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 proposed correctness criterion for McyLL proof nets can be considered as the non-commutative counterpart of the famous Danos contraction criterion for proof nets of the pure multiplicative fragment (MLL) of LL. Unlike the most part of current syntaxes for non-commutative proof nets, the proposed syntax allows a direct sequentialization for the full class of McyLL PNs, without requiring these latter must be cut-free. 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.