Lower bounds for witness-finding algorithms

Can a procedure that decides whether a Boolean formula has a satisfying assignment help to find such an assignment? The naïve adaptive “search-todecision” reduction uses a linear number of (quite weak) queries. Is there a lower bound on the number of queries required for a nonadaptive search-to-decision reduction? We report on lower bounds for various classes of queries. Most interesting types of queries require at least Ω(n2) queries to find a witness. We were hoping to use lower bounds on these types of witness-finding algorithms to show that randomized reductions to C=P are less powerful than randomized reductions to PP, but we were unable to do so. Figure 1 contains a summary of the upper and lower bounds for nonadaptive witness-finding algorithms. In addition, adaptive Q0 algorithms use Θ(n) queries, and adaptive Q1 witness-finding algorithms with high success probability use Θ(n2) queries. TODO Currently the lower bound for the query complexity of adaptive Q1 witness-finding algorithms with success probability 1 2 is Ω(n). Can we either improve this to Ω(n2) or provide a O(n) upper bound? TODO Compare these randomized witness-finding algorithms with “bounded query hierarchy” algorithms, in which the query generator and the witness generator are not randomized (but possibly nondeterministic), and the goal is to decide a language, not necessarily output a witness.