Minimal pairs of polynomial degrees with subexponential complexity

The goal of extending work on relative polynomial time computability from computations relative to sets of natural numbers to computations relative to arbitrary functions of natural numbers is discussed. The principal techniques used to prove that the honest subrecursive classes are a lattice are then used to construct a minimal pair of polynomial degrees with subexponential complexity; that is two sets computable by Turing machines in subexponential time but not in polynomial time are constructed such that any set computable from both in polynomial time can be computed directly in polynomial time.