Abstract
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 \(\mathcal{A}\).
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References
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)
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)
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)
Berthé, V., Holton, C., Zamboni, L.Q.: Initial powers of Sturmian sequences. Acta Arith. 122, 315–347 (2006)
Berthé, V., Labbé, S.: Factor complexity of S-adic words generated by the Arnoux-Rauzy-Poincaré algorithm. Adv. App. Math. 63, 90–130 (2015)
Berthé, V., Rigo, M. (eds.): Combinatorics, Automata and Number Theory, Encyclopedia of Mathematics and its Applications, vol. 135. Cambridge University Press, Cambridge (2010)
Ferenczi, S.: Rank and symbolic complexity. Ergod. Theor. Dyn. Syst. 16, 663–682 (1996)
Fici, G.: Special factors and the combinatorics of suffix and factor automata. Theor. Comput. Sci. 412, 3604–3615 (2011)
Leroy, J.: Contribution à la résolution de la conjecture \(S\)-adique. Doctoral thesis, Université de Picardie Jules Verne (2012)
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)
Leroy, J., Richomme, G.: A combinatorial proof of S-adicity for sequences with linear complexity. Integers 13, 19 (2013). Article #A5
Levé, F., Richomme, G.: Quasiperiodic Sturmian words and morphisms. Theor. Comput. Sci. 372(1), 15–25 (2007)
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)
Lothaire, M.: Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
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
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Many thanks to referees for their careful readings and their interesting suggestions and questions.
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Richomme, G. (2017). A Characterization of Infinite LSP Words. In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_24
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DOI: https://doi.org/10.1007/978-3-319-62809-7_24
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