On the density of honest subrecursive classes

Abstract

The relation of honest subrecursive classes to the computational complexity of the functions they contain is briefly reviewed. It is shown that the honest subrecursive classes are dense under the partial ordering of set inclusion. In fact, any countable partial ordering can be embedded in the gap between an effective increasing sequence of honest subrecursive classes and an honest subrecursive class which is properly above the sequence (or in the gap between an eflective decreasing sequence and a class which is properly below the sequence). Information is obtained about the possible existence of least upper bounds (greater lower bounds) of increasing (decreasing) sequences of honest subrecursive classes. Finally it is shown that for any two honest subrecursive classes, one properly containing the other, there exists a pair of incomparable honest subrecursive classes such that the greatest lower bound of the pair is the smaller of the first two classes and the least upper bound of the pair is the larger of the first two classes