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**Algebrization: A New Barrier in Complexity Theory**

P=BPP PNP Scott Aaronson (MIT) Avi Wigderson (IAS) NEXPP/poly NPSIZE(n) 4xyw-12yz+17xyzw-2x-2y-2z-2w IP=PSPACE MIP=NEXP NEXPP/polyNEXP=MA 13xw-44xz+x-7y+ -15xyz+43xy-5x PPSIZE(n) PPP/polyPP=MA PromiseMASIZE(n) MAEXPP/poly

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**What To Call It? SCOTT & AVI A NEW KIND OF ORACLE**

Algebraic Relativization? Algevitization? Algevization? Algebraicization? Algebraization? Algebrization?

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**Any proof of PNP will have to defeat two terrifying monsters…**

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

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**Vinodchandran’s Proof:**

Yet within the last decade, we’ve seen circuit lower bounds that overcome both barriers [Buhrman-Fortnow-Thierauf 1998]: MAEXP P/poly Furthermore, this separation doesn’t relativize [Vinodchandran 2004]: PP SIZE(nk) for every fixed k [A. 2006]: Vinodchandran’s result is non-relativizing Vinodchandran’s Proof: PP P/poly We’re done PP P/poly P#P = MA [LFKN] P#P = PP 2P PP [Toda] PP SIZE(nk) [Kannan] Non-Relativizing Non-Naturalizing [Santhanam 2007]: PromiseMA SIZE(nk) for fixed k

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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 PNP—or even P=RP or NEXPP/poly—will require non-algebrizing techniques

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**Algebrizing Relativizing Naturalizing**

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

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**Definitions The inclusion CD relativizes if CADA for all oracles A**

CA[poly]: Polynomial-size queries to A only CA[exp]: Exponential-size queries also allowed Given an oracle A={An} with An:{0,1}n{0,1}, an extension Ã of A is a collection of polynomials Ãn:ZnZ satisfying: (i) Ãn(x)=An(x) for all Boolean x{0,1}n, (ii) deg(Ãn)=O(n), (iii) size(Ãn(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 CD algebrizes if CADÃ for all oracles A and all extensions Ã of A**

Proving CD requires non-algebrizing techniques if there exist A,Ã such that CADÃ A separation CD algebrizes if CÃDA for all A,Ã Proving CD requires non-algebrizing techniques if there exist A,Ã such that CÃDA Notice we’ve 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 IPA=PSPACEA for all AO 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 coNPIP Algebrizes**

Recall the usual coNPIP proof of [LFKN]: Bullshit! The only time Arthur ever has to evaluate the polynomial p directly is in the very last round—when he checks that p(r1,…,rn) equals what Merlin said it does, for some r1,…,rn chosen randomly in the previous rounds.

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** A g x y How was the polynomial p produced?**

By starting from a Boolean circuit, then multiplying together terms that enforce “correct propagation” at each gate: A(x,y)g + (1-A(x,y))(1-g) Ã(x,y)g + (1-Ã(x,y))(1-g) xyg + (1-xy)(1-g) x y g A 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? He’d almost need oracle access to a low-degree extension Ã of A. Hey, wait…

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**Other Results That Algebrize**

PSPACEA[poly] IPÃ [Shamir] NEXPA[poly] MIPÃ [BFL] PPÃ PÃ/poly PPA MAÃ [LFKN] NEXPÃ[poly] PÃ/poly NEXPA[poly] MAÃ [IKW] MAEXPÃ[exp] PA/poly [BFT] PPÃ SIZEA(n) [Vinodchandran] PromiseMAÃ SIZEA(n) [Santhanam] OWF secure against PÃ NPA CZKÃ [GMW]

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**Proving PNP Will Require Non-Algebrizing Techniques**

Theorem: There exists an oracle A, and an extension Ã, such that NPÃPA. 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 NPÃ = PA = PSPACE. By the same argument, PNP can’t 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 RPAPÃ. What’s the difficulty here, compared to “standard” oracle separation theorems? Since Ã is a low-degree polynomial, we don’t have the freedom to toggle each Ã(x) independently. I.e. the algorithm we’re fighting is no longer looking for a needle in a haystack—it can also look in the haystack’s low-degree extension! We will defeat it anyway.

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

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Proof: Given a Boolean point z, let z be the unique multilinear polynomial that’s 1 at z and 0 at all other Boolean points. Then we can express any multilinear polynomial r as A standard diagonalization argument now yields the separation between P and RP we wanted—at least in the case of finite fields. Requiring r(y)=0 for all yY yields |Y| linear equations in 2n unknowns. Hence there exists a solution r such that r(z)0 for at least 2n-|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)).

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**Other Oracle Results We Can Prove By Building “Designer Polynomials”**

A,Ã : NPA coNPÃ A,Ã : NPA BPPÃ (only for finite fields, not integers) A,Ã : NEXPÃ[exp] PA/poly A,Ã : NPÃ SIZEA(n) By contrast, MAEXP 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 A0 A1 Truth table of a Boolean function A Alice and Bob’s Goal: Compute some property of the function A:{0,1}n{0,1}, using minimal communication Let Ã:FnF be the unique multilinear extension of A over a finite field F 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

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This argument works just as well in the randomized world, the nondeterministic world, the quantum world… Proof: Given any point yFn, we can write Also works with integer extensions (we didn’t have to use a finite field). The protocol is now as follows: Ã(y1)=Ã0(y1)+Ã1(y1) y1 (O(nlog|F|) bits) Ã1(y1) (O(log|F|) bits) 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 y2 (O(nlog|F|) bits)

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**The Harvest: Separations in Communication Complexity Imply Algebraic Oracle Separations**

(2n) randomized lower bound for Disjointness [KS 1987] [Razborov 1990] A,Ã : NPA BPPÃ (2n/2) quantum lower bound for Disjointness [Razborov 2002] A,Ã : NPA BQPÃ (2n/2) lower bound on MA-protocols for Disjointness [Klauck 2003] A,Ã : coNPA MAÃ Exponential separation between classical and quantum communication complexities [Raz 1999] A,Ã : BQPA BPPÃ Exponential separation between MA and QMA communication complexities [Raz-Shpilka 2004] A,Ã : QMAA MAÃ 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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**“Hardest” communication predicate?**

Can also go the other way: algebrization-inspired communication protocols [Klauck 2003]: Disjointness requires (N) communication, even if there’s a Merlin to prove Alice and Bob’s sets are disjoint “Obvious” Conjecture: Klauck’s 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} Claimed value S’ for S rRF Alice and Bob’s Goal: Compute First step: Let F be a finite field with |F|[N,2N]. Extend A and B to degree-(N-1) polynomials Now let If Merlin is honest, then But how to check S’=S? If S’S, then

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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 it’s fundamentally unable to resolve barrier problems like P vs. NP, or even P vs. BPP or NEXP vs. P/poly. Why? It “doesn’t 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 doesn’t 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 coNPA AMÃ? Improve PSPACEA[poly] IPÃ to PSPACEÃ[poly] = IPÃ 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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