Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent Deterministic Automata

  • Giovanna J. Lavado
  • Giovanni Pighizzini
  • Shinnosuke Seki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)

Abstract

We investigate the conversion of nondeterministic finite automata and context-free grammars into Parikh equivalent deterministic finite automata, from a descriptional complexity point of view.

We prove that for each nondeterministic automaton with n states there exists a Parikh equivalent deterministic automaton with \(e^{O(\sqrt{n \cdot \ln n})}\) states. Furthermore, this cost is tight. In contrast, if all the strings accepted by the given automaton contain at least two different letters, then a Parikh equivalent deterministic automaton with a polynomial number of states can be found.

Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with n variables there exists a Parikh equivalent deterministic automaton with \(2^{O(n^2)}\) states. Even this bound is tight.

Keywords

Finite automaton context-free grammar Parikh’s theorem descriptional complexity semilinear set Parikh equivalence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chrobak, M.: Finite automata and unary languages. Theoretical Computer Science 47, 149–158 (1986); Corrigendum, ibid. 302, 497–498 (2003) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Esparza, J., Ganty, P., Kiefer, S., Luttenberger, M.: Parikh’s theorem: A simple and direct automaton construction. Information Processing Letters 111(12), 614–619 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9, 350–371 (1962)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gruska, J.: Descriptional complexity of context-free languages. In: Proceedings of 2nd Mathematical Foundations of Computer Science, pp. 71–83 (1973)Google Scholar
  5. 5.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979)Google Scholar
  6. 6.
    Kopczyński, E., To, A.W.: Parikh images of grammars: Complexity and applications. In: Symposium on Login in Computer Science, pp. 80–89 (2010)Google Scholar
  7. 7.
    Lavado, G.J., Pighizzini, G.: Parikh’s Theorem and Descriptional Complexity. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 361–372. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Lupanov, O.: A comparison of two types of finite automata. Problemy Kibernet. 9, 321–326 (1963) (in Russian); German translation: Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, 329–335 (1966)Google Scholar
  9. 9.
    Meyer, A.R., Fischer, M.J.: Economy of description by automata, grammars, and formal systems. In: FOCS, pp. 188–191. IEEE (1971)Google Scholar
  10. 10.
    Moore, F.R.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Transactions on Computers C-20(10), 1211–1214 (1971)CrossRefGoogle Scholar
  11. 11.
    Parikh, R.J.: On context-free languages. Journal of the ACM 13(4), 570–581 (1966)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Pighizzini, G., Shallit, J., Wang, M.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. Journal of Computer and System Sciences 65(2), 393–414 (2002)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Rabin, M., Scott, D.: Finite automata and their decision problems. IBM J. Res. Develop. 3, 114–125 (1959)MathSciNetCrossRefGoogle Scholar
  14. 14.
    To, A.W.: Parikh images of regular languages: Complexity and applications, arXiv:1002.1464v2 (February 2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Giovanna J. Lavado
    • 1
  • Giovanni Pighizzini
    • 1
  • Shinnosuke Seki
    • 2
  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanoItaly
  2. 2.Department of Information and Computer ScienceAalto UniversityAaltoFinland

Personalised recommendations