Cyclic multiplicative-additive proof nets of linear logic with an application to language parsing
Résumé
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 syntax for PNs of the cyclic multiplicative and additive fragment of linear logic (CyMALL). Any proof structure (PS), in Girards style, is weighted by boolean monomial weights, moreover, its conclusions Γ (a sequence of formulas occurrences) are endowed with a cyclic order σ, i.e., σ(Γ). Naively, a CyMALL PS π with conclusions σ(Γ) is correct if, for any slice ϕ(π) (obtained by a boolean valuation ϕ of π) there exists an additive resolution (i.e. a multiplicative refinement of ϕ(π)) that is a CyMLL PN with conclusions σ(Γr), where Γr is an additive resolution of Γ (i.e. a choice of an additive subformula for each formula of Γ). In its turn, 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, 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 characterization of CyMALL PNs for the extended (MALL) Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences. In particular additive Lambek PNs allow to parse phrases containing words with syntactical ambiguity (i.e. words with polymorphic type).
Origine : Fichiers produits par l'(les) auteur(s)
Loading...