On the random-self-reducibility of complete sets

Informally, a function f is random-self-reducible if the evaluation of f at any given instance x can be reduced in polynomial time to the evaluation of f at one or more random instances y_i. A set is random-self-reducible if its characteristic function is. The authors generalize the previous formal definitions of random-self-reducibility. They show that, even under this very general definition, sets that are complete for any level of the polynomial hierarchy are not random-self-reducible, unless the hierarchy collapses. In particular, NP-complete sets are not random-self-reducible, unless the hierarchy collapses at the third level. By contrast, the authors show that sets complete for the classes PP and MOD_mP are random-self-reducible