Operations on Unambiguous Finite Automata

  • Jozef JirásekJr.
  • Galina Jirásková
  • Juraj Šebej
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9840)


A nondeterministic finite automaton is unambiguous if it has at most one accepting computation on every input string. We investigate the complexity of basic regular operations on languages represented by unambiguous finite automata. We get tight upper bounds for intersection (mn), left and right quotients (\(2^n-1\)), positive closure (\({3\over 4}\cdot 2^n-1\)), star (\({3\over 4}\cdot 2^n\)), shuffle (\(2^{mn}-1\)), and concatenation (\({3\over 4}\cdot 2^{m+n}-1\)). To prove tightness, we use a binary alphabet for intersection and left and right quotients, a ternary alphabet for star and positive closure, a five-letter alphabet for shuffle, and a seven-letter alphabet for concatenation. We also get some partial results for union and complementation.


State Complexity Regular Language Finite Automaton Input String Tree Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Allauzen, C., Mohri, M., Rastogi, A.: General algorithms for testing the ambiguity of finite automata and the double-tape ambiguity of finite-state transducers. Int. J. Found. Comput. Sci. 22(4), 883–904 (2011). MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Birget, J.: Partial orders on words, minimal elements of regular languages and state complexity. Theor. Comput. Sci. 119(2), 267–291 (1993). MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetMATHGoogle Scholar
  4. 4.
    Čevorová, K.: Kleene star on unary regular languages. In: Jurgensen, H., Reis, R. (eds.) DCFS 2013. LNCS, vol. 8031, pp. 277–288. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  5. 5.
    Chan, T., Ibarra, O.H.: On the finite-valuedness problem for sequential machines. Theor. Comput. Sci. 23, 95–101 (1983). MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Colcombet, T.: Unambiguity in automata theory. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 3–18. Springer, Heidelberg (2015). CrossRefGoogle Scholar
  7. 7.
    Eliáš, P.: Fooling sets for complements of UFAs. Unpublished manuscript (2016)Google Scholar
  8. 8.
    Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inf. Process. Lett. 59(2), 75–77 (1996). MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Holzer, M., Kutrib, M.: Nondeterministic descriptional complexity of regular languages. Int. J. Found. Comput. Sci. 14(6), 1087–1102 (2003). MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hromkovič, J., Schnitger, G.: Ambiguity and communication. Theory Comput. Syst. 48(3), 517–534 (2011). MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hromkovič, J., Seibert, S., Karhumäki, J., Klauck, H., Schnitger, G.: Communication complexity method for measuring nondeterminism in finite automata. Inf. Comput. 172(2), 202–217 (2002). MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ibarra, O.H., Ravikumar, B.: On sparseness, ambiguity and other decision problems for acceptors and transducers. In: Monien, B., Vidal-Naquet, G. (eds.) STACS 1986. LNCS, vol. 210, pp. 171–179. Springer, Heidelberg (1986). Google Scholar
  13. 13.
    Jirásková, G.: State complexity of some operations on binary regular languages. Theor. Comput. Sci. 330(2), 287–298 (2005). MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Leiss, E.L.: Succint representation of regular languages by boolean automata. Theor. Comput. Sci. 13, 323–330 (1981). MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Leung, H.: Separating exponentially ambiguous finite automata from polynomially ambiguous finite automata. SIAM J. Comput. 27(4), 1073–1082 (1998). MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Leung, H.: Descriptional complexity of NFA of different ambiguity. Int. J. Found. Comput. Sci. 16(5), 975–984 (2005). MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lupanov, O.B.: A comparison of two types of finite automata. Problemy Kibernetiki 9, Kibernetiki (1963), (in Russian) German translation: Über den Vergleich zweier Typen endlicher Quellen. Probleme der. Kybernetik 6, 328–335 (1966)Google Scholar
  18. 18.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Doklady 11, 1373–375 (1970)MATHGoogle Scholar
  19. 19.
    Okhotin, A.: Unambiguous finite automata over a unary alphabet. Inf. Comput. 212, 15–36 (2012). MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ravikumar, B., Ibarra, O.H.: Relating the type of ambiguity of finite automata to the succinctness of their representation. SIAM J. Comput. 18(6), 1263–1282 (1989). MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Schmidt, E.M.: Succinctness of description of context-free, regular, and finite languages. Ph. D. thesis. Cornell University (1978)Google Scholar
  22. 22.
    Sipser, M.: Introduction to the Theory of Computation. PWS Publishing Company, Boston (1997)MATHGoogle Scholar
  23. 23.
    Stearns, R.E., Hunt, H.B.: On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata. SIAM J. Comput. 14(3), 598–611 (1985). MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Weber, A., Seidl, H.: On the degree of ambiguity of finite automata. Theor. Comput. Sci. 88(2), 325–349 (1991). MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994). MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jozef JirásekJr.
    • 1
    • 2
  • Galina Jirásková
    • 1
  • Juraj Šebej
    • 2
  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia
  2. 2.Faculty of Science, Institute of Computer ScienceP.J. Šafárik UniversityKošiceSlovakia

Personalised recommendations