On one query, self-reducible sets

The authors study one-word-decreasing self-reducible sets, which are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query to a word lexicographically smaller than the input. It is first shown that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact it is proved that, for any class K chosen from a certain set of complexity classes, it holds that (1) if there is a sparse polynomial-time bounded-truth-table-hard set for K, then K=P, and (2) if there is a sparse strongly nondeterministic bounded-truth-table-hard set for K, then K⊆NP∩co-NP. The main result also shows that the same facts hold for the class PSPACE