Relativizing Small Complexity Classes and their Theories

Existing definitions of the relativizations of \NCOne, \L\ and \NL\ do not preserve the inclusions $\NCOne \subseteq \L$, $\NL\subseteq \ACOne$. We start by giving the first definitions that preserve them. Here for \L\ and \NL\ we define their relativizations using Wilson's stack oracle model, but limit the height of the stack to a constant (instead of $\log(n)$). We show that the collapse of any two classes in ${\ACZm, \TCZ, \NCOne, \L, \NL}$ implies the collapse of their relativizations. Next we exhibit an oracle $\alpha$ that makes $\ACk(\alpha)$ a proper hierarchy. This strengthens and clarifies the separations of the relativized theories in [Takeuti, 1995]. The idea is that a circuit whose nested depth of oracle gates is bounded by $k$ cannot compute correctly the $(k+1)$ compositions of every oracle function. Finally we develop theories that characterize the relativizations of subclasses of \Ptime\ by modifying theories previously defined by the second two authors. A function is provably total in a theory iff it is in the corresponding relativized class, and hence the oracle separations imply separations for the relativized theories.