Comparing Reductions to NP-Complete Sets

Under the assumption that NP does not have p-measure 0, we investigate reductions to NP-complete sets and prove the following: 1. Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turing-complete for NP but not truth-table-complete. 2. Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNP-complete for NP but not Turing-complete. 3. Every problem that is many-one complete for NP is complete under length-increasing reductions that are computed by polynomial-size circuits. The first item solves one of Lutz and Mayordomo’s “Twelve Problems in Resource-Bounded Measure” (1999). We also show that every many-one complete problem for NE is complete under one-to-one, length-increasing reductions that are computed by polynomial-size circuits.