Abstract
We contribute to the study of square-free words. The classical notion of a square-free word has a natural generalization to partial words, studied in several papers since 2008. We prove that the maximal density of wildcards in the ternary infinite square-free partial word is surprisingly big: 3/16. In addition, we introduce a related characteristic of infinite square-free words, called flexibility, and find its values for some interesting words and classes of words.
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As mentioned above, some sets do not have density; to avoid additional notions we postulate that upper (lower) bounds on flexibility should work for upper (resp., lower) densities of the corresponding wildcard sets.
References
Arshon, S.E.: Proof of the existence of asymmetric infinite sequences. Mat. Sbornik 2, 769–779 (1937). in Russian, with French abstract
Berstel, J., Boasson, L.: Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218, 135–141 (1999)
Blanchet-Sadri, F., Black, K., Zemke, A.: Unary pattern avoidance in partial words dense with holes. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 155–166. Springer, Heidelberg (2011)
Blanchet-Sadri, F., Hegstrom, R.A.: Partial words and a theorem of Fine and Wilf revisited. Theor. Comput. Sci. 270(1–2), 401–419 (2002)
Blanchet-Sadri, F., Mercaş, R., Scott, G.: A generalization of Thue freeness for partial words. Theoret. Comput. Sci. 410, 793–800 (2009)
Dejean, F.: Sur un théorème de Thue. J. Combin. Theory. Ser. A 13, 90–99 (1972)
Fischer, M., Paterson, M.: String matching and other products. SIAM-AMS Proc. 7, 113–125 (1974)
Halava, V., Harju, T., Kärki, T.: Square-free partial words. Inform. Process. Lett. 108(5), 290–292 (2008)
Manea, F., Mercaş, R.: Freeness of partial words. Theoret. Comput. Sci. 389(1–2), 265–277 (2007)
Muthukrishnan, S., Ramesh, H.: String matching under a general matching relation. In: Shyamasundar, R.K. (ed.) FSTTCS 1992. LNCS, vol. 652, pp. 356–367. Springer, Heidelberg (1992)
Petrova, E.A., Shur, A.M.: Constructing premaximal ternary square-free words of any level. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 752–763. Springer, Heidelberg (2012)
Petrova, E.A., Shur, A.M.: On the tree of ternary square-free words. In: Manea, F., Nowotka, D. (eds.) WORDS 2015. LNCS, vol. 9304, pp. 223–236. Springer, Heidelberg (2015)
Petrova, E.A.: Avoiding letter patterns in ternary square-free words. Electr. J. Comb. 23(1), P1.18 (2016)
Shur, A.M.: On ternary square-free circular words. Electronic J. Combinatorics 17, R140 (2010)
Shur, A.M., Konovalova, Y.V.: On the periods of partial words. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 657–665. Springer, Heidelberg (2001)
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Gasnikov, D., Shur, A.M. (2016). Ternary Square-Free Partial Words with Many Wildcards. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_15
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DOI: https://doi.org/10.1007/978-3-662-53132-7_15
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