algebrization: a new barrier in complexity theory scott aaronson (mit) avi wigderson (ias)...

Algebrization: A New Barrier in Complexity Theory

Scott Aaronson (MIT)

Avi Wigderson (IAS)

4xyw-12yz+17xyzw-2x-2y-2z-2w

IP=PSPACE

MAEXPP/poly

MIP=NEXP

PPSIZE(n)

PromiseMASIZE(n)

PPP/polyPP=MA

NEXPP/polyNEXP=MA

RG=EXP

NEXPP/poly NPSIZE(n)

-15

xy

z+4

3x

y-5

x1

3x

w-4

4x

z+x

-7y

+

PNPP=BPP

What To Call It?

Algebraic Relativization?

Algevitization?

Algevization?

Algebraicization?

Algebraization?

Algebrization?

SCOTT &

AVIA NEW

KIND OF

ORACLE

Any proof of PNP will have to defeat two terrifying

monsters…

PNP

A

Relativization[Baker-Gill-Solovay 1975]

Natural Proofs[Razborov-Rudich 1993]

AR

ITHM

ETIZATIO

N

Furthermore, even our best weapons seem to

work against one monster but not the other…

DIA

GO

NA

LIZ

AT

ION

Yet within the last decade, we’ve seen circuit lower bounds that overcome both barriers[Buhrman-Fortnow-Thierauf 1998]: MAEXP P/polyFurthermore, this separation doesn’t relativize

[Vinodchandran 2004]: PP SIZE(nk) for every fixed k[A. 2006]: Vinodchandran’s result is non-relativizing

[Santhanam 2007]: PromiseMA SIZE(nk) for fixed k

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

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

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?]

Outline

A New Kind of Relativization

Why Existing Results AlgebrizeExample: coNPIP

The Need for Non-Algebrizing Techniques…to prove PNP…to prove P=RP

Connections to Communication ComplexityO(N log N) MA-protocol for Inner Product

Conclusions and Open Problems

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”

x1 x2 x3

A

x4

Given a complexity class C, CA is the class of problems solvable by a C machine with oracle access to A

The inclusion CD relativizes if CADA for all oracles A

CA[poly]: Polynomial-size queries to A onlyCA[exp]: Exponential-size queries also allowed

To prove an oracle separation between C and D means to construct an A such that CADA.

In designing A, we get to be arbitrarily devious—toggling each bit independently.

This idea (plus diagonalization over all P machines) was used by [Baker-Gill-Solovay] to create an A such that PANPA.

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 x’s the machine queries, set A(x)=0 for all of them, and set A(x)=1 for some x that wasn’t queried.

An extension of a Boolean function A:{0,1}n{0,1} is an integer-valued polynomial Ã:ZnZ that agrees with A on all 2n Boolean points.

Given a set of Boolean functions A={An} (one for each input size n), we’ll be interested in sets of polynomials Ã={Ãn} with the following properties:

(i) Ãn is an extension of An for every n.(ii) deg(Ãn) cn for some constant c.(iii) size(Ãn(x)) p(size(x)) for some polynomial p, where

.log1:size1

2

n

iixx

Note: Can also consider extensions over finite fields instead of the integers. Will tell

you when this distinction matters.

Note: Can also consider extensions over finite fields instead of the integers. Will tell

you when this distinction matters.

Now for the “new” relativization

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.

Why not require CÃDÃ? Because we don’t know how to prove things like PSPACEÃIPÃ!

Why not require CÃDÃ? Because we don’t know how to prove things like PSPACEÃIPÃ!

Notice we’ve defined things so that every relativizing result is also algebrizing.

Notice we’ve defined things so that every relativizing result is also algebrizing.

Related Work

Low-degree oracles have been studied before for various reasons

[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.)

He didn’t consider algebraic oracle separations (indeed, proving separations in his model seems much harder than in ours)

Outline






Why coNPIP Algebrizes

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.

Recall the usual coNPIP proof of [LFKN]:

0,,1,0,,1

1

nxx

nxxp

Bullshit!

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

He’d almost need oracle access to a low-degree extension Ã of A.

Ã(x,y)g + (1-Ã(x,y))(1-g)

Hey, wait…

Other Results That Algebrize

PSPACEA[poly] IPÃ [Shamir]

NEXPA[poly] MIPÃ [BFL]

EXPA[poly] RGÃ (RG = Refereed Games) [FK]

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 ZKIPÃ [GMW]

Outline






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.

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.

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.

00

0

0

0

0

0

1

1

1

1

Y

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

.1,0

xxr zz

zn

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

.:0:1,0

zrz

z

n zr

xxrxp

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)).

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 wanted—at least in the case of finite fields.

A standard diagonalization argument now yields the separation between P and RP we wanted—at least in the case of finite fields.

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 PA/poly

A,Ã : NPÃ SIZEA(n)

By contrast, MAEXP P/poly and PromiseMA SIZE(n)

algebrize!

By contrast, MAEXP P/poly and PromiseMA SIZE(n)

algebrize!

Since MAEXP 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.

Since MAEXP 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.

Outline






From Algebraic Query Algorithms to Communication Protocols

A(000)=1A(001)=0A(010)=0A(011)=1

A(100)=0A(101)=0A(110)=1A(111)=1

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.

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:FnF be the unique multilinear extension of A over a finite field F.

A0 A1

Proof: Given any point yFn, 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.

.~~:

10

~

10

1,01

1,00

1,0

11

yAyA

yxAyxA

yxAyA

nn

n

xx

xx

xx

The protocol is now as follows:

y1 (O(nlog|F|) bits)

Ã1(y1) (O(log|F|) bits)

y2 (O(nlog|F|) bits)

This argument works just as well in the randomized world, the nondeterministic

world, the quantum world…

This argument works just as well in the randomized world, the nondeterministic

world, the quantum world…

Also works with integer extensions (we didn’t have to use a finite field).

Also works with integer extensions (we didn’t have to use a finite field).

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

Advantage of this approach: Ã is just the multilinear extension of A!

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).

Disadvantage: The functions achieving the separations are more contrived (e.g. Disjointness instead of OR).

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=2n), even with a Merlin around to prove Alice and Bob’s sets are disjoint.

“Obvious” Conjecture: Klauck’s 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?

O(N log N) MA-protocol for Inner Product

A:[N][N]{0,1} B:[N][N]{0,1}

Alice and Bob’s Goal: Compute

N

yx

yxByxAIP1,

.,,

First step: Let F be a finite field with |F|[N,2N]. Extend A and B to degree-(N-1) polynomials .:

~,~ 2 FFBA

.,~

,~

:1

yxByxAxSN

y

If Merlin is honest, then .1

N

x

xSIP But how to check S’=S?

If S’S, then

.

2'deg,

~,

~'Pr

1 N

N

F

SyrByrArS

N

yr

Now let

NrBrBr ,~,,1,

~,

rRFClaimed value S’ for S

Outline






ConclusionsArithmetization 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.

Open ProblemsDevelop non-algebrizing techniques!

Do there exist A,Ã such that coNPA AMÃ?

Can we improve PSPACEA[poly] IPÃ to PSPACEÃ[poly] = IPÃ?

Is MAEXPÃ[poly] PA/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”?

algebrization: a new barrier in complexity theory scott aaronson (mit) avi wigderson (ias)...

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