Wikipedia states on sparse languages that There is a Turing reduction (as opposed to the Karp reduction from Mahaney's theorem) from a NP-complete language to a sparse language iff NP $\subseteq$ P/poly. Is that correct in that it is so far proven for an arbitrary Turing reduction and not a limited P-time reduction? (Could it be true for P-time reductions also, but maybe not proven so far?) It cites this without reference. What is a reference for this?
A detail on variant of Mahaney's theorem about reductions of sparse languages vs P/NP
complexity theory;reference request;np complete;reductions;p vs np
No, what you suggest would be meaningless. The Turing reduction is also polytime. For a proof, see Theorem 3.2 in Oded Goldreich's lecture notes.