Direct-product testing,and

a new 2-query PCP

Russell Impagliazzo (IAS & UCSD)Valentine Kabanets (SFU)

Avi Wigderson (IAS)

Direct Product: Definition For f : U R, the k -wise direct

product fk : Uk Rk is fk (x1,…, xk) = ( f(x1), …, f(xk) ).

[Impagliazzo’02, Trevisan’03]: DP Code

TT ( fk ) is DP Encoding of TT ( f )

Rate and distance of DP Code are “bad”, but the code is still very useful in Complexity …

DP Code: Two Basic Questions

- Decoding: Given C ¼ fk, “find” f. (useful for Hardness Amplification)

- Testing: Given C, test if C ¼ fk.(useful for PCP constructions)

C is given as oracle Decoding vs. Testing Promise on C no promise Search problem decision problem Small # queries Minimal # queries

Decoding: Hardness Amplification

fk is harder to compute on average than fMotivation: Cryptography

Pseudorandomness, Computational Complexity, PCPs

DP Theorem/ XOR Lemma: [Yao82, Levin87, GL89, I94, GNW95, IW97, T03, IJK06, IJKW08]

If C computes fk on ² of all (x1,…, xk) Uk Then C’ computes f on 1-δ of all x U ² = exp(-δk)

Direct-Product Testing Given an oracle C : Uk Rk

Test makes some queries to C, and(1) Accept if C = fk.(2) Reject if C is “far away” from any fk

(2’) If Test accepts C with “high” probability ², then C must be “close” to some fk.

- Want to minimize number of queries to C.- Want to minimize acceptance probability ²

DP Testing History Given an oracle C : Uk Rk, is C¼ gk ? #queries acc. prob.Goldreich-Safra 00* 20 .99Dinur-Reingold 06 2 .99Dinur-Goldenberg 08 2 1/kα

Dinur-Goldenberg 08 2 1/kNew 3 exp(-kα)New* 2 1/kα

* Derandomization

/

Consistency testsTest: Query C(S1), C(S2), … check consistency on common values.Thm: If Test accepts oracle C with prob ²then there is a function g: U R such thatfor ≈ ² of k-tuples S, C (S) ¼ gk (S) [C(S) = gk(S) in all but 1/k elements in S]

Proof: g(x) = Plurality { C (S)x | x 2 S}g(x) = Plurality { C (S)x | x 2 S & C(S)A=a }

Unique Decoding

List Decoding

Consistency tests

V-Test [GS00,FK00,DR06,DG08] Pick two random k-sets S1 = (B1,A), S2 =

(A,B2) with m = k1/2 common elements A.

Check if C(S1)A = C(S2)A

B1 B2

A

Theorem [DG08]: If V-Test accepts withprobability ² > 1/k,

then there is g : U Rs.t. C ¼ gk on at least ² fraction of k-sets.

When ² < 1/k, the V-Test does not work.

S1 S2

Z-Test Pick three random k-sets S1 =(B1, A1),

S2=(A1,B2), S3=(B2, A2) with |A1| = |A2| = m = k1/2.

Check if C(S1)A1= C(S2)A1

and C(S2)B2 = C(S3)B2

Theorem (main result):

If Z-Test accepts withprobability ² > exp(-k), then there is g : U Rs.t. C ¼ gk on at least ² fraction of k-sets.

B1

B2

A1

A2

S1 S2

S3

Proof Ideas

Flowers, cores, petalsFlower: determined by S=(A,B)

Core: A

Core values: α=C(A,B)A

Petals: ConsA,B = { (A,B’) | C(A,B’)A =α }

In a flower, all petals agree on core values!

[IJKW08]:Flower analysis

B

B4

AA B2

B3

B1

B5

V-Test ) Structure (similar to [FK, DG])

Suppose V-Test accepts with probability ².

ConsA,B = { (A,B’) | C(A,B’)A = C(A,B)A }

(1) Largeness: Many (²/2)flowers (A,B) have many (²/2) petals ConsA,B

(2) Harmony: In every large flower, almost all pairs of overlapping sets in Cons are almost perfectly consistent.

