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Algebrization: A New Barrier in Complexity Theory Scott Aaronson (MIT) Avi Wigderson (IAS) 4xyw-12yz+17xyzw-2x-2y-2z-2w IP=PSPACE MA EXP P/poly MIP=NEXP PP SIZE(n) PromiseMA SIZE(n) PP P/poly PP=MA NEXP P/poly NEXP=MA NEXP P/poly NP SIZE(n) -15xyz+43xy-5x 13xw-44xz+x-7y+ P NP P=BPP

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What To Call It? Algebraic Relativization? Algevitization? Algevization? Algebraicization? Algebraization? Algebrization? SCOTT & AVI A NEW KIND OF ORACLE

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Any proof of P NP will have to defeat two terrifying monsters… P NP A Relativization [Baker-Gill-Solovay 1975] Natural Proofs [Razborov-Rudich 1993] ARITHMETIZATION Furthermore, even our best weapons seem to work against one monster but not the other… DIAGONALIZATION

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Yet within the last decade, weve seen circuit lower bounds that overcome both barriers [Buhrman-Fortnow-Thierauf 1998]: MA EXP P/poly Furthermore, this separation doesnt relativize [Vinodchandran 2004]: PP SIZE(n k ) for every fixed k [A. 2006]: Vinodchandrans result is non-relativizing [Santhanam 2007]: PromiseMA SIZE(n k ) for fixed k Vinodchandrans Proof: PP P/poly Were done PP P/poly P #P = MA [LFKN] P #P = PP 2 P PP [Toda] PP SIZE(n k ) [Kannan] Non-Relativizing Non-Naturalizing

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Bottom Line: Relativization and natural proofs, even taken together, are no longer insuperable barriers to circuit lower bounds Obvious Question [Santhanam 2007]: Is there a third barrier? This Talk: Unfortunately, yes. Algebrization: A generalization of relativization where the simulating machine gets access not only to an oracle A, but also a low-degree extension Ã of A over a finite field or ring We show: Almost all known techniques in complexity theory algebrize Any proof of P NPor even P=RP or NEXP P/polywill require non-algebrizing techniques

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Algebrizing Relativizing Naturalizing [Toda], [Impagliazzo- Wigderson], [Valiant- Vazirani], [Kannan], hundreds more [BFT], [Vinodchandran], [Santhanam], … [Furst-Saxe-Sipser], [Razborov-Smolensky], [Raz], dozens more [Your result here] [GMW?]

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Definitions The inclusion C D relativizes if C A D A for all oracles A C A[poly] : Polynomial-size queries to A only C A[exp] : Exponential-size queries also allowed Given an oracle A={A n } with A n :{0,1} n {0,1}, an extension Ã of A is a collection of polynomials Ã n :Z n Z satisfying: (i) Ã n (x)=A n (x) for all Boolean x {0,1} n, (ii) deg(Ã n )=O(n), (iii) size(Ã n (x)) p(size(x)) for some polynomial p, where Note: Can also consider extensions over finite fields instead of the integers. Will tell you when this distinction matters.

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A complexity class inclusion C D algebrizes if C A D Ã for all oracles A and all extensions Ã of A Proving C D requires non-algebrizing techniques if there exist A,Ã such that C A D Ã A separation C D algebrizes if C Ã D A for all A,Ã Proving C D requires non-algebrizing techniques if there exist A,Ã such that C Ã D A Notice weve defined things so that every relativizing result is also algebrizing.

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Related Work Low-degree oracles have been studied before for various reasons (recently by [JKRS07]) [Fortnow94] defined a class O of oracles such that IP A =PSPACE A for all A O Since he wanted the same oracle A on both sides, he had to define A recursively (take a low-degree extension, then reinterpret as a Boolean function, then take another low-degree extension, etc.) Proving separations in his model seems extremely hard

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Why coNP IP Algebrizes The only time Arthur ever has to evaluate the polynomial p directly is in the very last roundwhen he checks that p(r 1,…,r n ) equals what Merlin said it does, for some r 1,…,r n chosen randomly in the previous rounds. Recall the usual coNP IP proof of [LFKN]: Bullshit!

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How was the polynomial p produced? By starting from a Boolean circuit, then multiplying together terms that enforce correct propagation at each gate: xyg + (1-xy)(1-g) x y g Arthur and Merlin then reinterpret p not as a Boolean function, but as a polynomial over some larger field. But what if the circuit contained oracle gates? Then how could Arthur evaluate p over the larger field? A(x,y)g + (1-A(x,y))(1-g) A Hed almost need oracle access to a low-degree extension Ã of A. Ã(x,y)g + (1-Ã(x,y))(1-g) Hey, wait…

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Other Results That Algebrize PSPACE A[poly] IP Ã [Shamir] NEXP A[poly] MIP Ã [BFL] PP Ã P Ã /poly PP A MA Ã [LFKN] NEXP Ã[poly] P Ã /poly NEXP A[poly] MA Ã [IKW] MA EXP Ã[exp] P A /poly[BFT] PP Ã SIZE A (n)[Vinodchandran] PromiseMA Ã SIZE A (n)[Santhanam] OWF secure against P Ã NP A CZK Ã [GMW]

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Proving P NP Will Require Non- Algebrizing Techniques Theorem: There exists an oracle A, and an extension Ã, such that NP Ã P A. Proof: Let A be a PSPACE-complete language, and let Ã be the unique multilinear extension of A. Then Ã is also PSPACE-complete [BFL]. Hence NP Ã = P A = PSPACE. By the same argument, P NP cant even be proved with double-algebrizing or triple- algebrizing techniques…

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Harder Example: Proving P=RP Will Require Non-Algebrizing Techniques (hence P=NP as well) Theorem: There exist A,Ã such that RP A P Ã. Whats the difficulty here, compared to standard oracle separation theorems? Since Ã is a low-degree polynomial, we dont have the freedom to toggle each Ã(x) independently. I.e. the algorithm were fighting is no longer looking for a needle in a haystackit can also look in the haystacks low-degree extension! We will defeat it anyway.

