Cyclic Multiplicative Proof Nets of Linear Logic with an Application to Language Parsing

This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e. de-sequentialized proofs, of linear logic (LL). In particular, it presents a simple and intuitive syntax for PNs of the cyclic multiplicative fragment of linear logic (CyMLL). The proposed correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of LL. The main intuition relies on the fact that any DR-switching (i.e. any correction or test graph for a given PN) can be naturally viewed as a seaweed, i.e. a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Dislike the most part of current syntaxes for non-commutative PNs, our syntax allows a sequentialization for the full class of CyMLL PNs, without requiring these latter must be cut-free. Moreover, we give a simple characterization of CyMLL PNs for Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences by means of Lambek PNs.