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

Learn about direct product testing and a new 2-query PCP by Russell Impagliazzo, Valentine Kabanets, Avi Wigderson. Understand the direct product definition, coding, decoding, and testing for complexity applications. Discover the DP theorem and XOR lemma for secure computations and complexity theory.

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

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

  2. 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 …

  3. 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

  4. Decoding: Hardness Amplification fkis harder to computeon 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)

  5. 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 ofqueries to C. - Want to minimize acceptance probability ²

  6. 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 /

  7. 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

  8. Consistency tests

  9. 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 S1 S2 A

  10. 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 S1 S2 B2 A2 S3

  11. Proof Ideas

  12. 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

  13. 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

  14. 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 Implication: Restricted to Cons, an approx V-Test on E accepts almost surely: Unique Decode! A D2 A E

  15. 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 Idea: Symmetry arguments. Largness guarantees that random selections are near-uniform. x B’

  16. 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

  17. 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

  18. 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

  19. 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)

  20. 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

  21. 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 (

  22. 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.

  23. Derandomization

  24. 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

  25. 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!

  26. Application: PCPs

  27. 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 )

  28. 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

  29. 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 Q2k-1 Q2 Q4 Q2k Q1 Q2

  30. 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

  31. 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, we get exponential decay (as sequential-repetition)in soundness ! Combine tests to minimize # of queries. Replace Z-test by V-test (local DP suffices)

  32. 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

  33. The 2-query PCP amplification 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

  34. Analysis of our PCP construction

  35. 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

  36. 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

  37. 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 ?

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