On one query self-reducible sets

We study one word-decreasing self-reducible sets, which were introduced by Lozano and Torán. These are usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query and this is a word lexico-graphically smaller than the input. We show first 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 we can prove that for any class K chosen from {NP, PP, C [sub =] P, MOD [sub 2] P, MOD [sub 3] P, ...} it holds that (1) if there is a sparse [= sub btt super P] -hard set for K then K = P, and (2) if there is a sparse [= sub btt super SN] -hard set for K then K C NP ¿ co-NP. The main result also shows that the same facts hold for the class PSPACE. This generalizes a recent result from Ogiwara and Watanabe to the mentioned complexity classes.Postprint (published version