Recursion does not always help

06/16/2022

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by   Gordon Plotkin, et al.

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We show that, under mild assumptions, adding recursion does not increase the total functions definable in the typed λβη-calculus or (under no assumptions) the, possibly partial, functions definable in the λΩ-calculus. As a consequence, adding recursion does not increase the class of partial or total definable functions on free algebras and so, in particular, on the natural numbers.