Computational foundations of basic recursive function theory

The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and theorems in recursion theory may be classified according to which axioms of computation are needed to prove them. The theory is about typed functions over the natural numbers, and it includes theorems showing that there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept, so important in computer science, serves to distinguish between classical and constructive type theories (in a different way than does the decidability concept as expressed in the law of excluded middle). The implications of these ideas for the logical foundations of computer science are discussed, particularly in the context of recent interest in using constructive type theory in programming.

Authors

• Robert L. Constable
• Scott F. Smith

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