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CS151 Complexity Theory
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  1. CS151Complexity Theory Lecture 14 May 13, 2004

  2. Outline • accidentally omitted material on PH • non-uniformity and the PH • BPP and the PH • resume interactive proofs and their power CS151 Lecture 14

  3. Karp-Lipton • we know that P = NP implies SAT has polynomial-size circuits. • (showing SAT does not have poly-size circuits is one route to proving P ≠ NP) • suppose SAT has poly-size circuits • any consequences? • might hope: SAT  P/poly PHcollapses to P, same as if SAT  P CS151 Lecture 14

  4. Karp-Lipton Theorem (KL): if SAT has poly-size circuits then PH collapses to the second level. • Proof: • suffices to show Π2 Σ2 • L Π2 implies: L = {x : y z (x, y, z)  R} with R  P. CS151 Lecture 14

  5. Karp-Lipton L = {x : y z (x, y, z)  R} • “z (x, y, z)  R?” is in NP • pretend C solves SAT, use self-reducibility • Claim: if SAT  P/poly, then L = { x : C y [use C repeatedly to find some z for which (x, y, z)  R; accept iff (x, y, z)  R] } poly time CS151 Lecture 14

  6. Karp-Lipton L = {x : y z (x, y, z)  R} {x : C y [use C repeatedly to find some z for which (x,y,z)  R; accept iff (x,y,z)  R] } • x  L: • some C decides SAT  Cy […] accepts • x  L: • yz (x, y, z)  R  Cy[…] rejects CS151 Lecture 14

  7. BPP PH • Recall: don’t know BPP different from EXP Theorem (S,L,GZ): BPP (Π2Σ2) • don’t know Π2Σ2 different from EXP but believe muchweaker CS151 Lecture 14

  8. BPP PH • Proof: • BPP language L: p.p.t. TM M: x  L  Pry[M(x,y) accepts] ≥ 2/3 x  L  Pry[M(x,y) rejects] ≥ 2/3 • strong error reduction: p.p.t. TM M’ • use n random bits (|y’| = n) • # strings y’ for which M’(x, y’) incorrect is at most 2n/3 • (can’t achieve with naïve amplification) CS151 Lecture 14

  9. BPP PH so many ones, some disk is all ones so few ones, not enough for whole disk • view y’ = (w, z), each of length n/2 • consider output of M’(x, (w, z)): w =000…00000…01000…10 … 111…11 xL 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 … 0 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 1 xL 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 … 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CS151 Lecture 14

  10. BPP PH • proof (continued): • strong error reduction: # bad y’ < 2n/3 • y’ = (w, z) with |w| = |z| = n/2 • Claim: L = {x : w z M’(x, (w, z)) = 1 } • xL: suppose wz M’(x, (w, z)) = 0 • implies  2n/2 0’s; contradiction • xL: suppose wz M’(x, (w, z)) = 1 • implies  2n/2 1’s; contradiction CS151 Lecture 14

  11. BPP PH • given BPP language L: p.p.t. TM M: x  L  Pry[M(x,y) accepts] ≥ 2/3 x  L  Pry[M(x,y) rejects] ≥ 2/3 • showed L = {x : wz M’(x, (w, z)) = 1} • thus BPP  Σ2 • BPP closed under complement BPP  Π2 • conclude: BPP (Π2Σ2) CS151 Lecture 14

  12. Interactive Proofs • interactive proof system for L is an interactive protocol (P, V) common input: x Prover Verifier . . . # rounds = poly(|x|) accept/reject CS151 Lecture 14

  13. Interactive Proofs • interactive proof system for L is an interactive protocol (P, V) • completeness: x  L  Pr[V accepts in (P, V)(x)]  2/3 • soundness: x  L   P* Pr[V accepts in (P*, V)(x)]  1/3 • efficiency: V is p.p.t. machine • repetition: can reduce error to any ε CS151 Lecture 14

  14. Interactive Proofs IP = {L : L has an interactive proof system} • Observations/questions: • philosophically interesting: captures more broadly what it means to be convinced a statement is true • clearly NP  IP. Potentially larger. How much larger? • if larger, randomness is essential (why?) CS151 Lecture 14

  15. Graph Isomorphism • graphs G0 = (V, E0) and G1 = (V, E1) are isomorphic (G0 G1) if exists a permutation π:V  V for which (x, y)  E0  (π(x), π(y))  E1 1 1 1 4 2 3 4 3 2 3 4 2 CS151 Lecture 14

  16. Graph Isomorphism • GI = {(G0, G1) : G0 G1 } • in NP • not known to be in P, or NP-complete • GNI = complement of GI • not known to be in NP Theorem (GMW): GNI  IP • indication IP may be more powerful than NP CS151 Lecture 14

  17. GNI in IP • interactive proof system for GNI: input: (G0, G1) Prover Verifier H = π(Gc) flip coin c  {0,1}; pick random π if H  G0 r = 0, else r = 1 r accept iff r = c CS151 Lecture 14

  18. GNI in IP • completeness: • if G0 not isomorphic to G1 then H is isomorphic to exactly one of (G0, G1) • prover will choose correct r • soundness: • if G0 G1 then prover sees same distribution on H for c = 0, c = 1 • no information on c  any prover P* can succeed with probability at most 1/2 CS151 Lecture 14

