Feasible functionals and intersection of ramified types

We show that the basic feasible functions of Cook and Urquhart’s BFF [8,9] are precisely the functionals definable in a natural system of ramified recurrence that uses type intersection (for tier-variants of a common type). This further confirms the stability of BFF as a notion of computational feasibility in higher type. It also suggests the potential usefulness of type-intersection restricted to sort-variants of a common type. INTRODUCTION: COMPUTABILITY AND FEASIBILITY IN HIGHER TYPE Computable higher type functionals have been studied for about a century, for several intertwined reasons. One of the first to explicitly consider feasibility of functionals was Robert Constable, who in [6] introduced a machine model for functionals, and considered the definability of the functionals computable therein in a certain function algebra. 3 Melhorn [18] refined Constable’s algebraic approach by lifting to second order types the characterization given by Cobham [5] of the class FP of functions computable in polynomial time. A corresponding machine model was defined by Kapron and Cook in [13], and shown to be equivalent to Mehlhorn’s class. Another thread in the evolution of the subject was concerned with functional interpretation of proofs in Buss’s Bounded Arithmetic. In [2] Buss introduced a system IS 2 of arithmetic and showed that its definable functions form precisely FP. In [3] Buss considered the intuitionistic variant of IS 2 , and 1 Research partially supported by NSF grant CCR-0105651. 2 Email: [email protected] 3 See [4] for a correction. c ©2003 Published by Elsevier Science B. V.