Computability of Recursive Functions

As a result of the work of Turing, Post, Kleene and Church [1, 2, 3, 9, 10, l l 12, 17, 18] it is now widely accepted ~ that the concept of "computable" as applied to a function ~ of natural numbers is correctly identified with the concept of "partial recursive." One half of this equivalence, that all functions computable by any finite, discrete, deterministic device supplied with unlimited storage are partial recursive, is relatively straightforward 3 once the elements of recursive function theory have been established. All tha t is necessary is to number the configurations of machine-plus-storage medium, show that the changes of configuration number caused by each "move" are given by partial recursive functions, and then use closure properties of the class of partial recursive functions to deduce that the function computed by the complete sequence of moves is partial recursive. Until recently all proofs [4, 6, 12, 13, 19, 20] of the converse half of the equivalence, namely, that all partial recursive functions are computable, have consisted of proofs tha t all partial recursive functions can be computed by Turing machines, ~ which are certainly machines in the above sense. Although