  19. The power of IP • GNI  IP suggests IP more powerful than NP, since GNI not thought to be in NP • GNI in coNP Theorem (LFKN): coNP  IP CS151 Lecture 14

  20. The power of IP • Proof idea: input: φ(x1, x2, …, xn) • prover: “I claim φ has k satisfying assignments” • true iff • φ(0, x2, …, xn) has k0 satisfying assignments • φ(1, x2, …, xn) has k1 satisfying assignments • k = k0 + k1 • prover sends k0, k1 • verifier sends random c {0,1} • prover recursively proves “φ’ = φ(c, x2, …, xn) has kc satisfying assignments” • at end, verifier can check for itself. CS151 Lecture 14

  21. The power of IP • Analysis of proof idea: • Completeness: φ(x1, x2, …, xn) has k satisfying assignments  accept with prob. 1 • Soundness: φ(x1, x2, …, xn) does not have k satisfying assigns.  accept prob.  1 – 2-n • Why? It is possible that k is only off by one; verifier only catches prover if coin flips c are successive bits of this assignment CS151 Lecture 14

  22. The power of IP • Solution to problem (ideas): • replace {0,1}n with (Fq)n • verifier substitutes random field element at each step • vast majorityof field elements catch cheating prover (rather than just 1) Theorem: L = { (φ, k): CNF φ has exactly k satisfying assignments} is in IP CS151 Lecture 14

  23. The power of IP • First step: arithmetization • transform φ(x1, … xn) into polynomial pφ(x1, x2, … xn) of degree d over a field Fq; q prime > 2n • recursively: • xi xiφ (1 - pφ) • φ φ’  (pφ)(pφ’) • φ φ’  1 - (1 - pφ)(1 - pφ’) • for all x  {0,1}n we have pφ(x) =φ(x) • degree d  |φ| • can compute pφ(x) in poly time from φ and x CS151 Lecture 14

  24. The power of IP • Prover wishes to prove: k = Σx1 = 0, 1Σx2 = 0,1 …Σxn = 0, 1pφ(x1, x2, …, xn) • Define: kz = Σx2 = 0,1 …Σxn = 0, 1pφ(z, x2, …, xn) • prover sends: kz for all z  Fq • verifier: • checks that k0 + k1 = k • sends random z  Fq • continue with proof that kz = Σx2 = 0,1 …Σxn = 0, 1pφ(z, x2, …, xn) • at end: verifier checks for itself CS151 Lecture 14

  25. The power of IP • Prover wishes to prove: k = Σx1 = 0, 1Σx2 = 0,1 …Σxn = 0, 1pφ(x1, x2, …, xn) • Define: kz = Σx2 = 0,1 …Σxn = 0, 1pφ(z, x2, …, xn) • a problem: can’t send kz for all z  Fq • solution: send the polynomial ! • recall degree d  |φ| CS151 Lecture 14

  26. The actual protocol input: (φ, k) Prover Verifier p1(x) p1(x) = Σx2,…,xn {0,1} pφ(x, x2, …, xn) p1(0)+p1(1)=k? pick random z1 in Fq z1 p2(x) = Σx3,…,xn {0,1} pφ(z1, x, x3, …, xn) p2(x) p2(0)+p2(1)=p1(z1)? pick random z2 in Fq z2 p3(x) = Σx4,…,xn {0,1} pφ(z1, z2, x, x4 …, xn) p3(x) p3(0)+p3(1)=p2(z2)? pick random z3 in Fq . . pn(x) pn(0)+pn(1)=pn-1(zn-1)? pick random zn in Fq pn(zn) = pφ(z1, z2, …, zn)? CS151 Lecture 14

  27. Analysis of protocol • Completeness: • if (φ, k)  L then honest prover on previous slide will always cause verifier to accept CS151 Lecture 14

  28. Analysis of protocol • Soundness: • let pi(x) be the correct polynomials • let pi*(x) be the polynomials sent by (cheating) prover • (φ, k)  Lp1(0) + p1(1) ≠ k • either p1*(0) + p1*(1) ≠ k (and V rejects) • or p1* ≠ p1 Prz1[p1*(z1) = p1(z1)]  d/q  |φ|/2n • assume (pi+1(0)+pi+1(1)= )pi(zi) ≠ pi*(zi) • either pi+1*(0) + pi+1*(1) ≠ pi*(zi) (and V rejects) • or pi+1* ≠ pi+1 Przi+1[pi+1*(zi+1) = pi+1(zi+1)]  |φ|/2n CS151 Lecture 14

  29. Analysis of protocol • Soundness (continued): • if verifier does not reject, there must be some i for which: pi* ≠ pi and yet pi*(zi) = pi(zi) • for each i, probability is  |φ|/2n • union bound: probability that there exists an i for which the bad event occurs is • n|φ|/2n  poly(n)/2n << 1/3 CS151 Lecture 14

  30. Analysis of protocol • Conclude: L = { (φ, k): CNF φ has exactly k satisfying assignments} is in IP • L is coNP-hard, so coNP IP • Question remains: • NP, coNP IP. Potentially larger. How much larger? CS151 Lecture 14

  31. Shamir’s Theorem Theorem: IP = PSPACE • Note: IP PSPACE • enumerate all possible interactions, explicitly calculate acceptance probability • interaction extremely powerful ! • An implication: you can interact with master player of Generalized Geography and determine if she can win from the current configuration even if you do not have the power to compute optimal moves! CS151 Lecture 14