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TR05-040 | 13th April 2005 00:00

Oracles Are Subtle But Not Malicious


Authors: Scott Aaronson
Publication: 13th April 2005 22:14
Downloads: 1389


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 barrier 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 results due to Beigel and to
Buhrman, Fortnow, and Thierauf. More importantly, it implies the
first nonrelativizing separation of "traditional" complexity classes,
as opposed 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 that PP does
not even have quantum circuits of size n^k with quantum advice. To
our knowledge, this is the first nontrivial lower bound on quantum
circuit size.

Second, we study a beautiful algorithm of Bshouty et al. for learning
Boolean circuits in ZPP^NP. We show that the NP queries in
this algorithm cannot be parallelized by any relativizing technique,
by giving an oracle relative to which ZPP^||NP and even BPP^||NP
have linear-size circuits. On the other hand, we also show that the
NP queries could be parallelized if P=NP. Thus, classes such as
ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this
subject have implications for computational learning theory as well
as for the circuit minimization problem.

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