Inhomogeneities in the Polynomial-Time Degrees: The Degrees of Super Sparse Sets

The polynomial-time bounded counterparts of Turing (T) and many-one (m) reducibility introduced by Cook and Karp, respectively, provide a useful tool for classifying solvable but intractable problems according to their relative complexity. These concepts proved to be of particular value for classifying problems in NP. Ladner [5] started the study of the polynomial-time (p-) degrees induced by these reducibilities. For instance, he showed that the partial orderings R p and R p of the p-degrees (w.r.t. p-mand p-T-reducibility, respectively) of recursive sets form dense upper semilattices (usl) but not lattices. Moreover, assuming P :~ NP, the classes NPm p and NPx p of p-degrees of NP-sets form infinite dense usl's too. Ladner obtained his results on the degrees of NP-sets by global arguments: To show that NPm p or NP p has a certain property Q, he showed that, for any recursive set A ~ P, property Q applies to the degree classes