A Topological Study of Phrase-Structure Languages

It is proposed that structural equivalence of phrase-structure languages be defined by means of introducing, for each such language, a class of topological structures on the language. More specifically, given a phrase-structure language (either as a set of trees or as a set of strings), we introduce a class of topological spaces associated with finite sets of "phrases." A function from one language to another, where both are equipped with such classes of topological spaces, is said to be structurally continuous, if for any topological space belonging to the first, there is a space belonging to the second such that the function is continuous with respect ~o these spaces. Then phrase-structure languages, or grammars that generate such languages, may be classified into structurally homeomorphic types in the obvious way. Two different methods of topologizing phrase-structure languages (one dependent on the other) are considered, and it is shown that for the class of context-free languages, one method provides a finer classification of languages (or grammars) than the other. In Part 2 we apply the general theory to a particular subclass of context-free languages, the class of tree language counterparts of regular languages.