Relativizing small complexity classes and their theories

Existing definitions of the relativizations of NC 1, L and NL do not preserve the inclusions $${{\bf NC}^1 \subseteq {\bf L}, {\bf NL}\subseteq {\bf AC}^1}$$ . 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 $${\{{\bf AC}^0 (m), {\bf TC}^0, {\bf NC}^1, {\bf L}, {\bf NL}\}}$$ implies the collapse of their relativizations. Next we exhibit an oracle α that makes AC k (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. Finally, 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, and hence, the oracle separations imply separations for the relativized theories.