Tree Adjoining Grammars in Noncommutative Linear Logic

This paper presents a logical formalization of Tree-Adjoining Grammar (TAG). TAG deals with lexicalized trees and two operations are available: substitution and adjunction. Adjunction is generally presented as an insertion of a tree inside another, surrounding the subtree at the adjunction node. This seems to be contradictory with standard logical ability. We prove that some logic, namely a fragment of non-commutative intuitionistic linear logic (N-ILL), can serve this purpose. Briefly speaking, linear logic is a logic considering facts as resources. NILL can then be considered either as an extension of Lambek calculus, or as a restriction of linear logic. We model the TAG formalism in four steps: trees (initial or derived) and the way they are constituted, the operations (substitution and adjunction), and the elementary trees, i.e. the grammar. The sequent calculus is a restriction of the standard sequent calculus for N-ILL. Trees (initial or derived) are then obtained as the closure of the calculus under two rules that mimic the grammatical ones. We then prove the equivalence between the language generated by a TAG grammar and the closure under substitution and adjunction of its logical representation. Besides this nice property, we relate parse trees to logical proofs, and to their geometric representation: proofnets. We briefly present them and give examples of parse trees as proofnets. This process can be interpreted as an assembling of blocks (proofnets corresponding to elementary trees of the grammar).