University Press, UK (1997)zbMATHGoogle Scholar
  14. 14.
    Lothaire, M.: Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sciortino, M., Zamboni, L.Q.: Suffix automata and standard Sturmian words. In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 382–398. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73208-2_36 CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Univ. Paul-Valéry Montpellier 3, UFR 6, Dpt MIAp, Case J11Montpellier Cedex 5France
  2. 2.LIRMM (CNRS, Univ. Montpellier), UMR 5506 - CC 477Montpellier Cedex 5France

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