Motivated by the question of how to define an analog of interactive
proofs in the setting of logarithmic time- and space-bounded
computation, we study complexity classes defined in terms of
operators quantifying over oracles. We obtain new
characterizations of $\NCe$, $\L$, $\NL$, $\NP$, and $\NSC$
(the nondeterministic version of SC). In some cases, we prove
that our simulations are optimal (for instance, in bounding the
number of queries to the oracle).