Abstract
Pentus’ theorem states that any language generated by a Lambek grammar is context-free. We present a substitution that reduces the Lambek calculus enriched with the unit constant to the variant of the Lambek calculus that does not contain the unit (but still allows empty premises), and use this substitution to prove that any language generated by a categorial grammar based on the Lambek calculus with the unit is context-free.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bar-Hillel, Y., Gaifman, C., Shamir, E.: On categorial and phrase-structure grammars. Bull. Res. Council Israel Sect. F 9F, 1–16 (1960)
Buszkowski, W.: The equivalence of unidirectional Lambek categorial grammars and context-free grammars. Zeitschr. für math. Logik und Grundl. der Math. 31, 369–384 (1985)
Kuznetsov, S.: Lambek grammars with one division and one primitive type. Unpublished manuscript (2010)
Lambek, J.: The mathematics of sentence structure. American Math. Monthly 65(3), 154–170 (1958)
Lambek, J.: Deductive systems and categories II: Standard constructions and closed categories. In: Hilton, P. (ed.) Category Theory, Homology Theory and Their Applications I. Lect. Notes Math., vol. 86, pp. 76–122. Springer, Berlin (1969)
Métayer, F.: Polynomial equivalence among systems LLNC, LLNC a and LLNC0. Theor. Comput. Sci. 227(1), 221–229 (1999)
Pentus, M.: Lambek grammars are context free. In: Proc. of the 8th Annual IEEE Symposium on Logic in Computer Science, pp. 429–433. IEEE Computer Society Press, Los Alamitos (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kuznetsov, S. (2012). Lambek Grammars with the Unit. In: de Groote, P., Nederhof, MJ. (eds) Formal Grammar. FG FG 2010 2011. Lecture Notes in Computer Science, vol 7395. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32024-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-32024-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32023-1
Online ISBN: 978-3-642-32024-8
eBook Packages: Computer ScienceComputer Science (R0)