Separating Notions of Higher-Type Polynomial-Time

In his paper on machine characterizations of feasible functionals at all simple types, Seth [Set95] introduced the higher-type complexity class D and conjectured that this class is strictly larger than BFF, the basic feasible functionals of Cook and Urquhart [CU93]. In this paper we clarify the nature of D by characterizing D−, a subclass of D, through a typed programming formalism and show D− 6= BFF, confirming Seth’s conjecture. This separation result falls out of our program to better understand the basic feasible functionals on both complexity theoretic and semantic grounds. We are motivated by the following problems. In regards to complexity theory, Kapron and Cook [KC96] introduced (a) a notion of length for type-1 functions, (b) a notion of second-order polynomial, and (c) a type-2 machine model and they characterized the type-2 BFF’s as the class of functionals computable by the type-2 machines within a cost (second-order) polynomial in the lengths of the (type-1 and type-0) arguments. This parallels the standard definition of (type-1) complexity classes. In contrast, Seth’s machine characterization omits direct higher-type analogues of (a) and (b) and so we have only a partial complexity theoretic analysis of BFF. In regards to semantics, BFF is defined as the class of functionals computable by one of several programming formalism when inputs are drawn from Full, the full type hierarchy over N. But as Scott and Strachey [SS71] argued, it makes more computational sense to consider computation over continuous domains. In particular, there may be programming constructs that are not “feasible” with respect to Full, but are “feasible” with respect to (the much smaller and more structured) continuous domains. Our separation of D− and BFF hinges on such a construct. An Outline of Our Results. In the full paper [IKR00b] we introduce higher-type notions of length, polynomials, and machines for both the Full and continuous settings. Here we focus of the continuous setting and omit any treatment of machines. After establishing some notation, we first