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Monotonous and randomized reductions to sparse sets. (English) Zbl 0860.68051

Summary: An oracle machine is called monotonous, if after a negative answer the machine does not ask further queries to the oracle. For example, one-query truth-table, conjunctive, and Hausdorff reducibilities are monotonous. We study the consequences of the existence of sparse hard sets for different complexity classes under monotonous and randomized reductions. We prove trade-offs 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.

MSC:

68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)

Keywords:

oracle machine
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