Cyclic Multiplicative-Additive 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. It first provides a syntax for proof structures PSs of the cyclic multiplicative and additive fragment of linear logic CyMALL. A PS is an oriented graph, weighted by boolean monomial weights, whose conclusions $$\varGamma $$ are endowed with a cyclic order $$\sigma $$. Roughly, a PS $$\pi $$ with conclusions $$\sigma \varGamma $$ is correct so, it is a proof net, if any slice $$\varphi \pi $$, obtained by a boolean valuation $$\varphi $$ of $$\pi $$, is a multiplicative CyMLL PN with conclusions $$\sigma \varGamma _r$$, where $$\varGamma _r$$ is an additive resolution of $$\varGamma $$, i.e. a choice of an additive subformula for each formula of $$\varGamma $$. The 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, that is, a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Unlike the most part of current syntaxes for non-commutative PNs, our syntax allows a sequentialization for the full class of CyMALL PNs, without requiring these latter to be cut-free. One of the main contributions of this paper is to provide a characterization of CyMALL PNs for the additive Lambek Calculus and thus a geometrical non inductive way to parse sentences containing words with syntactical ambiguity i.e., with polymorphic type.