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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 RG=EXP 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 [LFKN], [Shamir], [BFL], [BFT], [Vinodchandran], [Santhanam], [IKW], … [Furst-Saxe-Sipser], [Razborov-Smolensky], [Raz], dozens more [Your result here] [GMW?]

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Outline A New Kind of Relativization Why Existing Results Algebrize Example: coNP IP The Need for Non-Algebrizing Techniques …to prove P NP …to prove P=RP Connections to Communication Complexity O( N log N) MA-protocol for Inner Product Conclusions and Open Problems

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Outline A New Kind of Relativization Why Existing Results Algebrize Example: coNP IP The Need for Non-Algebrizing Techniques …to prove P NP …to prove P=RP Connections to Communication Complexity O( N log N) MA-protocol for Inner Product Conclusions and Open Problems

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First, the old kind of relativization… An oracle is basically just a Boolean function A that we can insert into a circuit as a black box x1x1 x2x2 x3x3 A x4x4 Given a complexity class C, C A is the class of problems solvable by a C machine with oracle access to A 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

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To prove an oracle separation between C and D means to construct an A such that C A D A. In designing A, we get to be arbitrarily devioustoggling each bit independently. This idea (plus diagonalization over all P machines) was used by [Baker-Gill-Solovay] to create an A such that P A NP A. Example: Suppose a P machine is looking for an x {0,1} n such that A(x)=1. Then we can wait to see which xs the machine queries, set A(x)=0 for all of them, and set A(x)=1 for some x that wasnt queried.

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An extension of a Boolean function A:{0,1} n {0,1} is an integer-valued polynomial Ã:Z n Z that agrees with A on all 2 n Boolean points. Given a set of Boolean functions A={A n } (one for each input size n), well be interested in sets of polynomials Ã={Ã n } with the following properties: (i) Ã n is an extension of A n for every n. (ii) deg(Ã n ) cn for some constant c. (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. Now for the new relativization

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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. Why not require C Ã D Ã ? Because we dont know how to prove things like PSPACE Ã IP Ã ! 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 [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.) He didnt consider algebraic oracle separations (indeed, proving separations in his model seems much harder than in ours)

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Outline A New Kind of Relativization Why Existing Results Algebrize Example: coNP IP The Need for Non-Algebrizing Techniques …to prove P NP …to prove P=RP Connections to Communication Complexity O( N log N) MA-protocol for Inner Product Conclusions and Open Problems

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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] EXP A[poly] RG Ã (RG = Refereed Games) [FK] 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 ZKIP Ã [GMW]

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Outline A New Kind of Relativization Why Existing Results Algebrize Example: coNP IP The Need for Non-Algebrizing Techniques …to prove P NP …to prove P=RP Connections to Communication Complexity O( N log N) MA-protocol for Inner Product Conclusions and Open Problems

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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.

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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. 0 0 0 0 0 0 0 1 1 1 1 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 P A /poly A,Ã : NP Ã SIZE A (n) By contrast, MA EXP P/poly and PromiseMA SIZE(n) algebrize! Since MA EXP and MA are just above NEXP and NP respectively (indeed equal to them under derandomization assumptions), we seem to get a precise explanation for why progress on non-relativizing circuit lower bounds stopped where it did.

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Outline A New Kind of Relativization Why Existing Results Algebrize Example: coNP IP The Need for Non-Algebrizing Techniques …to prove P NP …to prove P=RP Connections to Communication Complexity O( N log N) MA-protocol for Inner Product Conclusions and Open Problems

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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).

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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 Advantage of this approach: Ã is just the multilinear extension of A! 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: use algebrization to get new communication protocols Recall that [Klauck 2003] showed Disjointness requires total communication ( N) (where N=2 n ), even with a Merlin around to prove Alice and Bobs sets are disjoint. Obvious Conjecture: Klaucks lower bound can be improved to (N). The obvious 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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Outline A New Kind of Relativization Why Existing Results Algebrize Example: coNP IP The Need for Non-Algebrizing Techniques …to prove P NP …to prove P=RP Connections to Communication Complexity O( N log N) MA-protocol for Inner Product Conclusions and Open Problems

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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 ? Can we improve PSPACE A[poly] IP Ã to PSPACE Ã[poly] = IP Ã ? Is MA EXP Ã[poly] P A /poly? Can our query complexity lower bound for integer extensions be generalized to queries of unbounded size? Do algebraic query complexity lower bounds ever imply communication complexity lower bounds? How far can we generalize our results to arbitrary error- correcting codes (not just low-degree extensions)? Can we construct pseudorandom low-degree polynomials?

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