 
Summary: Relativizing Small Complexity Classes and their
Theories
Klaus Aehlig , Stephen Cook, and Phuong Nguyen
University of Toronto
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
