PNP [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.