Oracles Are Subtle But Not Malicious

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Oracles Are Subtle But Not Malicious

• Scott Aaronson
• University of Waterloo

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Standard Whine
In the 60 years since Shannons counting
argument, we havent proven a superlinear circuit
lower bound for any explicit Boolean function!
Depends what we mean by explicit!
Kannan 1982 ?2 ? SIZE(n)
Köbler Watanabe 1998 ZPPNP ? SIZE(n)
Vinodchandran 2004 PP ? SIZE(n)
On the other hand, there are oracles where PNP
and MA have linear-size circuits
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My Whole Paper In One Slide
Oracle Results
Non-Oracle Results
In the real world, PP doesnt even have quantum
circuits of size nk, for any constant k
There exists an oracle relative to which PP has
linear-size circuits
PP and Perceptrons
Parallel NP and Learning
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Oracle Where PP?SIZE(n)Sales Pitch
In the real world, PP?SIZE(n)One of the only
nonrelativizing separations we have
I also get an oracle where PEXP ? P/poly ? PNP ?
?P
Subsumes several previous oraclesPNP ? PP
(Beigel) PP ? PSPACE (Aspnes et al.)MAEXP ?
P/poly (BFT) PNP NEXP (Buhrman et al.)
Same techniques yield a lower bound for k
perceptrons solving k ODDMAXBIT instances
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PP?BQSIZE(nk)Sales Pitch
Gives a new, self-contained proof of
Vinodchandrans result
First nontrivial quantum circuit lower bound
(outside the black-box model)
Along the way, I prove a Quantum Karp-Lipton
Theorem If PP has small quantum circuits, then
the Counting Hierarchy collapses to QMA
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Oracle Where Sales
Pitch
My result shows that their algorithm cant be
parallelized by any relativizing technique
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Learning Circuits In If PNPSales Pitch
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BATTLE MAP
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Alright, enough infomercial. Time for an oracle
where PP has linear-size circuits
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M1,M2, Enumeration of PP machines(Actually
PTIME(nlogn) machines) Goal Create an oracle
string A such thathave small circuits on inputs
of size nThen every Mi will be taken care of on
all but finitely many n, modulo a technicality
Idea Pick a random row, and encode there what
every Mi does on every input x
Then our linear-size circuit to simulate M1,,Mn
will just hardwire the location of that row
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Problem The PP machines are also watching the
oracle! As soon as we write down what the PP
machines do, we might change what they do, and so
on ad infinitum
Call a row r sensitive, if theres some change to
r that affects whether some PP machine accepts
some input
Strategy Define a progress measure Qgt0. Show
that (1) As long as every row is sensitive,
we can change the oracle string in a way that at
least doubles Q (2) Q can only double
finitely many times
Eventually, then, there must be an insensitive
rowand as soon as there is, we win!
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A WAR OF ATTRITION
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Our Weapon Polynomials
Let pi,x accepting paths of Mi(x)
? rejecting paths of Mi(x)
Basic Facts pi,x(A) is a multilinear polynomial
in the oracle bits, of degree at most nlogn
pi,x(A) ? 0 ? Mi(x) accepts
Lemma (follows from Nisan Szegedy 1994)Let p
be a multilinear polynomial in the bits of A.
Suppose there are ?(deg(p)2) rows we can modify
so as to change sgn(p(A)). Then there exists a
set of rows we can modify so as to at least
double p(A)
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What should the progress measure Q be?
First idea
Suppose every row is sensitive. Then by
pigeonhole, there exists an Mi(x) thats
sensitive to ?23n rows.
Is it possible that, whenever pi,x(A) becomes
large, the product of all the other terms somehow
covers for it by becoming small?
Meaning There are 23n rows we can modify so as
to change sgn(pi,x(A)) Hence, by Nisan-Szegedy,
theres a set of rows we can modify so as to
double pi,x(A)
Problem Our goal was to double Q(A)!
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Better idea
As before, theres some Mi(x) thats sensitive to
23n rows Let bMi(x) and k?log2pi,x(A)?. Then
well think of Q(A) as the product of two
polynomials
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By Nisan-Szegedy, that forces v(A) to become
large later on
u(A) increases sharply when we modify one row
v(A)
u(A)
This time a cushion keeps u(A) from getting too
small
To prevent Q(A)u(A)v(A) from doubling, v(A)
needs to nosedive
Number of modified rows
The product Q(A)u(A)v(A) is thereby forced to
double
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Open Problems
Prove better nonrelativizing circuit lower
bounds! (duhhhh...)
Does PEXP require exponential-size
circuits? Right now, can show it requires
circuits of half- exponential sizei.e. size f(n)
where f(f(n))2n
Bshouty et al.s algorithm only finds a circuit
within an O(n/log n) factor of minimal. Can we
improve this, or else give evidence that its
optimal?