Summary: Oracles Are Subtle But Not Malicious
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 low-degree 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 MAEXP.
For Vinodchandran showed, by a nonrelativizing argument,
that PP does not have circuits of size nk
for any fixed k.