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Lower and Upper Bounds for the Length of Joins in the Lambek Calculus

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Computer Science – Theory and Applications (CSR 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7913))

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Abstract

In 1993 Mati Pentus proved a criterion of conjoinability for the Lambek calculus and multiplicative cyclic linear logic. In 2011 Alexey Sorokin showed that any pair of conjoinable types in the Lambek calculus has the join type of quadratic length with respect to the length of the types in the pair. We prove that the lower bound on the length of joins in the Lambek calculus and multiplicative linear logic is also quadratic.

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References

  1. Abrusci, V.M.: Phase semantics and sequent calculus for pure non-commutative classical linear logic. Journal of Symbolic Logic 56(4), 1403–1451 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Béchet, D.: Parsing pregroup grammars and Lambek calculus using partial composition. Studia Logica 87(2), 199–224 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Foret, A.: Conjoinability and unification in Lambek categorial grammars. In: Proceedings of the 5th ROMA Workshop on New Perspectives in Logic and Formal Linguistics, Bulzoni, Roma (2001)

    Google Scholar 

  4. Foret, A.: On the computation of joins for non-associative Lambek categorial grammars. In: Proceedings of the 17th International Workshop on Unification, UNIF, vol. 3, pp. 25–38 (2003)

    Google Scholar 

  5. Girard, J.-Y.: Linear logic. Theoretical Computer Science 1, 1–101 (1987)

    Article  Google Scholar 

  6. Grishin, V.: On a generalization of the Ajdukewicz-Lambek system. In: Studies on Nonclassical Logics and Formal Systems, Nauka, Moscow (1983) (in Russian); English translation in: Proceedings of the 5th ROMA Workshop on New Perspectives in Logic and Formal, Linguistics. Bulzoni Editore, Roma (2001)

    Google Scholar 

  7. Lambek, J.: Type grammar revisited. In: Lecomte, A., Perrier, G., Lamarche, F. (eds.) LACL 1997. LNCS (LNAI), vol. 1582, pp. 1–27. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  8. Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  9. Moortgat, M., Pentus, M.: Type similarity for the Lambek-Grishin calculus. In: Proceedings of the 12th Conference on Formal Grammar, Dublin (2007)

    Google Scholar 

  10. Moot, R., Retoré, C. (eds.): The Logic of Categorial Grammars. LNCS, vol. 6850. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  11. Pentus, M.: Equivalent types in Lambek calculus and linear logic. Preprint. Department of mathematical logic, Steklov Mathematical Institute, Moscow 2 (1992)

    Google Scholar 

  12. Pentus, M.: The conjoinability relation in Lambek calculus and linear logic. ILLC Prepublication Series ML-93-03. Institute for Logic, Language and Computation, University of Amsterdam (1993)

    Google Scholar 

  13. Pentus, M.: Lambek grammars are context-free. In: Proceedings of the Logic in Computer Science, LICS 1993, pp. 429–433 (1993)

    Google Scholar 

  14. Pentus, M.: Models of the Lambek calculus. Annals of Pure and Applied Logic 75(1), 179–213 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pentus, M.: Free monoid completeness of the Lambek calculus allowing empty premises. In: Larrazabal, J.M., Lascar, D., Mints, G. (eds.) Proceedings of Logic Colloquium 1996, pp. 171–209. Springer, Berlin (1998)

    Google Scholar 

  16. Roorda, D.: Resource logic: proof theoretical investigations. PhD thesis, FWI, Universiteit van Amsterdam (1991)

    Google Scholar 

  17. Safiullin, A.: Derivability of admissible rules with simple premises in the Lambek calculus. Moscow University Mathematics Bulletin 62(4), 168–171 (2007)

    Article  MathSciNet  Google Scholar 

  18. Sorokin, A.: On the length of joins in Lambek calculus. Moscow University Mathematics Bulletin 66(3), 101–104 (2011)

    Article  MathSciNet  Google Scholar 

  19. Yetter, D.: Quantales and (noncommutative) linear logic. Journal of Symbolic Logic 55(1), 41–64 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow Institute of Physics and Technology, Moscow State University, Russia

    Alexey Sorokin

Authors
  1. Alexey Sorokin
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Editor information

Editors and Affiliations

  1. School of Computing Science, Simon Fraser University, 8888 University Drive, V5A 1S6, Burnaby, BC, Canada

    Andrei A. Bulatov

  2. Institute of Mathematics and Computer Science, Ural Federal University, 51 Lenina Ave., 620083, Ekaterinburg, Russia

    Arseny M. Shur

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Sorokin, A. (2013). Lower and Upper Bounds for the Length of Joins in the Lambek Calculus. In: Bulatov, A.A., Shur, A.M. (eds) Computer Science – Theory and Applications. CSR 2013. Lecture Notes in Computer Science, vol 7913. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38536-0_13

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  • DOI: https://doi.org/10.1007/978-3-642-38536-0_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38535-3

  • Online ISBN: 978-3-642-38536-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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