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
S1S2
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
B
D2
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
B
AAD2
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 !
B
A 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
B
AA
B3
AAB1
AA
Counterexample [DG]
For every x 2 U pick a random gx: U R
For every k-subset S pick a random x(S) 2 S
Define 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 )B2
But 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 error
Subspaces: 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
games
Mix 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 ?