Descriptional Complexity of Power and Positive Closure on Convex Languages

  • Michal HospodárEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11601)


We study the descriptional complexity of the k-th power and positive closure operations on the classes of prefix-, suffix-, factor-, and subword-free, -closed, and -convex regular languages, and on the classes of right, left, two-sided, and all-sided ideal languages. We show that the upper bound kn on the nondeterministic complexity of the k-th power in the class of regular languages is tight for closed and convex classes, while in the remaining classes, the tight upper bound is \(k(n-1)+1\). Next we show that the upper bound n on the nondeterministic complexity of the positive closure operation in the class of regular languages is tight in all considered classes except for classes of factor-closed and subword-closed languages, where the complexity is one. All our worst-case examples are described over a unary or binary alphabet, except for witnesses for the k-th power on subword-closed and subword-convex languages which are described over a ternary alphabet. Moreover, whenever a binary alphabet is used for describing a worst-case example, it is optimal in the sense that the corresponding upper bounds cannot be met by a language over a unary alphabet. The most interesting result is the description of a binary factor-closed language meeting the upper bound kn for the k-th power. To get this result, we use a method which enables us to avoid tedious descriptions of fooling sets. We also provide some results concerning the deterministic state complexity of these two operations on the classes of free, ideal, and closed languages.


  1. 1.
    Birget, J.: Intersection and union of regular languages and state complexity. Inf. Process. Lett. 43(4), 185–190 (1992). Scholar
  2. 2.
    Brzozowski, J., Jirásková, G., Li, B., Smith, J.: Quotient complexity of bifix-, factor-, and subword-free regular languages. Acta Cybernet. 21(4), 507–527 (2014). Scholar
  3. 3.
    Brzozowski, J.A., Jirásková, G., Li, B.: Quotient complexity of ideal languages. Theor. Comput. Sci. 470, 36–52 (2013). Scholar
  4. 4.
    Brzozowski, J.A., Jirásková, G., Zou, C.: Quotient complexity of closed languages. Theory Comput. Syst. 54(2), 277–292 (2014). Scholar
  5. 5.
    Brzozowski, J.A., Sinnamon, C.: Complexity of left-ideal, suffix-closed and suffix-free regular languages. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 171–182. Springer, Cham (2017). Scholar
  6. 6.
    Čevorová, K.: Square on ideal, closed and free languages. In: Shallit and Okhotin [27], pp. 70–80. Scholar
  7. 7.
    Čevorová, K.: Square on closed languages. In: Bordihn, H., Freund, R., Nagy, B., Vaszil, G. (eds.) NCMA 2016., vol. 321, pp. 121–130. Österreichische Computer Gesellschaft (2016)Google Scholar
  8. 8.
    Čevorová, K., Jirásková, G., Mlynárčik, P., Palmovský, M., Šebej, J.: Operations on automata with all states final. In: Ésik, Z., Fülöp, Z. (eds.) Proceedings of 14th International Conference on Automata and Formal Languages, AFL 2014. EPTCS, vol. 151, pp. 201–215 (2014). Scholar
  9. 9.
    Cmorik, R., Jirásková, G.: Basic operations on binary suffix-free languages. In: Kotásek, Z., Bouda, J., Černá, I., Sekanina, L., Vojnar, T., Antoš, D. (eds.) MEMICS 2011. LNCS, vol. 7119, pp. 94–102. Springer, Heidelberg (2012). Scholar
  10. 10.
    Domaratzki, M., Okhotin, A.: State complexity of power. Theor. Comput. Sci. 410(24–25), 2377–2392 (2009). Scholar
  11. 11.
    Han, Y., Salomaa, K.: Nondeterministic state complexity for suffix-free regular languages. In: McQuillan and Pighizzini [24], pp. 189–196. Scholar
  12. 12.
    Han, Y., Salomaa, K., Wood, D.: Nondeterministic state complexity of basic operations for prefix-free regular languages. Fund. Inform. 90(1–2), 93–106 (2009). Scholar
  13. 13.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. Int. J. Found. Comput. Sci. 14(6), 1087–1102 (2003). Scholar
  14. 14.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  15. 15.
    Hospodár, M., Jirásková, G., Mlynárčik, P.: Nondeterministic complexity of operations on closed and ideal languages. In: Han, Y.-S., Salomaa, K. (eds.) CIAA 2016. LNCS, vol. 9705, pp. 125–137. Springer, Cham (2016). Scholar
  16. 16.
    Hospodár, M., Jirásková, G., Mlynárčik, P.: Nondeterministic complexity of operations on free and convex languages. In: Carayol, A., Nicaud, C. (eds.) CIAA 2017. LNCS, vol. 10329, pp. 138–150. Springer, Cham (2017). Scholar
  17. 17.
    Hospodár, M., Jirásková, G., Mlynárčik, P.: A survey on fooling sets as effective tools for lower bounds on nondeterministic complexity. In: Böckenhauer, H.-J., Komm, D., Unger, W. (eds.) Adventures Between Lower Bounds and Higher Altitudes. LNCS, vol. 11011, pp. 17–32. Springer, Cham (2018). Scholar
  18. 18.
    Hospodár, M., Jirásková, G., Mlynárčik, P.: Nondeterministic complexity in subclasses of convex languages. Theor. Comput. Sci. (2019).
  19. 19.
    Jirásková, G.: State complexity of some operations on binary regular languages. Theor. Comput. Sci. 330(2), 287–298 (2005). Scholar
  20. 20.
    Jirásková, G., Krausová, M.: Complexity in prefix-free regular languages. In: McQuillan and Pighizzini [27], pp. 197–204. Scholar
  21. 21.
    Jirásková, G., Mlynárčik, P.: Complement on prefix-free, suffix-free, and non-returning NFA languages. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 222–233. Springer, Cham (2014). Scholar
  22. 22.
    Jirásková, G., Olejár, P.: State complexity of intersection and union of suffix-free languages and descriptional complexity. In: Bordihn, H., Freund, R., Holzer, M., Kutrib, M., Otto, F. (eds.) NCMA 2009., vol. 256, pp. 151–166. Österreichische Computer Gesellschaft (2009)Google Scholar
  23. 23.
    Jirásková, G., Palmovský, M., Šebej, J.: Kleene closure on regular and prefix-free languages. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 226–237. Springer, Cham (2014). Scholar
  24. 24.
    McQuillan, I., Pighizzini, G. (eds.): DCFS 2010, EPTCS, vol. 31 (2010).
  25. 25.
    Mlynárčik, P.: Complement on free and ideal languages. In: Shallit and Okhotin [27], pp. 185–196. Scholar
  26. 26.
    Rampersad, N.: The state complexity of \({L}^2\) and \({L}^k\). Inf. Process. Lett. 98(6), 231–234 (2006). Scholar
  27. 27.
    Shallit, J., Okhotin, A. (eds.): DCFS 2015. LNCS, vol. 9118. Springer, Cham (2015). Scholar
  28. 28.
    Sinnamon, C.: Complexity of proper suffix-convex regular languages. In: Câmpeanu, C. (ed.) CIAA 2018. LNCS, vol. 10977, pp. 324–338. Springer, Cham (2018). Scholar
  29. 29.
    Sipser, M.: Introduction to the Theory of Computation. Cengage Learning, Boston (2012)zbMATHGoogle Scholar
  30. 30.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 41–110. Springer, Heidelberg (1997). Scholar
  31. 31.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994). Scholar

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Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia

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