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Input-Driven Pushdown Automata with Limited Nondeterminism

(Invited Paper)
  • Alexander Okhotin
  • Kai Salomaa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8633)

Abstract

It is known that determinizing a nondeterministic input-driven pushdown automaton (NIDPDA) of size n results in the worst case in a machine of size \(2^{\Theta(n^2)}\) (R. Alur, P. Madhusudan, “Adding nesting structure to words”, J.ACM 56(3), 2009). This paper considers the special case of k-path NIDPDAs, which have at most k computations on any input. It is shown that the smallest deterministic IDPDA equivalent to a k-path NIDPDA of size n is of size Θ(n k ). The paper also gives an algorithm for deciding whether or not a given NIDPDA has the k-path property, for a given k; if k is fixed, the problem is P-complete.

Keywords

Regular Language Input String Input Symbol Path Property Descriptional Complexity 
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.

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References

  1. 1.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: ACM Symposium on Theory of Computing, STOC 2004, Chicago, USA, June 13-16, pp. 202–211 (2004)Google Scholar
  2. 2.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. Journal of the ACM 56(3) (2009)Google Scholar
  3. 3.
    Björklund, H., Martens, W.: The tractability frontier of NFA minimization. J. Comput. System Sci. 78, 198–210 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    von Braunmühl, B., Verbeek, R.: Input driven languages are recognized in logn space. North-Holland Mathematics Studies 102, 1–19 (1985)CrossRefGoogle Scholar
  5. 5.
    Chervet, P., Walukiewicz, I.: Minimizing variants of visibly pushdown automata. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 135–146. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Chistikov, D., Majumdar, R.: A uniformization theorem for nested word to word transductions. In: Konstantinidis, S. (ed.) CIAA 2013. LNCS, vol. 7982, pp. 97–108. Springer, Heidelberg (2013)Google Scholar
  7. 7.
    Crespi-Reghizzi, S., Mandrioli, D.: Operator precedence and the visibly pushdown property. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 214–226. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Debarbieux, D., Gauwin, O., Niehren, J., Sebastian, T., Zergaoui, M.: Early nested word automata for XPath query answering on XML streams. In: Konstantinidis, S. (ed.) CIAA 2013. LNCS, vol. 7982, pp. 292–305. Springer, Heidelberg (2013)Google Scholar
  9. 9.
    Dymond, P.W.: Input-driven languages are in logn depth. Information Processing Letters 26, 247–250 (1988)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gauwin, O., Niehren, J., Roos, Y.: Streaming tree automata. Information Processing Letters 109, 13–17 (2008)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldstine, J., Kappes, M., Kintala, C.M.R., Leung, H., Malcher, A., Wotschke, D.: Descriptional complexity of machines with limited resources. Journal of Universal Computer Science 8, 193–234 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Goldstine, J., Kintala, C.M.R., Wotschke, D.: On measuring nondeterminism in regular languages. Information and Computation 86(2), 179–194 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Holzer, M., Salomaa, K., Yu, S.: On the state complexity of k-entry deterministic finite automata. J. Automata, Languages, and Combinatorics 6, 453–466 (2001)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Hromkovič, J., Seibert, S., Karhumäki, J., Klauck, H., Schnitger, G.: Communication complexity method for measuring nondeterminism in finite automata. Information and Computation 172, 202–217 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lange, M.: P-hardness of the emptiness problem for visibly pushdown automata. Inf. Proc. Lett. 111(7), 338–341 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Leung, H.: Separating exponentially ambiguous finite automata from polynomially ambiguous finite automata. SIAM J. Comput. 27, 1073–1082 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Leung, H.: Descriptional complexity of NFA of different ambiguity. Internat. J. Foundations Comput. Sci. 16, 975–984 (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–435. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  19. 19.
    Okhotin, A.: Unambiguous finite automata over a unary alphabet. Inform. Comput. 212, 15–36 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Okhotin, A., Piao, X., Salomaa, K.: Descriptional complexity of input-driven pushdown automata. In: Bordihn, H., Kutrib, M., Truthe, B. (eds.) Languages Alive 2012. LNCS, vol. 7300, pp. 186–206. Springer, Heidelberg (2012)Google Scholar
  21. 21.
    Okhotin, A., Salomaa, K.: Descriptional complexity of unambiguous nested word automata. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 414–426. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Okhotin, A., Salomaa, K.: Complexity of input-driven pushdown automata. In: Hemaspaandra, L.A. (ed.) SIGACT News Complexity Theory Column 82. SIGACT News (to appear, 2014)Google Scholar
  23. 23.
    Palioudakis, A., Salomaa, K., Akl, S.G.: State complexity and limited nondeterminism. In: Kutrib, M., Moreira, N., Reis, R. (eds.) DCFS 2012. LNCS, vol. 7386, pp. 252–265. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Palioudakis, A., Salomaa, K., Akl, S.G.: State complexity of finite tree width NFAs. J. Automata, Languages and Combinatorics 17, 245–264 (2012)Google Scholar
  25. 25.
    Palioudakis, A., Salomaa, K., Akl, S.G.: Comparisons between measures of nondeterminism on finite automata. In: Jurgensen, H., Reis, R. (eds.) DCFS 2013. LNCS, vol. 8031, pp. 217–228. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  26. 26.
    Salomaa, K.: Limitations of lower bound methods for deterministic nested word automata. Information and Computation 209, 580–589 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Shallit, J.: A Second Course in Formal Languages and Automata Theory. Cambridge University Press (2009)Google Scholar
  28. 28.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, pp. 41–110 (1997)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alexander Okhotin
    • 1
  • Kai Salomaa
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.School of ComputingQueen’s UniversityKingstonCanada

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