The classic theory of computation initiated by Turing and his contemporaries provides a theory of effective procedures - algorithms that can be executed by the human mind, deploying cognitive processes constituting the conscious rule interpreter. The cognitive processes constituting the human intuitive processor potentially call for a different theory of computation. Assuming that important functions computed by the intuitive processor can be described abstractly as symbolic recursive functions and symbolic grammars, we ask which symbolic functions can be computed by the human intuitive processor, and how those functions are best specified - given that these functions must be computed using neural computation. Characterizing the automata of neural computation, we begin the construction of a class of recursive symbolic functions computable by these automata, and the construction of a class of neural networks that embody the grammars defining formal languages.