Monotonous and Randomized Reductions to Sparse Sets

An oracle machine is called monotonous, if after a negative answer the machine does not ask further queries to the oracle. For example, one truth-table, conjunctive, and Hausdorr reducibilities are monotonous. We study the consequences of the existence of sparse hard sets for diierent complexity classes under monotonous and randomized reductions. We prove trade-oos between the randomized time complexity of NP sets that reduce to a set B via such reductions and the density of B as well as the number of queries made by the monotonous reduction. As a consequence, bounded Turing hard sets for NP are not co-rp reducible to a sparse set unless RP = NP. We also prove similar results under the apparently weaker assumption that some solution of the promise problem (1SAT; SAT) reduces via the mentioned reductions to a sparse set.