Deterministic Pushdown Automata and Unary Languages

  • Giovanni Pighizzini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5148)


The simulation of deterministic pushdown automata defined over a one letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states which is exponential in s. We prove that this simulation is tight. Furthermore, its cost cannot be reduced even if it is performed by a two-way nondeterministic automaton. We also prove that there are unary languages for which deterministic pushdown automata cannot be exponentially more succinct than finite automata. In order to state this result, we investigate the conversion of deterministic pushdown automata into context-free grammars. We prove that in the unary case the number of variables in the resulting grammar is strictly lower than the number of variables needed in the case of nonunary alphabets.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Althöfer, I.: Tight lower bounds for the length of word chains. Information Processing Letters 34, 275–276 (1990)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berstel, J., Carton, O.: On the complexity of Hopcroft’s State Minimizazion Algorithm. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 35–44. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    de Bruijn, N.: A combinatorial problem. Proc. Kon. Nederl. Akad. Wetensch 49, 758–764 (1946)Google Scholar
  4. 4.
    Domaratzki, M., Pighizzini, G., Shallit, J.: Simulating finite automata with context-free grammars. Information Processing Letters 84, 339–344 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ginsburg, S., Greibach, S.: Deterministic context-free languages. Information and Control 9, 563–582 (1966)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ginsburg, S., Rice, H.: Two families of languages related to ALGOL. Journal of the ACM 9, 350–371 (1962)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Goldstine, J., Price, J., Wotschke, D.: A pushdown automaton or a context-free grammar – Which is more economical? Theoretical Computer Science 18, 33–40 (1982)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Reading (1978)MATHGoogle Scholar
  9. 9.
    Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  10. 10.
    Knuth, D.: On the translation of languages from left to right. Information and Control 8, 607–639 (1965)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Mereghetti, C., Pighizzini, G.: Two-way automata simulations and unary languages. J. Aut.Lang.Combin. 5, 287–300 (2000)MATHMathSciNetGoogle Scholar
  12. 12.
    Mereghetti, C., Pighizzini, G.: Optimal simulations between unary automata. SIAM J. Comput. 30, 1976–1992 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Meyer, A., Fischer, M.: Economy of description by automata, grammars, and formal systems. In: Proc.12th Ann. IEEE Symp.on Switching and Automata Theory, pp. 188–191 (1971)Google Scholar
  14. 14.
    Pighizzini, G., Shallit, J., Wang, M.-W.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. Journal of Computer and System Sciences 65, 393–414 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sénizergues, G.: The equivalence problem for deterministic pushdown automata is decidable. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 671–682. Springer, Heidelberg (1997)Google Scholar
  16. 16.
    Stearns, R.: A regularity test for pushdown machines. Information and Control 11, 323–340 (1967)MATHCrossRefGoogle Scholar
  17. 17.
    Valiant, L.: Regularity and related problems for deterministic pushdown automata. Journal of the ACM 22, 1–10 (1975)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Giovanni Pighizzini
    • 1
  1. 1.Dipartimento di Informatica e ComunicazioneUniversità degli Studi di MilanoMilanoItaly

Personalised recommendations