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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.

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