P^NP [log n] and Sparse Turing-Complete Sets for NP

pNP[log n] is the class of languages recognizable by deterministic polynomial time machines that make O(log n) queries to an oracle for NP. Our main result is that if there exists a sparse set S ∈ NP such that co - NP ⊆ NP ^s , then the polynomial hierarchy (PH) is contained in P ^{NP[log n]} . Thus if there exists a sparse ≤ $_\text{T}^\text{P}$ -complete set for NP, PH ⊆ P ^{NP[log n]} We show that this collapse is optimal by showing for any function f(n) with f(n) < O(log n), there exists a relativized world where NP has a sparse ≤ $_\text{T}^\text{P}$ -complete set and yet P ^{NP[log n]} ⊈ P ^NP[f(n)] . We also discuss complete problems for P ^{NP[log n]} and show languages related to the optimal solution size of Clique and K-SAT are ≤ $_\text{m}^\text{p}$ -complete. In related research, we give a characterization of the sets C for which P ^{C[log n]} equals the class of languages ≤ $_\text{m}^\text{p}$ -reducible to C.