Parikh Image of Pushdown Automata

  • Pierre Ganty
  • Elena GutiérrezEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with \(n\) states and \(p \ge 2n + 4\) stack symbols that accepts one single long word for which every equivalent context-free grammar needs \(\varOmega (n^2(p-2n-4))\) variables. This family shows that the classical algorithm for converting a PDA into an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family. We conclude that, when assuming this weaker equivalence, the conversion algorithm is also optimal. Second, Parikh’s theorem motivates the comparison of PDAs against finite state automata. In particular, the same family of unary PDAs gives a lower bound on the number of states of every Parikh-equivalent finite state automaton. Finally, we look into the case of unary deterministic PDAs. We show a new construction converting a unary deterministic PDA into an equivalent context-free grammar that achieves best known bounds.



We thank Pedro Valero for pointing out the reference on smallest grammar problems [2]. We also thank the anonymous referees for their insightful comments and suggestions.


  1. 1.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997). doi: 10.1007/3-540-63141-0_10 CrossRefGoogle Scholar
  2. 2.
    Charikar, M., Lehman, E., Liu, D., Panigrahy, R., Prabhakaran, M., Sahai, A., Shelat, A.: The smallest grammar problem. IEEE Trans. Inf. Theory 51(7), 2554–2576 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chistikov, D., Majumdar, R.: Unary pushdown automata and straight-line programs. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 146–157. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43951-7_13 Google Scholar
  4. 4.
    Esparza, J., Ganty, P., Kiefer, S., Luttenberger, M.: Parikh’s theorem: a simple and direct automaton construction. IPL 111(12), 614–619 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Esparza, J., Luttenberger, M., Schlund, M.: A brief history of strahler numbers. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 1–13. Springer, Cham (2014). doi: 10.1007/978-3-319-04921-2_1 CrossRefGoogle Scholar
  6. 6.
    Finkel, A., Willems, B., Wolper, P.: A direct symbolic approach to model checking pushdown systems (extended abstract). Electron. Notes Theoret. Comput. Sci. 9, 27–37 (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ganty, P., Gutiérrez, E.: Parikh image of pushdown automata (long version) (2017). Pre-print arXiv arXiv: 1706.08315
  8. 8.
    Goldstine, J., Price, J.K., Wotschke, D.: A pushdown automaton or a context-free grammar: which is more economical? Theoret. Comput. Sci. 18, 33–40 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 3rd edn. Addison-Wesley Longman Publishing Co., Inc., Boston (2006)zbMATHGoogle Scholar
  10. 10.
    Pighizzini, G.: Deterministic pushdown automata and unary languages. Int. J. Found. Comput. Sci. 20(04), 629–645 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rohit, J.P.: On context-free languages. J. ACM 13(4), 570–581 (1966)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.IMDEA Software InstituteMadridSpain
  2. 2.Universidad Politécnica de MadridMadridSpain

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