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

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Abstract

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 1⋯w 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 1⋯w 2m u 2m with w 1⋯w 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.

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Notes

  1. Around the same time as Kasami et al. [9] first introduced multiple context-free grammars, essentially the same formalism was proposed by Vijay-Shanker et al. [15] under the name linear context-free rewriting systems (LCFRS). In this paper, we mostly follow the terminology of Seki et al. [14].

  2. We let \(\mathbb {N}\) denote the set of natural numbers {0,1,2,…} and ε denote the empty string.

  3. Formally, a context is a tree with a single special leaf node (“hole”), which is labeled by □. When U[] is a context and T is a tree, U[T] denotes the tree that results from removing the hole of U[] and inserting T in its place.

  4. See footnote 10 of Radzinski [13]. Radzinski refers to the technical report [9] rather than the journal article [14] based on it, but the proof is the same in both papers.

  5. A string v is a factor of a string z if z=uvw for some strings u,w.

  6. As usual, the sets V,L,R are understood to be the components of the least solution to these equations.

  7. By part (i), part (ii) can be equivalently stated with a + and b + in place of a ∗ and b ∗, but it will turn out to be slightly more convenient in this form.

  8. We will appeal to Lemma 21 similarly in Cases 2–5 without explicitly going through this kind of reasoning.

References

  1. Berstel, J., Boasson, L.: Context-free languages. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 59–102. Elsevier, Amsterdam (1990)

    Google Scholar 

  2. Greibach, S.A.: Hierarchy theorems for two-way finite state transducers. Acta Inform. 11, 89–101 (1978)

    MATH  MathSciNet  Google Scholar 

  3. Greibach, S.A.: One-way finite visit automata. Theor. Comput. Sci. 6, 175–221 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Groenink, A.V.: Mild context-sensitivity and tuple-based generalizations of context-free grammar. Linguist. Philos. 20(6), 607–636 (1997)

    Article  Google Scholar 

  5. Groenink, A.V.: Surface without Structure. Ph.D. thesis, University of Utrecht (1997)

  6. Kanazawa, M.: The pumping lemma for well-nested multiple context-free languages. In: Diekert, V., Nowotka, D. (eds.) Developments in Language Theory: 13th International Conference, DLT 2009. Lecture Notes in Computer Science, vol. 5583, pp. 312–325. Springer, Berlin (2009)

    Chapter  Google Scholar 

  7. Kanazawa, M., Salvati, S.: Generating control languages with abstract categorial grammars. In: Preliminary Proceedings of FG-2007: The 12th Conference on Formal Grammar (2007)

    Google Scholar 

  8. Kanazawa, M., Salvati, S.: The copying power of well-nested multiple context-free grammars. In: Dediu, A.H., Fernau, H., Martín-Vide, C. (eds.) Language and Automata Theory and Applications, Fourth International Conference, LATA 2010. Lecture Notes in Computer Science, vol. 6031, pp. 344–355. Springer, Berlin (2010)

    Google Scholar 

  9. Kasami, T., Seki, H., Fujii, M.: Generalized context-free grammars, multiple context-free grammars and head grammars. Technical report, Osaka University (1987)

  10. Kracht, M.: The Mathematics of Language. de Gruyter, Berlin (2003)

    Book  MATH  Google Scholar 

  11. Michaelis, J.: On formal properties of minimalist grammars. Linguistics in Potsdam 13. Universitätsbibliothek, Publikationsstelle, Potsdam. Ph.D. thesis, ISBN 3-935024-28-2

  12. Palis, M.A., Shende, S.M.: Pumping lemmas for the control language hierarchy. Math. Syst. Theory 28(3), 199–213 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Radzinski, D.: Chinese number-names, tree adjoining languages, and mild context-sensitivity. Comput. Linguist. 17(3), 277–299 (1991)

    Google Scholar 

  14. Seki, H., Matsumura, T., Fujii, M., Kasami, T.: On multiple context-free grammars. Theor. Comput. Sci. 88(2), 191–229 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Vijay-Shanker, K., Weir, D.J., Joshi, A.K.: Characterizing structural descriptions produced by various grammatical formalisms. In: 25th Annual Meeting of the Association for Computational Linguistics, pp. 104–111 (1987)

    Chapter  Google Scholar 

  16. Weir, D.J.: A geometric hierarchy beyond context-free languages. Theor. Comput. Sci. 104(2), 235–261 (1992). doi:10.1016/0304-3975(92)90124-X

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Makoto Kanazawa

  2. Computation Institute and Department of Linguistics, University of Chicago, Chicago, IL, 60637, USA

    Gregory M. Kobele

  3. Fakultät für Linguistik und Literaturwissenschaft, Universität Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Jens Michaelis

  4. INRIA Bordeaux Sud-Ouest, LaBRI, 351, Cours de la Libération, 33405, Talence Cedex, France

    Sylvain Salvati

  5. Graduate School of Informatics, Kyoto University, 36-1 Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Ryo Yoshinaka

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  1. Makoto Kanazawa
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  5. Ryo Yoshinaka
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Correspondence to Makoto Kanazawa.

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Kanazawa, M., Kobele, G.M., Michaelis, J. et al. The Failure of the Strong Pumping Lemma for Multiple Context-Free Languages. Theory Comput Syst 55, 250–278 (2014). https://doi.org/10.1007/s00224-014-9534-z

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