Complete and tractable machine-independent characterizations of second-order polytime

The class of Basic Feasible Functionals BFF is the second-order counterpart of the class of first-order functions computable in polynomial time. We present several implicit characterizations of BFF based on a typed programming language of terms. These terms may perform calls to imperative procedures, which are not recursive. The type discipline has two layers: the terms follow a standard simply-typed discipline and the procedures follow a standard tier-based type discipline. BFF consists exactly of the second-order functionals that are computed by typable and terminating programs. The completeness of this characterization surprisingly still holds in the absence of lambda-abstraction. Moreover, the termination requirement can be specified as a completeness-preserving instance, which can be decided in time quadratic in the size of the program. As typing is decidable in polynomial time, we obtain the first tractable (i.e., decidable in polynomial time), sound, complete, and implicit characterization of BFF, thus solving a problem opened for more than 20 years.