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\begin{center}{\huge Ten Years of Bounded Queries in}\end{center}
\begin{center}{\huge Computability Theory}\end{center}
\begin{center}{\large Bill Gasarch}\end{center}
\begin{center}{\large The University of 
Maryland at College Park}\end{center}

\bigskip

For the last ten years there has been research on
the following question: How many queries (to various oracles)
are required to solve problems (of various sorts).
Here is a sample of the kinds of questions we have asked and
(mostly) answered.

1) Let $A$ be a set.
Given $n$ numbers clearly we can deterine 
WHICH ONES are in $A$ with $n$ queries to $A$.
For which $A$ can we do better?
How well can we do?

2) Let $A$ be a set.
Given $n$ numbers clearly we can deterine 
HOW MANY of them are in $A$
For which $A$ can we do better?
How well can we do?

3) Let $A$ be a set.
Given $n$ numbers clearly we can deterine 
if the number that are in $A$ is EVEN OF ODD.
For which $A$ can we do better?
How well can we do?

4) For which sets $A$ are there functions that can be computed 
with $n$ queries to $A$ that cannot be computed with $n-1$ queries to $A$.

5) Are there sets that can be decided with $n$ queries 
to $A$ that cannot be computed with $n-1$ queries

All of these questions can be varied by considering parallel queries,
sequential queries, and queries to sets other than $A$.
Techniques in this field have been a blend of combinatorics
and recursion theory.

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