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Proof nets for multiplicative cyclic linear logic and Lambek calculus

Published online by Cambridge University Press:  22 February 2019

V. Michele Abrusci  and
Roberto Maieli
V. Michele Abrusci
Affiliation:
Department of Mathematics and Physics, “Roma Tre” University, Rome, Italy
Roberto Maieli*
Affiliation:
Department of Mathematics and Physics, “Roma Tre” University, Rome, Italy
*
*Corresponding author. Email: maieli@uniroma3.it
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Abstract

This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

Keywords

Categorial grammarscyclic ordersLambek calculuslanguage parsinglinear logicnoncommutative logicproof netssequent calculus
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 6 , June 2019 , pp. 733 - 762
Copyright
© Cambridge University Press 2019 

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Footnotes

†

This work contains new original contributions and improvements w.r.t. the contents of a previous paper (Abrusci and Maieli, 2015a) on the same subject presented by the authors at the 22nd Workshop on Logic, Language, Information and Computation (WoLLIC2015), held at the Indiana University (Bloomington, USA) from the 20th to the 23rd of July 2015.

References

Abrusci, V. M. (2002). Classical conservative extensions of Lambek calculus. Studia Logica 71 (3) 277–314.CrossRefGoogle Scholar
Abrusci, V. M. and Maieli, R. (2015a). Cyclic multiplicative proof nets of linear logic with an application to language parsing. In: Proceedings of the Conference WoLLIC 2015, LNCS, vol. 9160, Bloomington, USA, 53–68.Google Scholar
Abrusci, V. M. and Maieli, R. (2015b). Cyclic multiplicative and additive proof nets of linear logic with an application to language parsing. In: Proceedings of the Conference FG 2015, LNCS, vol. 9804, Barcelona, Springer, Heidelberg, 43–59.Google Scholar
Abrusci, V. M. and Ruet, P. (2000). Noncommutative logic I: the multiplicative fragment. Annals of Pure and Applied Logic 101 (1) 29–64.CrossRefGoogle Scholar
Andreoli, J.-M. andPareschi, R. (1991). From Lambek Calculus to word-based parsing. In: Proceedings of Workshop on Substructural Logic and Categorial Grammar, CIS Munchen, Germany.Google Scholar
Bagnol, M., Doumane, A. and Saurin, A. (2015). On the dependencies of logical rules. In: Proceedings of the 18th International Conference, FoSSaCS 2015, London, UK, 436–450.Google Scholar
Danos, V. and Regnier, L. (1989). The structure of multiplicatives. AML 28 181–203.Google Scholar
Danos, V. (1990). La Logique Linéaire appliquée à l’étude de divers processus de normalisation (principalment du λ-calcul). Phd Thesis, Univ. Paris VII.Google Scholar
de Naurois, P.J. and Mogbil, V. (2007). Correctness of multiplicative (and exponential) proof structures is NL-complete. In: Proceedings of CSL 2007, LNCS, vol. 4646, 435–450.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1–102.CrossRefGoogle Scholar
Girard, J.-Y. (1995). Proof nets: the parallel syntax for proof theory. In: Ursini, Agliano (eds.) Logic and Algebra, New York, M. Dekker.Google Scholar
Guerrini, S. (2011). A linear algorithm for MLL proof net correctness and sequentialization. Theoretical Computer Science 412 (20) 1958–1978.CrossRefGoogle Scholar
Hughes, D. and van Glabbeek, R. (2003). Proof nets for unit-free multiplicative-additive linear logic. In: Proceedings of the 18th IEEE Logic in Computer Science, Los Alamitos.Google Scholar
Lambek, J. (1958). The mathematics of sentence structure. The American Mathematical Monthly 65 (3) 154–170.CrossRefGoogle Scholar
Maieli, R. (2003). A new correctness criterion for multiplicative non-commutative proof-nets. In: Archive for Mathematical Logic, vol. 42, Berlin Heidelberg, Springer-Verlag, 205–220.Google Scholar
Maieli, R. (2007). Retractile proof nets of the purely multiplicative and additive fragment of linear logic. In: Proceedings of the 14th International Conference LPAR, LNAI, vol. 4790, Berlin Heidelberg, Springer-Verlag, 363–377.Google Scholar
Maieli, R. (2014). Construction of retractile proof structures. In: Dowek, G. (ed.) Proceedings of the International Joint Conference RTA-TLCA, Vienna, LNCS, vol. 8560, Switzerland, Springer International Publishing, 319–333.Google Scholar
Melliès, P.-A. (2004). A topological correctness criterion for multiplicative non-commutative logic. In: Ehrhard, T., Girard, J.-Y., Ruet, P., and Scott, P. (eds.) Linear Logic in Computer Science, London Mathematical Society Lecture Note, vol. 316, chapter 8, UK, Cambridge University Press, 283–321.CrossRefGoogle Scholar
Mogbil, V. (2001). Quadratic correctness criterion for noncommutative logic. In: Proceedings of CSL 2001, Paris, France, LNCS, vol. 2142, Berlin Heidelberg, Springer-Verlag, 69–83.Google Scholar
Moot, R. and Retoré, Ch. (2012). The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics, LNCS, vol. 6850, Berlin Heidelberg, Springer-Verlag.CrossRefGoogle Scholar
Moot, R. (2002). Proof Nets for Linguistic Analysis. Phd thesis, Utrecht University.Google Scholar
Retoré, C. (1996). Calcul de Lambek et logique linéaire. Traitement Automatique des Langues 37 (2) 39–70.Google Scholar
Roorda, D. (1992). Proof nets for Lambek calculus. Journal of Logic and Computation 2 (2) 21–233.CrossRefGoogle Scholar