Abstract
Existing definitions of the relativizations of NC1, L and NL do not preserve the inclusions NC1 ⊆ L, NL ⊆ AC1. 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 {AC0(m), TC0, NC1, L, NL} implies the collapse of their relativizations. Next we develop theories that characterize the relativizations of subclasses of P 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. Finally we exhibit an oracle a that makes ACk(α) 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.
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