Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Abstract

Weir [34] proved a Chomsky-Schutzenberger-like representation theorem for the string languages of tree-adjoining grammars, where the Dyck language Dn in the Chomsky-Schutzenberger characterization is replaced by the intersection D2n ∩ g(D2n), where g is a certain bijection on the alphabet consisting of 2n pairs of brackets. This paper presents a generalization of this theorem to the string languages that are the yield images of the tree languages generated by simple (i.e., linear and non-deleting) context-free tree grammars. This result is obtained through a natural generalization of the original Chomsky-Schutzenberger theorem to the tree languages of simple context-free tree grammars. We use Rogers’s [24,23] notion of multi-dimensional trees to state this latter theorem in a very general, abstract form.