Oracles Are Subtle But Not Malicious

Oracles Are Subtle But Not Malicious Scott Aaronson University of Waterloo

Where exactly do these results hit a brick wall—and can we knock it down? Standard Whine “In the 60 years since Shannon’s counting argument, we haven’t 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…

If P=NP, then we could exactly learn any poly-size circuit C in There’s an oracle where has linear-size circuits My Whole Paper In One Slide Oracle Results Non-Oracle Results In the real world, PP doesn’t 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

(I thereby solve four open problems of ) 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 oracles:PNP PP (Beigel) PP  PSPACE (Aspnes et al.)MAEXP P/poly (BF&T) PNP = NEXP (Buhrman et al.) Same techniques yield a lower bound for k perceptrons solving k ODDMAXBIT instances

PPBQSIZE(nk):Sales Pitch Gives a new, self-contained proof of Vinodchandran’s 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

Bshouty et al. (1994) gave a beautiful algorithm to learn any polynomial-size circuit C in I also get a new result about the ancient problem of circuit minimization: There exists an oracle A, such that circuits with oracle access to A can’t even be approximately minimized in (By contrast, suffices) Oracle Where :Sales Pitch My result shows that their algorithmcan’t be parallelizedby any relativizing technique

Shows that the difficulty of learning circuits in is “merely” computational (not information-theoretic) Learning Circuits In If P=NP:Sales Pitch

Successful nonrelativizing incursion RELATIVIZATION BARRIER BLACK-BOX BARRIER “BATTLE MAP”

Alright, enough infomercial. Time for an oracle where PP has linear-size circuits…

n2n columns i,x 25n rows r 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

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 there’s some change to r that affects whether some PP machine accepts some input Strategy: Define a “progress measure” Q>0. 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 row—and as soon as there is, we win!

“A WAR OF ATTRITION”

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)|

What should the progress measure Q be? First idea: Suppose every row is sensitive. Then by pigeonhole, there exists an Mi(x) that’s 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, there’s a set of rows we can modify so as to double |pi,x(A)| Problem: Our goal was to double Q(A)!

Better idea: As before, there’s some Mi(x) that’s sensitive to 23n rows Let b=Mi(x) and k=log2|pi,x(A)|. Then we’ll think of Q(A) as the product of two polynomials:

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

Can we learn circuits in , under some computational assumption that we actually believe? 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 size—i.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 it’s optimal?

Oracles Are Subtle But Not Malicious