Relativizing small complexity classes and their theories

Existing definitions of the relativizations of NC 1 , L and NL do not preserve the inclusions NC 1 ⊆ L , NL ⊆ 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 { AC 0 ( m ) , TC 0 , NC 1 , L , 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.