NFA-to-DFA Trade-Off for Regular Operations

  • Galina Jirásková
  • Ivana KrajňákováEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)


We examine the operational state complexity assuming that the operands of a regular operation are represented by nondeterministic finite automata, while the language resulting from the operation is required to be represented by a deterministic finite automaton. We get tight upper bounds \(2^n\) for complementation, reversal, and star, \(2^m\) for left and right quotient, \(2^{m+n}\) for union and symmetric difference, \(2^{m+n}-2^m-2^n+2\) for intersection, \(2^{m+n}-2^n+1\) for difference, \(\frac{3}{4}2^{m+n}\) for concatenation, and \(2^{mn}\) for shuffle. We use a binary alphabet to describe witnesses for complementation, reversal, star, and left and right quotient, and a quaternary alphabet otherwise. Whenever we use a binary alphabet, it is always optimal.



We would like to kindly thank Michal Hospodár for his valuable notes and comments.


  1. 1.
    Birget, J.: Intersection and union of regular languages and state complexity. Inf. Process. Lett. 43(4), 185–190 (1992). Scholar
  2. 2.
    Birget, J.: The state complexity of \(\overline{\Sigma ^*\overline{L}}\) and its connection with temporal logic. Inf. Process. Lett. 58(4), 185–188 (1996). Scholar
  3. 3.
    Brzozowski, J.A.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010). Scholar
  4. 4.
    Câmpeanu, C., Salomaa, K., Yu, S.: Tight lower bound for the state complexity of shuffle of regular languages. J. Autom. Lang. Comb. 7(3), 303–310 (2002). Scholar
  5. 5.
    Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47(3), 149–158 (1986). Scholar
  6. 6.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. Int. J. Found. Comput. Sci. 14(6), 1087–1102 (2003). Scholar
  7. 7.
    Hospodár, M., Jirásková, G., Mlynárčik, P.: Descriptional complexity of the forever operator. Int. J. Found. Comput. Sci. 30(1), 115–134 (2019). Scholar
  8. 8.
    Jirásek, J.Š., Jirásková, G., Szabari, A.: Operations on self-verifying finite automata. In: Beklemishev, L.D., Musatov, D.V. (eds.) CSR 2015. LNCS, vol. 9139, pp. 231–261. Springer, Cham (2015). Scholar
  9. 9.
    Jirásek Jr., J., Jirásková, G., Šebej, J.: Operations on unambiguous finite automata. Int. J. Found. Comput. Sci. 29(5), 861–876 (2018). Scholar
  10. 10.
    Jirásková, G.: State complexity of some operations on binary regular languages. Theor. Comput. Sci. 330(2), 287–298 (2005). Scholar
  11. 11.
    Maslov, A.N.: Estimates of the number of states of finite automata. Sov. Math. Dokl. 11(5), 1373–1375 (1970)zbMATHGoogle Scholar
  12. 12.
    Salomaa, A., Salomaa, K., Yu, S.: State complexity of combined operations. Theor. Comput. Sci. 383(2–3), 140–152 (2007). Scholar
  13. 13.
    Šebej, J.: Reversal of regular languages and state complexity. In: Pardubská, D. (ed.) Proceedings of the Conference on Theory and Practice of Information Technologies, ITAT 2010. CEUR Workshop Proceedings, vol. 683, pp. 47–54. (2010).
  14. 14.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 41–110. Springer, Heidelberg (1997). Scholar
  15. 15.
    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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© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.Mathematical Institute, Slovak Academy of SciencesKošiceSlovakia

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