Lambek Calculus and Linear Logic: Proof Nets as Parse Structures

  • Richard Moot
  • Christian Retoré
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6850)


This chapter, a large part of which is a translation of (Retoré, 1996), deals with the connection between Lambek categorial grammar and linear logic, the main objective being the presentation of proof nets which are excellent parse structures, because they identify linguistically equivalent analyses of a given sentence.

This graphical notation for proofs that are parse structures in categorial grammar is a not a mere variation for convenience. On a technical ground, it avoids the so-called spurious ambiguity problem of categorial grammars (the fact that we can find many different proofs/parse structures for what corresponds to a single analysis or lambda term). Conceptually, this proof syntax is a justification of the use of the expression parsing as deduction often associated with categorial grammar. Indeed proof nets only distinguish between proofs which correspond to different syntactic analyses.

We first give a rather complete presentation of the correspondence between the Lambek calculus and variants of multiplicative linear logic, since the Lambek calculus can be defined as non-commutative intuitionistic multiplicative linear logic without empty antecedents.

Next we define proof nets and establish their correspondence with the more traditional sequent calculus, present parsing as proof net construction and present some recent descriptions of non commutative proof nets.

As an evidence of their linguistic relevance, we explain how they provide a formal account of some performance questions, like the complexity of the processing of several intricate syntactic constructs, like center embedded relatives, garden path phenomena and preferred readings.


