Adjunction As Substitution: An Algebraic Formulation of Regular Context-Free and Tree Adjoining Languages

There have been many attempts to give a coherent formulation of a hierarchical progression that would lead to a refined partition of the vast area stretching from the context-free to the context-sensitive languages. The purpose of this note is to describe a theory that seems to afford a promising method of interpreting the tree adjoining languages as the natural third step in a hierarchy that starts with the regular and the context-free languages. The formulation of this theory is inspired by two sources: the categorical concept of an algebraic theory and the powerful tool of macro variables which is well established within the framework of program schemes. The rough idea is that according to their intended interpretation objects of algebraic theories are sets of derived operations and that macro variables range over these sets. Guided by this conception we show how monadic macro variables provide a perspective from which both the context-free and the tree adjoining languages become realizations of the same general notion when the relevant underlying algebra is specified. The context-free languages are determined by an inductive process in which monadic macro variables are replaced by derived operations of an algebra all of whose operations are unary. In the case of the tree adjoining languages the same substitution process is applied to derived operations over an arbitrary algebra. The central notion in this account is that of a higher-order substitution. Whereas in traditional presentations of rule systems for language families the emphasis has been on a first-order substitution process in which auxiliary variables are replaced by elements of the carrier of the proper algebra— concatenations of terminal and auxiliary category symbols in the string case—we lift this process to the level of operations defined on the elements of the carrier of the algebra. Our own view is that this change of emphasis provides the adequate platform for a better understanding of the operation of adjunction. To put it in a nutshell: Adjoining is not a first-order, but a second-order substitution operation. This is not the first place that macro productions are put to use outside the field of program schemes. It has been known since the pioneering work of M.J. Fischer that macro grammars (without the restriction to monadic variables) are weakly equivalent to indexed grammars. The paper by Maibaum also contains one direction of the equivalence between context-free grammars