Connections between the complexity of unique satisfiability and the threshold behavior of randomized reductions

The present research is motivated by new results on the complexity of the unique satisfiability problem (USAT). Some new results are obtained, using the concept of randomized reductions. The proofs use only the fact that USAT is complete for D^P under randomized reductions, even though the probability bound of these reductions may be low. Furthermore, the results show that the structural complexities of USAT and D^P many-one complete sets are very similar, lending support to the argument that even sets complete under `weak´ randomized reductions can capture the properties of the many-one complete sets. The authors generalize these results for the Boolean hierarchy and give upper and lower bounds on the thresholds for these classes