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Theorem: Let F be a field, and let Y F n be the set of points queried by the algorithm. Then there exists a polynomial p:F n F, of degree at most 2n, such that (i) p(y)=0 for all y Y. (ii) p(z)=1 for at least 2 n -|Y| Boolean points z. (iii) p(z)=0 for the remaining Boolean points Y

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Proof: Given a Boolean point z, let z be the unique multilinear polynomial thats 1 at z and 0 at all other Boolean points. Then we can express any multilinear polynomial r as Requiring r(y)=0 for all y Y yields |Y| linear equations in 2 n unknowns. Hence there exists a solution r such that r(z) 0 for at least 2 n -|Y| Boolean points z. We now set In the integers case, we can no longer use Gaussian elimination to construct r. However, we (i.e. Avi) found a clever way around this problem using Chinese remaindering and Hensel lifting, provided every query y satisfies size(y)=O(poly(n)). A standard diagonalization argument now yields the separation between P and RP we wantedat least in the case of finite fields.

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Other Oracle Results We Can Prove By Building Designer Polynomials A,Ã : NP A coNP Ã A,Ã : NP A BPP Ã (only for finite fields, not integers) A,Ã : NEXP Ã[exp] P A /poly A,Ã : NP Ã SIZE A (n) By contrast, MA EXP P/poly and PromiseMA SIZE(n) algebrize! We seem to get a precise explanation for why progress on non-relativizing circuit lower bounds stopped where it did

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From Algebraic Query Algorithms to Communication Protocols A(000)=1 A(001)=0 A(010)=0 A(011)=1 A(100)=0 A(101)=0 A(110)=1 A(111)=1 Truth table of a Boolean function A Alice and Bobs Goal: Compute some property of the function A:{0,1} n {0,1}, using minimal communication Theorem: If a problem can be solved using T queries to Ã, then it can also be solved using O(Tnlog|F|) bits of communication between Alice and Bob Let Ã:F n F be the unique multilinear extension of A over a finite field F A0A0 A1A1

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Proof: Given any point y F n, we can write Theorem: If a problem can be solved using T queries to Ã, then it can also be solved using O(Tnlog|F|) bits of communication between Alice and Bob The protocol is now as follows: y 1 (O(nlog|F|) bits) Ã 1 (y 1 ) (O(log|F|) bits) y 2 (O(nlog|F|) bits) This argument works just as well in the randomized world, the nondeterministic world, the quantum world… Also works with integer extensions (we didnt have to use a finite field). Ã(y 1 )=Ã 0 (y 1 )+Ã 1 (y 1 )

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The Harvest: Separations in Communication Complexity Imply Algebraic Oracle Separations (2 n ) randomized lower bound for Disjointness [KS 1987] [Razborov 1990] A,Ã : NP A BPP Ã (2 n/2 ) quantum lower bound for Disjointness [Razborov 2002] A,Ã : NP A BQP Ã (2 n/2 ) lower bound on MA-protocols for Disjointness [Klauck 2003] A,Ã : coNP A MA Ã Exponential separation between classical and quantum communication complexities [Raz 1999] A,Ã : BQP A BPP Ã Exponential separation between MA and QMA communication complexities [Raz- Shpilka 2004] A,Ã : QMA A MA Ã Advantages of this approach: Ã is just the multilinear extension of A! Works automatically with integer extensions Advantages of this approach: Ã is just the multilinear extension of A! Works automatically with integer extensions Disadvantage: The functions achieving the separations are more contrived (e.g. Disjointness instead of OR).

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Can also go the other way: algebrization- inspired communication protocols [Klauck 2003]: Disjointness requires ( N) communication, even if theres a Merlin to prove Alice and Bobs sets are disjoint Obvious Conjecture: Klaucks lower bound can be improved to (N) This conjecture is false! We give an MA-protocol for Disjointness (and indeed Inner Product) with total communication cost O( N log N) Hardest communication predicate?

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O( N log N) MA-protocol for Inner Product A:[ N] [ N] {0,1} B:[ N] [ N] {0,1} Alice and Bobs Goal: Compute First step: Let F be a finite field with |F| [N,2N]. Extend A and B to degree-( N-1) polynomials If Merlin is honest, then But how to check S=S? If S S, then Now let r R F Claimed value S for S

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Conclusions Arithmetization had a great run. It led to IP=PSPACE, the PCP Theorem, non-relativizing circuit lower bounds… Yet we showed its fundamentally unable to resolve barrier problems like P vs. NP, or even P vs. BPP or NEXP vs. P/poly. Why? It doesnt pry open the black-box wide enough. I.e. it uses a polynomial-size Boolean circuit to produce a low-degree polynomial, which it then evaluates as a black box. It doesnt exploit the small size of the circuit in any deeper way. To reach this conclusion, we introduced a new model of algebraic query complexity, which has independent applications (e.g. to communication complexity) and lots of nooks and crannies to explore in its own right.

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Open Problems Develop non-algebrizing techniques! Do there exist A,Ã such that coNP A AM Ã ? Improve PSPACE A[poly] IP Ã to PSPACE Ã[poly] = IP Ã The power of double algebrization Integer queries of unbounded size Algebraic query lower bounds communication lower bounds? Generalize to arbitrary error-correcting codes (not just low-degree extensions)? Test if a low-degree extension came from a small circuit?

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