**Authors:** Richard Beigel

**Abstract:**
Let *A* be any nonrecursive set. For each *k* ≥ 0, we show
that there are functions (*k*+1)-truth-table reducible to
*A* that are not *k*-weak-truth-table reducible to *A*;
and we show that there are functions (*k*+1)-Turing reducible to
*A* that are not *k*-Turing reducible to *A*.
Intuitively, this means that *k*+1 queries to a nonrecursive
oracle are strictly more useful than *k* queries to the same
oracle (for the sake of computing functions), and that *k*+1
oracle answers from a nonrecursive oracle provide information that
cannot be provided by any *k* answers from the same oracle.

The situation is different for decision problems. We say that a set
*A* is *parallel-supportive* if, for all *k*, there is
a language that is (*k*+1)-wtt reducible to *A* but not
*k*-wtt reducible to *A*. A set *A* is
*supportive* if, for all *k*, there is a language that is
(*k*+1)-Turing reducible to *A* but not *k*-Turing
reducible to *A*. We show that every degree above **0'**
contains a set that is supportive but not parallel-supportive. We
show that every hyperimmune-free degree contains a set that is neither
supportive nor parallel-supportive. We also construct a subclass of
the hyperimmune degrees with the same property.

On the other hand we show that parallel-supportive and supportive sets
exist in abundance: all jumps of sets, all generic sets, all
nonrecursive semi-recursive sets, all nonrecursive r.e. sets, all
nonrecursive *n*-r.e. sets, and almost all sets are supportive and
parallel-supportive. In particular every nonrecursive Turing degree
contains a set that is parallel-supportive and supportive. We also
show that all nonrecursive nonsuperterse sets are supportive.