The recent work on automata whose variables and parameters are real numbers (e.g., Blum, Shub, and Smale, 1989; Koiran, 1993; Bournez and Cosnard, 1996; Siegelmann, 1996; Moore, 1996) has focused largely on questions about computational complexity and tractability. It is also revealing to examine the metric relations that such systems induce on automata via the natural metrics on their parameter spaces. This brings the theory of computational classification closer to theories of learning and statistical modeling which depend on measuring distances between models. With this in mind, I develop a generalized method of identifying pushdown automata in one class of real-valued automata. I show how the real-valued automata can be implemented in neural networks. I then explore the metric organization of these automata in a basic example, showing how it fleshes out the skeletal structure of the Chomsky Hierarchy and indicates new approaches to problems in language learning and language typology.