  We study reductions that limit the extreme adaptivity of Turing reductions.
In particular, we study reductions that make a rapid, structured progression
through the set to which they are reducing: Each query is strictly longer
(shorter) than the previous one. We call these reductions query-increasing
(query-decreasing) Turing reductions. We also study query-nonincreasing
(query-nondecreasing) Turing reductions. These are Turing reductions in which
the sequence of query lengths is nonincreasing (nondecreasing). We ask whether
these restrictions in fact limit the power of reductions. We prove that
query-increasing and query-decreasing Turing reductions are incomparable with
(that is, are neither strictly stronger than nor strictly weaker than)
truth-table reductions and are strictly weaker than Turing reductions. In
addition, we prove that query-nonincreasing and query-nondecreasing Turing
reductions are strictly stronger than truth-table reductions and strictly
weaker than Turing reductions. Despite the fact that we prove query-increasing
and query-decreasing Turing reductions to in the general case be strictly
weaker than Turing reductions, we identify a broad class of sets A for which
any set that Turing reduces to A will also reduce to A via both
query-increasing and query-decreasing Turing reductions. In particular, this
holds for all tight paddable sets, where a set is said to be tight paddable
exactly if it is paddable via a function whose output length is bounded tightly
both from above and from below in the length of the input. We prove that many
natural NP-complete problems such as satisfiability, clique, and vertex cover
are tight paddable.