B

B4

AA B2

B3

B1

B5

V-Test: HarmonyFor random B1 = (E,D1) and B2 = (E,D2) (|E|=|

A|)Pr [B1 2 Cons & B2 2 Cons & C(A, B1)E C(A, B2)E ] < ²4

<< ²

BD2

D1

AE

Proof: Symmetry between A and E (few errors in AuE )Chernoff: ² ¼ exp(-kα) E

A

Implication: Restricted to Cons, an approxV-Test on E accepts almost surely: Unique Decode!

Harmony ) Local DPMain Lemma: Assume (A,B) is harmonious. Define

g(x) = Plurality { C(A,B’)x | B’2 Cons & x 2 B’ }

Then C(A,B’)B’ ¼ gk (B’), for almost all B’ 2 Cons

BAA D2

D1

EIntuition: g = g(A,B) isthe unique (approximate) decoding of C on Cons(A,B)B’

x Idea: Symmetry arguments.Largness guarantees thatrandom selections are near-uniform.

Proof SketchMain Lemma: Assume (A,B) is harmonious. Define

g(x) = Plurality { C(A,B’)x | B’2 Cons & x 2 B’ }

Then C(A,B’)B’ ¼ gk (B’), for almost all B’ 2 Cons

Proof: Assume otherwise.A random B1 in Cons has many “minority” elements x where C(B1)x g(x).

A random E ½ B1 has many “minority” elements [Chernoff]

A random B2=(E,D2) is likely s.t. C(B2)E ¼ g(E) [def of g]

Then C(B1)E C(B2)E, Hence no harmony !

BA D2

D1

E

Local DP structureField of flowers (Ai,Bi)

For each, gi s.tC(S) ¼ gi

k (S) ifS2 Cons(Ai,Bi)

Global g?

B2

AA Bi

AA

BAA

B3

AAB1

AA

Counterexample [DG]For every x 2 U pick a random gx: U RFor every k-subset S pick a random x(S) 2

SDefine C(S) = gx(S)(S)C(S1)A=C(S2)A “iff” x(S1)=x(S2)

V-test passes with high prob:² = Pr[C(S1)A=C(S2)A] ~ m/k2

No global g if ² < 1/k2

B1 B2

AS1 S2

From local DP to global DP How to “glue” local solutions?

² > 1/kα “double excellence” (2 queries) [DG]² > exp(-kα) Z-test (3 queries)

Local to Global DP: small ² Lemma: (A1,B1) random (Cons large w.p. ²/2).

Define g(x) = Plurality { C (A1,B’)x | B’2 Cons & x 2 B’ }

(local)Then C(S) ¼ gk (S), for ¼ ²/4 of all S

(global)B1

B2

A1

A2

B1

B2

A1

A2

B1

B2

A1

A2

Local to Global DP: Z-testProof: Cons = ConsA1,B1. Define

g(x) = Plurality { C(A1,B’)x | B’2 Cons & x 2 B’ }

Harmony implies C(A1,B’)B’ ¼ gk (B’), for almost all B’2Cons B1

B2

A1

A2

Can assume Flower (A1, B1 ) is large, (otherwise V-Test rejects)

So (A1, B1) harmonious have g.

Pick random S=(B2, A2). May assume B2 in Cons (otherwise V-Test rejects)

If g(S) very different from C(S), then g(B2) C(S )B2But g(B2) ¼ C(A1,B2)B2

Z-Test rejects (

S

Local to Global DP: large ² “double harmony”

B1 A1

A2B2

S

Three events all happen withprobability > poly(m/k)

(1) (A1, B1) is harmonious, g1

(2) (A2, B2) is harmonious, g2

(3) S is consistent with both• Get that g1 (x) = g2 (x) for most x2 U.

Derandomization

Inclusion graphs are Inclusion graphs are SamplersSamplers

Most lemmas analyze sampling properties

m-subsets

A

Subsets: Chernoff bounds – exponential errorSubspaces: Chebychev bounds – polynomial error

Cons

S

k-subsets

x

elements of U

Derandomized DP Test Derandomized DP: fk (S), for linear subspaces S (similar to [IJKW08] ) .

