Oracles Are Subtle But Not Malicious Scott Aaronson University of Waterloo

Standard Whine In the 60 years since Shannons counting argument, we havent proven a superlinear circuit lower bound for any explicit Boolean function! Kannan 1982: 2 SIZE(n) Köbler & Watanabe 1998: ZPP NP SIZE(n) Vinodchandran 2004: PP SIZE(n) On the other hand, there are oracles where P NP and MA have linear-size circuits… Depends what we mean by explicit! Where exactly do these results hit a brick walland can we knock it down?

My Whole Paper In One Slide Oracle ResultsNon-Oracle Results PP and Perceptrons Parallel NP and Learning There exists an oracle relative to which PP has linear-size circuits In the real world, PP doesnt even have quantum circuits of size n k, for any constant k If P=NP, then we could exactly learn any poly- size circuit C in Theres an oracle where has linear-size circuits

Oracle Where PP SIZE(n): Sales Pitch Subsumes several previous oracles: P NP PP (Beigel)PP PSPACE (Aspnes et al.) MA EXP P/poly (BF&T)P NP = NEXP (Buhrman et al.) I also get an oracle where PEXP P/poly P NP P Same techniques yield a lower bound for k perceptrons solving k ODDMAXBIT instances In the real world, PP SIZE(n) One of the only nonrelativizing separations we have (I thereby solve four open problems of )

PP BQSIZE(n k ): Sales Pitch First nontrivial quantum circuit lower bound (outside the black-box model) Gives a new, self-contained proof of Vinodchandrans result Along the way, I prove a Quantum Karp-Lipton Theorem: If PP has small quantum circuits, then the Counting Hierarchy collapses to QMA

Oracle Where : Sales Pitch My result shows that their algorithm cant be parallelized by any relativizing technique 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 cant even be approximately minimized in Bshouty et al. (1994) gave a beautiful algorithm to learn any polynomial-size circuit C in (By contrast, suffices)

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

Successful nonrelativizing incursion BLACK-BOX BARRIER RELATIVIZATION BARRIER BATTLE MAP

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

M 1,M 2,…: Enumeration of PP machines (Actually PTIME(n logn ) machines) Goal: Create an oracle string A such that have small circuits on inputs of size n Then every M i will be taken care of on all but finitely many n, modulo a technicality 2 5n rows r n2 n columns i,x Idea: Pick a random row, and encode there what every M i does on every input x Then our linear-size circuit to simulate M 1,…,M n 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 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 Call a row r sensitive, if theres some change to r that affects whether some PP machine accepts some input Eventually, then, there must be an insensitive row and as soon as there is, we win!

A WAR OF ATTRITION

Our Weapon: Polynomials Let p i,x := | {accepting paths of M i (x)} | | {rejecting paths of M i (x)} | Basic Facts: p i,x (A) is a multilinear polynomial in the oracle bits, of degree at most n logn p i,x (A) 0 M i (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)|

First idea: Suppose every row is sensitive. Then by pigeonhole, there exists an M i (x) thats sensitive to 2 3n rows. Meaning: There are 2 3n rows we can modify so as to change sgn(p i,x (A)) Hence, by Nisan-Szegedy, theres a set of rows we can modify so as to double |p i,x (A)| Problem: Our goal was to double Q(A)! What should the progress measure Q be? Is it possible that, whenever |p i,x (A)| becomes large, the product of all the other terms somehow covers for it by becoming small?

Better idea: As before, theres some M i (x) thats sensitive to 2 3n rows Let b=M i (x) and k= log 2 |p i,x (A)|. Then well think of Q(A) as the product of two polynomials:

u(A) This time a cushion keeps u(A) from getting too small v(A) To prevent Q(A)=u(A)v(A) from doubling, v(A) needs to nosedive u(A) increases sharply when we modify one row Number of modified rows By Nisan-Szegedy, that forces v(A) to become large later on The product Q(A)=u(A)v(A) is thereby forced to double

Open Problems Prove better nonrelativizing circuit lower bounds! (duhhhh...) 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? 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))~2 n Can we learn circuits in, under some computational assumption that we actually believe?