Characterizing small depth and small space classes by operators of higher types

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 NC1, 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)