Proof nets for multiplicative cyclic linear logic and Lambek calculus

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 linear logic. 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.

[1]  Rob J. van Glabbeek,et al.  Proof nets for unit-free multiplicative-additive linear logic , 2005, TOCL.

[2]  V. Michele Abrusci,et al.  Cyclic Multiplicative Proof Nets of Linear Logic with an Application to Language Parsing , 2015, WoLLIC.

[3]  Richard Moot,et al.  The Logic of Categorial Grammars , 2012, Lecture Notes in Computer Science.

[4]  Vincent Danos.  La Logique Linéaire appliquée à l'étude de divers processus de normalisation (principalement du Lambda-calcul) , 1990 .

[5]  T. Jandrok,et al.  Le point aveugle , 2009 .

[6]  V. Michele Abrusci,et al.  Cyclic Multiplicative-Additive Proof Nets of Linear Logic with an Application to Language Parsing , 2015, FG.

[7]  C. Retoré,et al.  Handsome Non-Commutative Proof-Nets: perfect matchings, series-parallel orders and Hamiltonian circuits , 2004 .

[8]  Stefano Guerrini,et al.  A linear algorithm for MLL proof net correctness and sequentialization , 2011, Theor. Comput. Sci..

[9]  Roberto Maieli,et al.  Retractile Proof Nets of the Purely Multiplicative and Additive Fragment of Linear Logic , 2007, LPAR.

[10]  Virgile Mogbil,et al.  Correctness of Multiplicative (and Exponential) Proof Structures is NL -Complete , 2007, CSL.

[11]  J. Girard.  PROOF-NETS : THE PARALLEL SYNTAX FOR PROOF-THEORY , 1996 .

[12]  Vincent Danos,et al.  The structure of multiplicatives , 1989, Arch. Math. Log..

[13]  C. Retoré.  Calcul de Lambek et logique linéaire , 1996 .

[14]  Roberto Maieli.  Construction of Retractile Proof Structures , 2014, RTA-TLCA.

[15]  Richard Moot.  Proof nets for lingusitic analysis , 2002 .

[16]  Christian Retoré.  A Semantic Characterisation of the Correctness of a Proof Net , 1997, Math. Struct. Comput. Sci..

[17]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[18]  Richard Moot.  The Logic of Categorial Grammars: A deductive account of natural language syntax and semantics , 2012 .

[19]  V. Michele Abrusci.  Classical Conservative Extensions of Lambek Calculus , 2002, Stud Logica.

[20]  Virgile Mogbil.  Quadratic Correctness Criterion for Non-commutative Logic , 2001, CSL.

[21]  J. Girard,et al.  1 A topological correctness criterion for multiplicative non-commutative logic , 2016 .

[22]  Alexis Saurin,et al.  On the Dependencies of Logical Rules , 2015, FoSSaCS.

[23]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[24]  Roberto Maieli.  A new correctness criterion for multiplicative non-commutative proof nets , 2003, Arch. Math. Log..

[25]  Paul Ruet,et al.  Non-Commutative Logic I: The Multiplicative Fragment , 1999, Ann. Pure Appl. Log..

[26]  Dirk Roorda.  Proof Nets for Lambek Calculus , 1992, J. Log. Comput..