Theory of Computing Systems

, Volume 55, Issue 1, pp 250–278 | Cite as

The Failure of the Strong Pumping Lemma for Multiple Context-Free Languages

  • Makoto Kanazawa
  • Gregory M. Kobele
  • Jens Michaelis
  • Sylvain Salvati
  • Ryo Yoshinaka


Seki et al. (Theor. Comput. Sci. 88(2):191–229, 1991) showed that every m-multiple context-free language L is weakly 2m-iterative in the sense that either L is finite or L contains a subset of the form \(\{ u_{0} w_{1}^{i} u_{1} \cdots w_{2m}^{i} u_{2m} \mid i \in \mathbb {N}\}\), where w 1w 2n ε. Whether every m-multiple context-free language L is 2m-iterative, that is to say, whether all but finitely many elements z of L can be written as z=u 0 w 1 u 1w 2m u 2m with w 1w 2m ε and \(\{ u_{0} w_{1}^{i} u_{1} \cdots w_{2m}^{i} u_{2m} \mid i \in \mathbb {N}\} \subseteq L\), has been open. We show that there is a 3-multiple context-free language that is not k-iterative for any k.


Multiple context-free grammar Pumping lemma 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Makoto Kanazawa
    • 1
  • Gregory M. Kobele
    • 2
  • Jens Michaelis
    • 3
  • Sylvain Salvati
    • 4
  • Ryo Yoshinaka
    • 5
  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.Computation Institute and Department of LinguisticsUniversity of ChicagoChicagoUSA
  3. 3.Fakultät für Linguistik und LiteraturwissenschaftUniversität BielefeldBielefeldGermany
  4. 4.INRIA Bordeaux Sud-OuestTalence CedexFrance
  5. 5.Graduate School of InformaticsKyoto UniversityKyotoJapan

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