  A set is autoreducible if it can be reduced to itself by a Turing machine
that does not ask its own input to the oracle. We use autoreducibility to
separate the polynomial-time hierarchy from polynomial space by showing that
all Turing-complete sets for certain levels of the exponential-time hierarchy
are autoreducible but there exists some Turing-complete set for doubly
exponential space that is not.
  Although we already knew how to separate these classes using diagonalization,
our proofs separate classes solely by showing they have different structural
properties, thus applying Post's Program to complexity theory. We feel such
techniques may prove unknown separations in the future. In particular, if we
could settle the question as to whether all Turing-complete sets for doubly
exponential time are autoreducible, we would separate either polynomial time
from polynomial space, and nondeterministic logarithmic space from
nondeterministic polynomial time, or else the polynomial-time hierarchy from
exponential time.
  We also look at the autoreducibility of complete sets under nonadaptive,
bounded query, probabilistic and nonuniform reductions. We show how settling
some of these autoreducibility questions will also read to new complexity class
separations.
