Natural Deduction and Normalisation for Partially Commutative Linear Logic and Lambek Calculus with Product

Maxime Amblard (1, 2) , Christian Retoré (1, 2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 SIGNES - Linguistic signs, grammar and meaning: computational logic for natural language

This paper provides a natural deduction system for Partially Commutative Intuitionistic Multiplicative Linear Logic (PCIMLL) and establishes its normalisation and subformula property. Such a system in- volves both commutative and non commutative connectives and deals with context that are series-parallel multisets of formulæ. This calcu- lus is the extension of the one introduced by de Groote presented by the second order for modelling Petri net execution, with a full entropy which allow order to be relaxed into any suborder — as opposed to the Non Commutative Logic of Abrusci and Ruet. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Up to now PCIMLL with full entropy had no natural deduction. In particular for linguistic applications, such a syntax is much welcome to construct semantic representations from syntactic analyses.

Mots clés: logi; proof theory; linear logic; Lambek calculus

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Maxime Amblard, Christian Retoré. Natural Deduction and Normalisation for Partially Commutative Linear Logic and Lambek Calculus with Product. Computation and Logic in the Real World (Computing in Europe 2007), Jun 2007, Siena, Italy. pp.28--44. ⟨hal-00409486⟩