Computability and λ-definability

Abstract

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ- K -definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ- K -definable function is computable; that every λ-definable function is λ- K -definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ- K -definability. A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.