Cyclic Permutation Linear Logic Sequent Calculus Categorial Grammar Exchange Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Abrusci, V.M.: Non-commutative proof nets. In: Girard, et al., pp. 271–296 (1995)Google Scholar
  3. Asperti, A.: A linguistic approach to dead-lock. Tech. Rep. LIENS 91-15, Dép. Maths et Info, Ecole Normale Supérieure, Paris (1991)Google Scholar
  4. Asperti, A., Dore, G.: Yet Another Correctness Criterion for Multiplicative Linear Logic with Mix. In: Nerode, A., Matiyasevich, Y. (eds.) LFCS 1994. LNCS, vol. 813, pp. 34–46. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  5. Bellin, G., Scott, P.J.: On the π-calculus and linear logic. Theoretical Computer Science 135, 11–65 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. van Benthem, J.: Language in Action: Categories, Lambdas and Dynamic Logic. Studies in logic and the foundation of mathematics, vol. 130. North-Holland, Amsterdam (1991)zbMATHGoogle Scholar
  7. Bondy, J.A., Murty, U.S.R.: Graph Theory and Applications. Macmillan Press (1976)Google Scholar
  8. Dalrymple, M., Lamping, J., Pereira, F., Saraswat, V.: Linear logic for meaning assembly. In: Morrill, G., Oehrle, R. (eds.) Formal Grammar, pp. 75–93. FoLLI, Barcelona (1995)Google Scholar
  9. Danos, V.: La logique linéaire appliquée à l’étude de divers processus de normalisation et principalement du λ-calcul. Thèse de Doctorat, spécialité Mathématiques, Université Paris 7 (1990)Google Scholar
  10. Danos, V., Regnier, L.: The structure of multiplicatives. Archive for Mathematical Logic 28, 181–203 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Diestel, R.: Graph Theory, 4th edn. Springer (2010)Google Scholar
  12. Fleury, A.: La règle d’échange: logique linéaire multiplicative tréssée. Thèse de Doctorat, spécialité Mathématiques, Université Paris 7 (1996)Google Scholar
  13. Fleury, A., Retoré, C.: The mix rule. Mathematical Structures in Computer Science 4(2), 273–285 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Girard, J.Y.: Linear logic. Theoretical Computer Science 50(1), 1–102 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Girard, J.Y.: Linear logic: its syntax and semantics. In: Girard, et al., pp. 1–42 (1995)Google Scholar
  16. Girard, J.Y., Lafont, Y., Regnier, L. (eds.): Advances in Linear Logic. London Mathematical Society Lecture Notes, vol. 222. Cambridge University Press (1995)Google Scholar
  17. de Groote, P.: Linear Logic with Isabelle: Pruning the Proof Search Tree. In: Baumgartner, P., Posegga, J., Hähnle, R. (eds.) TABLEAUX 1995. LNCS (LNAI), vol. 918, pp. 263–277. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  18. de Groote, P.: A Dynamic Programming Approach to Categorial Deduction. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 1–15. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  19. de Groote, P., Lamarche, F.: Classical non-associative Lambek calculus. Studia Logica 71(3), 355–388 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. de Groote, P., Retoré, C.: Semantic readings of proof nets. In: Kruijff, G.J., Morrill, G., Oehrle, D. (eds.) Formal Grammar, pp. 57–70. FoLLI, Prague (1996)Google Scholar
  21. Guerrini, S.: Correctness of multiplicative proof nets is linear. In: 14th Symposium on Logic in Computer Science (LICS 1999), pp. 454–463. IEEE (1999)Google Scholar
  22. Guerrini, S.: A linear algorithm for MLL proof net correctness and sequentialization. Theoretical Computer Science 412(20), 1958–1978 (2011); Girard’s FestschriftMathSciNetCrossRefzbMATHGoogle Scholar
  23. Johnson, M.E.: Proof nets and the complexity of processing center-embedded constructions. Journal of Logic Language and Information Special Issue on Recent Advances in Logical and Algebraic Approaches to Grammar 7(4), 433–447 (1998)zbMATHGoogle Scholar
  24. Lamarche, F.: Proof nets for intuitionistic linear logic: Essential nets. 35 page technical report available by FTP from the Imperial College archives (1994)Google Scholar
  25. Lambek, J.: From categorial grammar to bilinear logic. In: Došen, K., Schröder-Heister, P. (eds.) Substructural Logics, pp. 207–237. Oxford University Press, Oxford (1993)Google Scholar
  26. Lincoln, P., Mitchell, J., Scedrov, A., Shankar, N.: Decision problems for propositional linear logic. Annals of Pure and Applied Logic 56(1-3), 239–311 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Melliès, P.A.: A topological correctness criterion for multiplicative non-commutative logic. In: Ehrhard, T., Girard, J.Y., Ruet, P., Scott, P. (eds.) Linear Logic in Computer Science. London Mathematical Society Lecture Note, vol. 316, ch. 8, pp. 283–321. Cambridge University Press (2004)Google Scholar
  28. Merenciano, J.M., Morrill, G.: Generation as Deduction on Labelled Proof Nets. In: Retoré, C. (ed.) LACL 1996. LNCS (LNAI), vol. 1328, pp. 310–328. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  29. Métayer, F.: Homology of proof-nets. Prépublication 39, Equipe de Logique, Université Paris 7 (1993)Google Scholar
  30. Moot, R.: Filtering axiom links for proof nets. In: Kallmeyer, L., Monachesi, P., Penn, G., Satta, G. (eds.) Proceedings of the 12th Conference on Formal Grammar (FG 2007). CSLI Publications, Dublin (2007) (to appear) ISSN 1935-1569 Google Scholar
  31. Morrill, G.: Memoisation of categorial proof nets: parallelism in categorial processing. In: Abrusci, V.M., Casadio, C. (eds.) Third Roma Workshop: Proofs and Linguistics Categories – Applications of Logic to the Analysis and Implementation of Natural Language. CLUEB, Bologna (1996)Google Scholar
  32. Morrill, G.: Incremental processing and acceptability. Computational Linguistics 26(3), 319–338 (2000); preliminary version: UPC Report de Recerca LSI-98-46-R (1998)CrossRefGoogle Scholar
  33. Morrill, G.: Categorial Grammar: Logical Syntax, Semantics, and Processing. Oxford University Press (2011)Google Scholar
  34. Morrill, G., Fadda, M.: Proof nets for basic discontinuous Lambek calculus. Journal of Logic and Computation 18(2), 239–256 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Murawski, A., Ong, C.H.: Dominator trees and fast verification of proof nets. In: 15th Symposium on Logic in Computer Science (LICS 2000), pp. 181–191. IEEE (2000)Google Scholar
  36. Pentus, M.: Lambek calculus is NP-complete. Theoretical Computer Science 357(1), 186–201 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Pogodalla, S.: Generation in the Lambek calculus framework: an approach with semantic proof nets. In: Proceedings of NAACL 2000 (2000a)Google Scholar
  38. Pogodalla, S.: Generation, Lambek calculus, montague’s semantics and semantic proof nets. In: Proceedings of Coling 2000 (2000b)Google Scholar
  39. Pogodalla, S.: Generation with semantic proof nets. Research Report 3878, INRIA (2000c),
  40. Pogodalla, S., Retoré, C.: Handsome non-commutative proof-nets: perfect matchings, series-parallel orders and hamiltonian circuits. Tech. Rep. RR-5409, INRIA, presented at Categorial Grammars (2004); to appear in the Journal of Applied LogicGoogle Scholar
  41. Regnier, L.: Lambda calcul et réseaux. Thèse de doctorat, spécialité mathématiques, Université Paris 7 (1992)Google Scholar
  42. Retoré, C.: Réseaux et séquents ordonnés. Thèse de Doctorat, spécialité Mathématiques, Université Paris 7 (1993)Google Scholar
  43. Retoré, C.: Calcul de Lambek et logique linéaire. Traitement Automatique des Langues 37(2), 39–70 (1996)Google Scholar
  44. Retoré, C.: Perfect matchings and series-parallel graphs: multiplicative proof nets as R&B-graphs. In: Girard, J.Y., Okada, M., Scedrov, A. (eds.) Linear 1996. Electronic Notes in Theoretical Science, vol. 3. Elsevier (1996)Google Scholar
  45. Retoré, C.: A semantic characterisation of the correctness of a proof net. Mathematical Structures in Computer Science 7(5), 445–452 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  46. Retoré, C.: Handsome proof-nets: perfect matchings and cographs. Theoretical Computer Science 294(3), 473–488 (2003), complete version RR-3652,
  47. Savateev, Y.: Product-Free Lambek Calculus Is NP-Complete. In: Artemov, S., Nerode, A. (eds.) LFCS 2009. LNCS, vol. 5407, pp. 380–394. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  48. Yetter, D.N.: Quantales and (non-commutative) linear logic. Journal of Symbolic Logic 55, 41–64 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Richard Moot
  • Christian Retoré

There are no affiliations available

Personalised recommendations