Ramified recurrence and computational complexity III: Higher type recurrence and elementary complexity Original Research Article

We consider the functionals defined using an extension to higher types of ramified recurrence, which was introduced independently in [4,18,21] and [35]. Three styles of functional programs over free algebras are examined: equational recurrence, applicative programs with recurrence operators and purely applicative higher-type programs. We show that for every free algebra A and each one of these styles, the functions defined by ramified recurrence in finite types are precisely the functions over A computable in a number of steps elementary in the size of the input. This should be contrasted with unrestricted higher type recurrence which yields, for numeric computing, all provably recursive functions of first order arithmetic. This paper is revised and expanded from the proceedings paper [23]. The research project of which it is part is closely related to Rohit Parikhʹs longstanding interest in conceptual delineation of feasibility.