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]:
TT ( fk )
DP Code
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
Promise on C
Search problem
Small # queries
vs.
Testing
no promise
decision problem
Minimal # queries
Decoding: Hardness Amplification
fk is harder to compute on average than f
Motivation: 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
.99
Dinur-Reingold 06
2
.99
Dinur-Goldenberg 08
2
1/kα
Dinur-Goldenberg 08
2
1/k
New
3
exp(-kα)
New*
2
1/kα
* Derandomization
/
Consistency tests
Test: Query C(S1), C(S2), …
check consistency on common values.
Thm: If Test accepts oracle C with prob ²
then there is a function g: U R such that
for ≈ ² of k-tuples S, C (S) ¼ gk (S)
k
S]
[C(S) = g (S) in all but 1/k(1) elements in
Unique Decoding
List Decoding
Proof: g(x) = Plurality { C (S)x | x 2 S}
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
Theorem [DG08]:
If V-Test accepts with
probability ² > 1/k(1) ,
B1
S1
B2
A
S2
then there is g : U R
s.t. C ¼ gk on at least
² fraction of k-sets.
When ² < 1/k, the V-Test
does not work.
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
B1
S1
B2 A
2
S3
A1
S2
Theorem (main result):
If Z-Test accepts with
probability ² > exp(-k(1)),
then there is g : U R
s.t. C ¼ gk on at least
² fraction of k-sets.
Proof Ideas
Flowers, cores, petals
Flower: determined by S=(A,B)
Core: A
B
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
B1
B2
A
B3
B5
B4
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 }
B
(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.
B1
B2
A
B3
B5
B4
V-Test: Harmony
For 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
D1
B
A
E

E
A

D2
Proof: Symmetry
between A and E
(few errors in AuE )
Chernoff: ² ¼ exp(-kα)
Implication: Restricted
to Cons, an approx
V-Test on E accepts
almost surely:
Unique Decode!
Harmony ) Local DP
Main 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
D1
B
x
B’
A
E
D2
Intuition: g = g(A,B) is
the unique (approximate)
decoding of C on Cons(A,B)
Idea: Symmetry arguments.
Largness guarantees that
random selections are
near-uniform.
Proof Sketch
Main 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
D1
B
A
E
D2
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 !
Local DP structure
Field of flowers (Ai,Bi)
For each, gi s.t
C(S) ¼ gik (S) if
S2 Cons(Ai,Bi)
Global g?
B1
A
B2
B3
A
A
Bi
B
A
A
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
S1
B2
A
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’ }
Then C(S) ¼ gk (S), for ¼ ²/4 of all S
B1
B2 A
2
A1
B1
(local)
(global)
B1 A1
B2 AB2 A2
2
A1
Local to Global DP: Z-test
Proof: 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
A1
Can assume Flower (A1, B1 ) is large,
(otherwise V-Test rejects)
So (A1, B1) harmonious  have g.
S
B2 A
2
Z-Test rejects (
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
Local to Global DP: large ²
“double harmony”
B1
A1
S
B2
Three events all happen with
probability > poly(m/k)
(1) (A1, B1) is harmonious,  g1
(2) (A2, B2) is harmonious,  g2
A2 (3) S is consistent with both
• Get that g1 (x) = g2 (x)
for most x2 U.
Derandomization
Inclusion graphs are Samplers
Most lemmas analyze sampling properties
m-subsets

A
k-subsets
elements
of U
S
Cons
x
Subsets: Chernoff bounds – exponential error
Subspaces: Chebychev bounds – polynomial error
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-±)kQ1 Q3
Q2k-1
X # queries: 2k
Q2 Q4
 parallel repetition :
 # queries : 2
X soundness: ?
Q2k
proof F: Vk  §k
Q1
Q2
PCP Amplification History
 f: V  Σ,
F : Vk  Σk |V|=N , t= log |Σ|
size #queries
soundness
Sequential repetition
Verbitsky
Raz
Holenstein
Feige-Verbitsky
Rao
Projection
games
Raz
Feige-Kilian
New
N
Nk
Nk
Nk
Nk
Nk
Nk
Nk
Nk
Moshkovitz-Raz
N1+o(1)
2k
2
2
2
2
2
2
2
2
exp( - ± k )
very-slow(k)  0
exp( - ±32 k/ t)
exp( - ±3 k/ t)
t essential
exp( - ±2 k )
±2 essential
1/kα
exp ( - ± k1/2)
2
1/loglog N
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),
proof is CE : Ek  (§2)k
the PCP
Q1
Q2
Accept if
(1) CE (Q1) and CE (Q2) agree on common vertices, and
(2) all edge constraints are satisfied
The 2-query PCP amplification
Q1
Q2
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).
Analysis of our PCP
construction
PCP Analysis
Q1
Q2

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
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 ?
