On the Generative Power of

Mathematical modeling of phrase structure grammars has yielded many results of benefit to linguists in their investigation of these grammars, such as Chomsky’s characterization in terms of self-embedding of those context-free languages which are not regular. The recent shift of focus in linguistic theory to transformational grammars has not been accompanied by a similar application of mathematical techniques to transformations. Our present purpose is to foster such studies by providing general definitions which model grammatical transformations as mappings on trees (equivalently, labeled bracketings) and investigating questions of current linguistic interest, such as the recursiveness of languages generated by transformational grammars. The first result of our research is that, despite the linguistically motivated, complex restrictions placed on transformational grammars, every recursively enumerable set of strings is a transformational language (Theorem 5.1). We demonstrate that this power of transformational grammars to generate non-recursive languages results from their ability to cycle their rules, applying transformations an unbounded number of times (Corollary 6.6). Analysis of decision procedures for grammars with bounded cycling reveals a connection between the amount of cycling permitted by a grammar and the complexity of the recursive set it generates; if cycling is bounded by any elementary recursive function (primitive recursive function, function in &? for n > 3), then the language generated has characteristic function in the same class (Corollary 6.7). One application of these results provides empirical support for the notion that natural languages are recursively, in fact elementarily, decidable. Our results also isolate one feature which must be further restricted in a linguistically motivated way if transformational theory is to achieve its goal of delimiting precisely the natural languages.