A two-way nondeterministic finite transducer (2-NFT) is a finite automaton with a two-way input tape and a one-way output tape. The generated language of a 2-NFT is the set of all strings it can output (across all inputs). Whereas two-way nondeterministic finite acceptors (2-NFAs) accept only regular languages, 2-NFTs can generate languages which are not even context-free, e.g. \(\{\texttt{a}^n \texttt{b}^n \texttt{c}^n \mid n \geq 0\}\). We prove a pumping lemma for 2-NFT languages which strengthens and generalizes previous results. Our pumping lemma states that every 2-NFT language L is k-iterative for some k ≥ 1. That is, every string in L above a certain length can be expressed in the form x 1 y 1 x 2 y 2 ⋯ x k y k x k + 1, where the ys can be “pumped” to produce new strings in L of the form \(x_1 y_1^i x_2 y_2^i \dotsm x_k y_k^i x_{k+1}\).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ehrich, R., Yau, S.: Two-way sequential transductions and stack automata. Information and Control 18(5), 404–446 (1971)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Engelfriet, J.: Two-way automata and checking automata. In: de Bakker, J.W., van Leeuwen, J. (eds.) Foundations of Computer Science III Part 1. Mathematical Centre Tracts, vol. 108, pp. 1–69. Mathematisch Centrum, Amsterdam (1979)Google Scholar
  3. 3.
    Engelfriet, J., Hoogeboom, H.J.: MSO definable string transductions and two-way finite-state transducers. ACM Trans. Comput. Logic 2(2), 216–254 (2001)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Engelfriet, J., Hoogeboom, H.J.: Finitary compositions of two-way finite-state transductions. Fundam. Inf. 80(1-3), 111–123 (2007)MATHMathSciNetGoogle Scholar
  5. 5.
    Filiot, E., Gauwin, O., Reynier, P.A., Servais, F.: From two-way to one-way finite state transducers. In: LICS 2013, pp. 468–477. IEEE Computer Society (2013)Google Scholar
  6. 6.
    Greibach, S.A.: One way finite visit automata. Theoretical Computer Science 6(2), 175–221 (1978)MATHMathSciNetGoogle Scholar
  7. 7.
    Greibach, S.A.: The strong independence of substitution and homomorphic replication. RAIRO - Theoretical Informatics and Applications 12(3), 213–234 (1978)MATHMathSciNetGoogle Scholar
  8. 8.
    Greibach, S.: Checking automata and one-way stack languages. J. Comput. Syst. Sci. 3(2), 196–217 (1969)MATHMathSciNetGoogle Scholar
  9. 9.
    Kanazawa, M., Kobele, G., Michaelis, J., Salvati, S., Yoshinaka, R.: The failure of the strong pumping lemma for multiple context-free languages. Theory of Computing Systems, 1–29 (2014)Google Scholar
  10. 10.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)MathSciNetGoogle Scholar
  11. 11.
    Radzinski, D.: Chinese number-names, tree adjoining languages, and mild context-sensitivity. Comput. Linguist. 17(3), 277–299 (1991)Google Scholar
  12. 12.
    Rajlich, V.: Absolutely parallel grammars and two-way finite-state transducers. J. Comput. Syst. Sci. 6(4), 324–342 (1972)MATHMathSciNetGoogle Scholar
  13. 13.
    Rodriguez, F.: Une double hiérarchie infinie de langages vérifiables. RAIRO - Theoretical Informatics and Applications 9(R1), 5–19 (1975)Google Scholar
  14. 14.
    Smith, T.: On Infinite Words Determined by Stack Automata. In: FSTTCS 2013. Leibniz International Proceedings in Informatics (LIPIcs), vol. 24, pp. 413–424. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2013)Google Scholar
  15. 15.
    de Souza, R.: Uniformisation of two-way transducers. In: Dediu, A.-H., Martín-Vide, C., Truthe, B. (eds.) LATA 2013. LNCS, vol. 7810, pp. 547–558. Springer, Heidelberg (2013)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2014

Authors and Affiliations

  • Tim Smith
    • 1
  1. 1.Northeastern UniversityBostonUSA

Personalised recommendations