Let A be a set of functions. A classifier for A is a way of telling, g
iven a function f, if f is in A. We will define this notion formally.
We will then modify our definition in three ways: (1) allow the classi
fier to ask questions to an oracle A (thus increasing the classifiers
computational power), (2) allow the classifier to ask questions about
f (thus increasing the classifiers information access), and (3) restri
ct the number of times the classifier can change its mind (thus decrea
sing the classifiers information access). By varying these parameters
we will gain a better understanding of the contrast between computatio
nal power and informational access. We have determined exactly (1) whi
ch sets are classifiable (theorem 3.6), (2) which sets are classifiabl
e with queries to some oracle (theorem 3.2), (3) which sets are classi
fiable with queries to some oracle and queries about f (theorem 5.2),
and (4) which sets are classifiable with queries to some oracle, queri
es about f and a bounded number of mindchanges (theorem 5.2). The last
two items involve the Borel hierarchy.