Some connections between bounded query classes and non-uniform complexity

Let A(x) be the characteristic function of A. Consider the function CkA(x1,...,xk) = A(x1)...A(xk). We show that if CkA can be computed in polynomial time with fewer than k queries to some set X then A ∈ P/poly. A generalization of this result has applications to bounded query classes, circuits, and enumerability. In particular we obtain the following. (1) Assuming Σ3p ≠ Φ3p, there are functions computable using f(n) + 1 queries to SAT that are not computable using f(n) queries to SAT, for f(n) = O(log n). (2) If CkA, restricted to length n inputs, can be computed by an unbounded fanin oracle circuit of size s(n) and depth d(n), with k - 1 queries to some set X, then A can be computed with an unbounded fanin (non-oracle) circuit of size nO(k) s(n) and depth d(n) + O(1). (3) Assuming that PH ≠ Σ4p ∩ Φ4p, and e > 1, #SAT is not 2ne-enumerable.