Square, Power, Positive Closure, and Complementation on Star-Free Languages

  • Sylvie Davies
  • Michal HospodárEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)


We examine the deterministic and nondeterministic state complexity of square, power, positive closure, and complementation on star-free languages. For the state complexity of square, we get a non-trivial upper bound \((n-1)2^n - 2(n-2)\) and a lower bound of order \({\Theta }(2^n)\). For the state complexity of the k-th power in the unary case, we get the tight upper bound \(k(n-1)+1\). Next, we show that the upper bound kn on the nondeterministic state complexity of the k-th power is met by a binary star-free language, while in the unary case, we have a lower bound \(k(n-1)+1\). For the positive closure, we show that the deterministic upper bound \(2^{n-1}+2^{n-2}-1\), as well as the nondeterministic upper bound n, can be met by star-free languages. We also show that in the unary case, the state complexity of positive closure is \(n^2-7n+13\), and the nondeterministic state complexity of complementation is between \((n-1)^2+1\) and \(n^2-2\).


  1. 1.
    Brzozowski, J.A., Liu, B.: Quotient complexity of star-free languages. Internat. J. Found. Comput. Sci. 23(6), 1261–1276 (2012). Scholar
  2. 2.
    Brzozowski, J.A., Szykula, M.: Large aperiodic semigroups. Internat. J. Found. Comput. Sci. 26(7), 913–932 (2015). Scholar
  3. 3.
    Câmpeanu, C., Culik, K., Salomaa, K., Yu, S.: State complexity of basic operations on finite languages. In: Boldt, O., Jürgensen, H. (eds.) WIA 1999. LNCS, vol. 2214, pp. 60–70. Springer, Heidelberg (2001). Scholar
  4. 4.
    Domaratzki, M., Okhotin, A.: State complexity of power. Theoret. Comput. Sci. 410(24–25), 2377–2392 (2009). Scholar
  5. 5.
    Geffert, V.: Magic numbers in the state hierarchy of finite automata. Inform. Comput. 205(11), 1652–1670 (2007). Scholar
  6. 6.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. Internat. J. Found. Comput. Sci. 14(6), 1087–1102 (2003). Scholar
  7. 7.
    Holzer, M., Kutrib, M., Meckel, K.: Nondeterministic state complexity of star-free languages. Theoret. Comput. Sci. 450, 68–80 (2012). Scholar
  8. 8.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Boston (1979)zbMATHGoogle Scholar
  9. 9.
    Howie, J.M.: Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17(3), 223–236 (1971). Scholar
  10. 10.
    Jirásková, G.: State complexity of some operations on binary regular languages. Theoret. Comput. Sci. 330(2), 287–298 (2005). Scholar
  11. 11.
    Leiss, E.L.: Succint representation of regular languages by boolean automata. Theoret. Comput. Sci. 13, 323–330 (1981). Scholar
  12. 12.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Doklady 11, 1373–1375 (1970)zbMATHGoogle Scholar
  13. 13.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Internat. J. Found. Comput. Sci. 13(1), 145–159 (2002). Scholar
  14. 14.
    Rampersad, N.: The state complexity of \({L}^{2}\) and \({L}^{k}\). Inform. Process. Lett. 98(6), 231–234 (2006). Scholar
  15. 15.
    Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. Control 8(2), 190–194 (1965). Scholar
  16. 16.
    Sipser, M.: Introduction to the Theory of Computation. Cengage Learning (2012)Google Scholar
  17. 17.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 41–110. Springer, Heidelberg (1997). Scholar
  18. 18.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theoret. Comput. Sci. 125(2), 315–328 (1994). Scholar

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Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia

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