Summary: Oracles Are Subtle But Not Malicious
Scott Aaronson #
University of Waterloo
Theoretical computer scientists have been debating the
role of oracles since the 1970's. This paper illustrates both
that oracles can give us nontrivial insights about the bar
rier problems in circuit complexity, and that they need not
prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear
sized circuits, by proving a new lower bound for perceptrons
and lowdegree threshold polynomials. This oracle settles
a longstanding open question, and generalizes earlier re
sults due to Beigel and to Buhrman, Fortnow, and Thierauf.
More importantly, it implies the first provably nonrelativiz
ing separation of ``traditional'' complexity classes, as op
posed to interactive proof classes such as MIP and MA EXP .
For Vinodchandran showed, by a nonrelativizing argument,
that PP does not have circuits of size n k for any fixed k.
We present an alternative proof of this fact, which shows