Tree Adjoining Grammars in a Fragment of the Lambek Calculus

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 one tree inside another, surrounding the subtree at the adjunction node. This seems to contradict standard logical ability. We prove that some logical formalisms, namely a fragment of the Lambek calculus, can handle adjunction.We represent objects and operations of the TAG formalism in four steps: first trees (initial or derived) and the way they are constituted, then the operations (substitution and adjunction), and finally the elementary tree, i.e., the grammar. Trees (initial or derived) are 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).