Relativizing Small Complexity Classes and Their Theories

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 α 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...

• Type: Conference
• Year: 2007
• Where: CSL
• Authors: Klaus Aehlig, Stephen Cook, Phuong Nguyen
• Keywords: Corresponding Relativized Class, CSL 2007, Wilson’s Stack Oracle, first Definitions