Theorem (Derandomized V-Test): If derandomized V-Test accepts C with probability ² > poly(1/k), then there is a function g : U R such that C (S) ¼ gk (S) on poly(²) of subspaces S.

Corollary: Polynomial rate testable DP-code with [DG] parameters!

Application: PCPs

Constraint Satisfaction Problem

A graph CSP over alphabet §: • Given a graph G=(V,E) on n nodes,

and edge constraints Áe: §2 {0,1} ( e2 E ),

• is there an assignment f: V § that satisfies all edge constraints.

Example: 3-Colorability ( § = {1,2,3}, Áe (a,b) = 1 iff a b )

PCP Theorem [AS,ALMSS]

For some constant 0<±<1 and constant-size alphabet §, it is NP-hard to distinguish between

satisfiable graph CSPs over §, and ±-unsatisfiable ones (where every assignment

violates at least ± fraction of edge constraints). 2-query PCP ( with completeness 1, soundness 1-± ) : PCP proof = assignment f: V §,Verifier: Accept if f satisfies a random edge Q1

Q2

Decreasing soundness by repetition

sequential repetition : proof f: V § soundness : 1-± (1-±)k

X # queries: 2k

parallel repetition : proof F: Vk §k

# queries : 2X soundness: ?

Q1

Q2

Q3

Q4

Q2k-1

Q2k

Q1

Q2

PCP Amplification History f: V Σ, F : Vk Σk |V|=N , t= log |Σ| size #queries soundnessSequential repetition N 2k exp( - ± k )Verbitsky Nk 2 very-slow(k) 0 Raz Nk 2 exp( - ±32 k/ t)Holenstein Nk 2 exp( - ±3 k/ t)Feige-Verbitsky Nk 2 t essentialRao Nk 2 exp( - ±2 k )Raz Nk 2 ±2 essentialFeige-Kilian Nk 2 1/kα

New Nk 2 exp ( - ± k1/2)Moshkovitz-Raz N1+o(1) 2 1/loglog N

ParallelrepetitionProjection

gamesMix N’Match

Ideas: DP-Test of the PCP proof

Given F : Vk § k, test if F = fk for some f: V § and test random constraints!

If F close to fk, we get exponential decay (as sequential-repetition) in soundness !

Combine tests to minimize # of queries.

Replace Z-test by V-test (local DP suffices)

A New 2-Query PCP (similar to [FK])

For a regular CSP graph G = (V, E), the PCP proof is CE : Ek (§2)k

Accept if (1) CE (Q1) and CE (Q2) agree on common vertices, and (2) all edge constraints are satisfied

Q1

Q2

The 2-query PCP amplification

Theorem: If CSP G=(V,E) is satisfiable, there is a proof

CE that is accepted with probability 1. If CSP is ± – unsatisfiable, then no CE is

accepted with probability > exp ( - ± k1/2).

Corollary: A 2-query PCP over §k, of size nk, perfect completeness, and soundness exp(- k1/2).

Q1

Q2

Analysis of our PCP construction

PCP Analysis

From CE : Ek §2)k to the vertex proof C : Vk §k : C(v1,…, vk) = CE( e1,…, ek) for random incident edges Consistency of CE , Consistency of C Main Lemma for C yields local DP function g : V § Back to CE: g is also local DP for CE (symmetry) g (Q2) ¼ CE (Q2) (since Q2 2 ConsQ1) g(Q2) violates > ± edges (by soundness of G & Chernoff) Hence, CE (Q2) violates some edges, and Test rejects

Q1

Q2

Summary Direct Product Testing: 3 queries &

exponentially small acceptance probability

Derandomized DP Testing: 2 queries & polynomially small acceptance probability

( derandomized V-Test of [DG08] )

PCP: 2-Prover parallel k-repetition for restricted games, with exponential in k1/2 decrease in soundness

Open Questions

Better dependence on k in our Parallel Repetition Theorem : exp ( - ± k) ?

Derandomized 2-Query PCP : Obtaining / improving

[Moshkovitz-Raz’08, Dinur-Harsha’09] via DP-testing ?