Direct-product testing, and a new 2-query PCP

2. Direct-product testing,and a new 2-query PCP Russell Impagliazzo (IAS & UCSD) Valentine Kabanets (SFU) Avi Wigderson (IAS)

3. Direct Product: Definition • For f : U  R, the k -wise direct productfk : Uk Rk is fk (x1,…, xk) = ( f(x1), …, f(xk) ). [Impagliazzo’02, Trevisan’03]: DP Code TT (fk)is DP Encoding ofTT ( f ) Rate and distance of DP Code are “bad”, but the code is still very useful in Complexity …

4. 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

5. Decoding: Hardness Amplification Motivation: Cryptography Pseudorandomness, Computational Complexity, PCPs fkis harder to computeon average than f 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)

6. 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, then C must be “close” to some fk. - Want to minimize number ofqueries to C. - Want small acceptance probability

7. 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 New3 exp(-kα) New*2 1/kα * Derandomization /

8. Consistency tests Test: QueryC(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-tuplesS, C (S)¼ gk (S) [C(S) = gk(S) in all but 1/k(1) 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

9. Consistency tests

10. V-Test[GS00,FK00,DR06,DG08] Pick two random k-setsS1 = (B1,A), S2 = (A,B2) withm = k1/2common elements A. Check if C(S1)A = C(S2)A Theorem[DG08]: IfV-Testaccepts with probability ² > 1/k(1) , then there is g : U R s.t. C ¼gkon at least ²fraction of k-sets. When² < 1/k, theV-Testdoes not work. B1 B2 A

11. Z-Test Pick three random k-setsS1 =(B1, A1), S2=(A1,B2), S3=(B2, A2) with|A1| = |A2| = m = k1/2. Check if C(S1)A1= C(S2)A1andC(S2)B2 = C(S3)B2 Theorem (main result): IfZ-Testaccepts with probability² > exp(-k(1)), then there is g : U R s.t. C ¼gkon at least ²fraction of k-sets. B1 A1 B2 A2

12. Proof Ideas

13. Flowers, cores, petals Flower: 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 B1 B B2 A A B3 B5 B4

14. 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 } • Largeness:Many(²/2) • flowers (A,B)have many (²/2) petals ConsA,B • Harmony:In every large flower, almost • all pairs of overlapping sets in Consare almost perfectly consistent. B1 B B2 A A B3 B5 B4

15. V-Test: Harmony For random B1 = (E,D1) andB2 = (E,D2) (|E|=|A|) Pr [B12 Cons &B22 Cons &C(A, B1)E C(A, B2)E] < ²4<< ² Proof: Symmetry between A and E (few errors in AuE) Chernoff: ²¼ exp(-kα) D1 B E Intuition: Restricted to Cons, an approx V-Test on E accepts almost surely: Unique Decode! A D2 A E

16. 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 Intuition: g = g(A,B) is the unique (approximate) decoding of C on Cons(A,B) E D2 A A x B’

17. 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’¼gk (B’), for almost all B’ 2 Cons Proof: Assume otherwise. A random B1inConshas many “minority” elements xwhere C(B1)x g(x). A random E ½ B1has many “minority” elements [Chernoff] A random B2=(E,D2) is likely s.t. C(B2)E¼ g(E) [def of g] Then C(B1)E C(B2)E, Hence no harmony ! D1 B D2 E A

18. Local DP structure Field of flowers (Ai,Bi) For each gi s.t C(S)¼gik (S) if S2 Cons(Ai,Bi) Global g? B2 B1 B3 Bi B A A A A A A A A A A

19. 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 S1 S2 A

20. 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)

21. 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)¼gk (S), for ¼ ²/4 of all S (global) B1 A1 B1 A1 B1 A1 B2 B2 B2 A2 A2 A2

22. 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’¼gk (B’), for almost all B’2Cons Can assume Flower (A1, B1) islarge, (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 B1 A1 S B2 A2 Z-Test rejects (

23. Local to Global DP: large ²“double harmony” • Three events all happen with • probability > poly(m/k) • (1) (A1, B1) is harmonious,  g1 • (2) (A2, B2) is harmonious,  g2 • S is consistent with both A1 B1 S B2 A2 • Get that g1 (x) = g2 (x) for most x2 U.

24. Derandomization

25. m-subsets  A Inclusion graphs are Samplers Most lemmas analyze sampling properties of U k-subsets elements S Cons x Subsets: Chernoff bounds – exponential error Subspaces: Chebychev bounds – polynomial error

26. Derandomized DP Test Derandomized DP: fk (S), for linear subspaces S(as in [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 code.

27. Application: PCPs

28. 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 )

29. 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 fsatisfies a random edge Q2 Q1

30. Decreasing soundness by repetition • sequential repetition : proof f: V  § • soundness : 1-± (1-±)k • # queries: 2k • parallel repetition : proof F: Vk  §k • # queries : 2 • soundness: ? Q1 Q3 Qk-1 Q2 Q4 Qk Q1 Q2

31. PCP Amplification History • f: V  Σ,F : Vk Σk |V|=N , t= log |Σ| size #queries soundness Sequential repetition N 2kexp( - ± k ) Verbitsky Nk2 very-slow(k)  0 Raz Nk2exp( - ±32 k/ t) HolensteinNk2exp( - ±3 k/ t) Feige-Verbitsky Nk2tessential Rao Nk2 exp( - ±2 k ) Raz Nk2±2essential Feige-Kilian Nk21/kα NewNk2 exp ( - ± k1/2) Moshkovitz-Raz N1+o(1) 2 1/loglog N Parallel repetition Projection games Mix N’ Match

32. Ideas: DP-Test of the PCP proof Given F : Vk§k, test if F = fkfor some f: V §and test random constraints! If F close to fk, getting exponential decay (as sequential-repetition)in soundness ! Combine tests to minimize # of queries. Replace Z-test by V-test (local DP suffices)

33. A New 2-Query PCP (similar to [FK]) • For a regular CSP graph G = (V, E), the PCP proof is CE : Ek (§2)k Q1 Q2 Accept if (1) CE (Q1) and CE (Q2) agree on common vertices, and (2) all edge constraints are satisfied

34. The 2-query PCP Theorem Theorem: • If CSP G=(V,E) is satisfiable, there is a proof CEthat 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

35. Analysis of our PCP construction

36. 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 Q22 ConsQ1) • g(Q2) violates > ± edges (by soundness of G & Chernoff) • Hence, CE (Q2) violates some edges, and Test rejects Q1 Q2

37. 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

38. 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] using similar techniques.