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On translating Lambek grammars with one division into context-free grammars

  • S. L. KuznetsovEmail author
Article

Abstract

We describe a method of translating a Lambek grammar with one division into an equivalent context-free grammar whose size is bounded by a polynomial in the size of the original grammar. Earlier constructions by Buszkowski and Pentus lead to exponential growth of the grammar size.

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References

  1. 1.
    A. V. Aho and J. D. Ullman, TheTheory of Parsing, Translation, and Compiling, Vol. 1: Parsing (Prentice-Hall, Englewood Cliffs, NJ, 1972).Google Scholar
  2. 2.
    Y. Bar-Hillel, C. Gaifman, and E. Shamir, “On categorial and phrase-structure grammars, ” Bull. Res. Council Israel, Sect. F, 9F, 1–16 (1960).MathSciNetzbMATHGoogle Scholar
  3. 3.
    W. Buszkowski, “The equivalence of unidirectional Lambek categorial grammars and context-free grammars, ” Z. Math. Logik Grundlagen Math. 31, 369–384 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    N. Chomsky, “Three models for the description of language, ” IRE Trans. Inf. Theory IT-2 (3), 113–124 (1956).CrossRefzbMATHGoogle Scholar
  5. 5.
    J. Evey, “Application of pushdown store machines, ” in Proc. 1963 Fall Joint Comput. Conf. (Spartan Books, Baltimore, MD, 1963), pp. 215–227.Google Scholar
  6. 6.
    J. E. Hopcroft, R. Motwani, and J. D. Ullman, Introductionto Automata Theory, Languages, and Computation, 2nd ed. (Addison-Wesley, Reading, MA, 2001).zbMATHGoogle Scholar
  7. 7.
    G. Jäger, “On the generative capacity of multi-modal categorial grammars, ” Res. Lang. Comput. 1 (1–2), 105–125 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Kanazawa, “The Lambek calculus enriched with additional connectives, ” J. Log. Lang. Inf. 1 (2), 141–171 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Kanazawa and S. Salvati, “The string-meaning relations definable by Lambek grammars and context-free grammars, ” in Formal Grammar: Proc. 17th and 18th Int. Confs., FG 2012/2013, Ed. by G. Morrill and M.-J. Nederhof (Springer, Berlin, 2013), Lect. Notes Comput. Sci. 8036, pp. 191–208.CrossRefGoogle Scholar
  10. 10.
    S. L. Kuznetsov, “On translating context-free grammars into Lambek grammars, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 72–79 (2015) [Proc. Steklov Inst. Math. 290, 63–69 (2015)].MathSciNetzbMATHGoogle Scholar
  11. 11.
    S. L. Kuznetsov and N. S. Ryzhkova, “A fragment of the Lambek calculus with iteration, ” in Mal’tsev Meeting 2015: Coll. Abstr. Int. Conf. (Novosibirsk, 2015), p. 213.Google Scholar
  12. 12.
    J. Lambek, “The mathematics of sentence structure, ” Am. Math. Mon. 65 (3), 154–170 (1958).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Moortgat, “Multimodal linguistic inference, ” J. Log. Lang. Inf. 5 (3–4), 349–385 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    G. V. Morrill, CategorialGrammar: Logical Syntax, Semantics, and Processing (Oxford Univ. Press, Oxford, 2011).Google Scholar
  15. 15.
    A. G. Oettinger, “Automatic syntactic analysis and the pushdown store, ” in Structure of Language and Its Mathematical Aspects (Am. Math. Soc., Providence, RI, 1961), Proc. Symp. Appl. Math. 12, pp. 104–129.Google Scholar
  16. 16.
    A. E. Pentus and M. R. Pentus, MathematicalTheory of Formal Languages (Binom, Moscow, 2006) [in Russian].zbMATHGoogle Scholar
  17. 17.
    M. R. Pentus, “Lambek calculus and formal grammars, ” Fundam. Prikl. Mat. 1 (3), 729–751 (1995). Engl. transl. in Provability, Complexity, Grammars (Am. Math. Soc., Providence, RI, 1999), AMS Transl., Ser. 2, 192, pp. 57–86.MathSciNetzbMATHGoogle Scholar
  18. 18.
    M. R. Pentus, “Completeness of the Lambek syntactic calculus, ” Fundam. Prikl. Mat. 5 (1), 193–219 (1999); see also “Models for the Lambek calculus, ” Ann. Pure Appl. Log. 75 (1–2), 179–213 (1995).MathSciNetzbMATHGoogle Scholar
  19. 19.
    M. Pentus, “Lambek calculus is NP-complete, ” Theor. Comput. Sci. 357, 186–201 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. Pentus, “A polynomial-time algorithm for Lambek grammars of bounded order, ” Linguist. Anal. 36, 441–471 (2010).Google Scholar
  21. 21.
    V. V. Podolskii, “Circuit complexity meets ontology-based data access, ” in Computer Science—Theory and Applications: Proc. 10th Int. Comput. Sci. Symp. Russia, 2015 (Springer, Cham, 2015), Lect. Notes Comput. Sci. 9139, pp. 7–26.Google Scholar
  22. 22.
    Yu. Savateev, “Lambek grammars with one division are decidable in polynomial time, ” in Computer Science—Theory and Applications: Proc. 3rd Int. Comput. Sci. Symp. Russia, 2008 (Springer, Berlin, 2008), Lect. Notes Comput. Sci. 5010, pp. 273–282.Google Scholar
  23. 23.
    Yu. V. Savateev, “Recogintion of derivability for the Lambek calculus with one division, ” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 2, 59–62 (2009) [Moscow Univ. Math. Bull. 64 (2), 73–75 (2009)].MathSciNetzbMATHGoogle Scholar
  24. 24.
    Yu. V. Savateev, “Algorithmic complexity of fragments of the Lambek calculus, ” Cand. Sci. (Phys.–Math.) Dissertation (Moscow State Univ., Moscow, 2009).Google Scholar
  25. 25.
    Yu. Savateev, “Product-free Lambek calculus is NP-complete, ” Ann. Pure Appl. Log. 163 (7), 775–788 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D. S. Shamkanov, “Circular proofs for the Gödel–Löb provability logic, ” Mat. Zametki 96 (4), 609–622 (2014) [Math. Notes 96, 575–585 (2014)].MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    D. Shamkanov, “Nested sequents for provability logic GLP, ” Log. J. IGPL 23 (5), 789–815 (2015).MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. P. Schützenberger, “On context-free languages and push-down automata, ” Inf. Control 6 (3), 246–264 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    L. G. Valiant, “General context-free recognition in less than cubic time, ” J. Comput. Syst. Sci. 10, 308–315 (1975).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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