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CS151 Complexity Theory

CS151 Complexity Theory. Lecture 13 May 11, 2004. Outline. Natural complete problems for PH and PSPACE proof systems interactive proofs and their power. Simpler version of MIN DNF. Theorem (U): MIN DNF is Σ 2 -complete. we’ll consider a simpler variant:

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CS151 Complexity Theory

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  1. CS151Complexity Theory Lecture 13 May 11, 2004

  2. Outline • Natural complete problems for PH and PSPACE • proof systems • interactive proofs and their power CS151 Lecture 13

  3. Simpler version of MIN DNF Theorem (U): MIN DNF is Σ2-complete. • we’ll consider a simpler variant: • IRREDUNDANT: given DNF φ, integer k; is there a DNF φ’ consisting of at most k terms of φ computing same function φ does? CS151 Lecture 13

  4. Simpler version of MIN DNF • analogy with an NP-complete problem: • SET COVER: given subsets S1,S2,...,Sm U, integer k, is there a collection of at most k sets that cover U. φ-1(0) φ-1(1) U terms of φ sets Si {0,1}n CS151 Lecture 13

  5. SSC is Σ2-complete • “sets” in IRREDUNDANT lie in an exponentially larger universe; they are represented succinctly by terms of φ • helpful intermediate problem: • SUCCINCT SET COVER (SSC): given 3-DNFs S = {φ1, φ2, φ3,..., φm} on n variables, integer k; is there a collection S’  S of size at most k for which φS’  1 (S’ is a cover)? CS151 Lecture 13

  6. SSC is Σ2-complete Theorem: SSC is Σ2-complete. • Proof: • in Σ2 (why?) “ S’  S  x [φS’(x) = 1]” • reduce from QSAT2 • instance: AB φ(A, B) = 1 • assume |A| = |B| = n CS151 Lecture 13

  7. SSC is Σ2-complete AB φ(A, B) = 1 • Proof (continued): • 2 new sets of variables S, T • |S| = |T| = n • Define: wt(S, T) = # of 1s in S and T together • “(S,T) encodes A”means i(si=ai)  (ti=ai) CS151 Lecture 13

  8. SSC is Σ2-complete • From φ(A, B)we define function f(S, T, B): 0 if wt(S, T) < n 0 if wt(S, T) = n and (S,T) does not encode any A 0 if wt(S, T) = n and (S,T) encodes A for which φ(A, B) = 0 1 if wt(S, T) = n and (S,T) encodes A for which φ(A, B) = 1 1 if wt(S, T) > n • verify: poly(n) size circuit C computes f • verify: f is monotone in S and T CS151 Lecture 13

  9. SSC is Σ2-complete 0 if wt(S, T) < n 0 if wt(S, T) = n and (S,T) does not encode any A 0 if wt(S, T) = n and (S,T) encodes A : φ(A,B) = 0 1 if wt(S, T) = n and (S,T) encodes A : φ(A,B) = 1 1 if wt(S, T) > n • Proof (continued): • produce an instance of SSC: • # of sets m = 2n + 1 • φi= (si) φi+n= (ti) • from C get a 3-DNF φm(S,T,B,W): f(S,T,B) = 1  W φm(S,T,B,W) = 1 CS151 Lecture 13

  10. SSC is Σ2-complete 0 if wt(S, T) < n 0 if wt(S, T) = n and (S,T) does not encode any A 0 if wt(S, T) = n and (S,T) encodes A : φ(A,B) = 0 1 if wt(S, T) = n and (S,T) encodes A : φ(A,B) = 1 1 if wt(S, T) > n Claim: AB φ(A, B) = 1 implies S’ = {φi | ai = 1}  {φi+n | ai = 0}  {φm} is a cover of size n+1. Proof: consider each fixed point (S,T,B,W) • if there exists (S’,T’) that encodes A and we have (S’,T’)  (S,T) then φm(S,T,B,W) = 1 • else, i (ai = 1 and si = 0) or (ai = 0 and ti = 0) =(ti) =(si) CS151 Lecture 13

  11. SSC is Σ2-complete Claim: cover S’ of size n+1 implies AB φ(A, B) = 1 Proof: • S’ must contain φm; otherwise it fails to cover the all-ones point • consider the pair (S*, T*) for which: • si = 1 if φi S’ and 0 otherwise • ti = 1 if φi+n S’ and 0 otherwise • must have: BW φm(S*, T*, B, W) = 1 • implies: B f(S*, T*, B) = 1 CS151 Lecture 13

  12. SSC is Σ2-complete 0 if wt(S, T) = n and (S,T) does not encode any A 0 if wt(S, T) = n and (S,T) encodes A : φ(A,B) = 0 1 if wt(S, T) = n and (S,T) encodes A : φ(A,B) = 1 • defined the pair (S*, T*) as follows: • si = 1 if φi S’ and 0 otherwise • ti = 1 if φi+n S’ and 0 otherwise • concluded: B f(S*, T*, B) = 1 • Note: wt(S*, T*) = n • (S*,T*) must encode A s.t. B φ(A, B) = 1 • Conclude: AB φ(A, B) = 1 CS151 Lecture 13

  13. IRR is Σ2-complete • Recall: IRREDUNDANT: given DNF φ, integer k; is there a DNF φ’ consisting of at most k terms of φ computing same function φ does? Theorem: IRR is Σ2-complete. • Proof: • in Σ2: “ φ’  x [φ’(x) = φ(x)]” CS151 Lecture 13

  14. IRR is Σ2-complete • reduce from SSC • instance:S = {φ1, φ2, φ3,..., φm} • may assume • φ1, φ2,..., φm-1 single literals • φm necessary in any cover • S is a cover • write out terms: φm = t1  t2  t3  …  tn • produce an instance of IRR: φ=i=1…n(z1…zi-1 zi+1…zn ti)  j=1…m-1(z1…znφj) CS151 Lecture 13

  15. IRR is Σ2-complete S = {φ1, φ2, φ3,..., φm = t1  …  tn } φ=i=1…n(z1…zi-1 zi+1…zn ti)  j=1…m-1(z1…znφj) • Proof (continued): Claim: if S’  S is a cover of size k then φ’=i=1…n(z1…zi-1 zi+1…zn ti)  j<m,φj S’(z1z2…zn φj) is equivalent to φ and has k+n-1 terms. Proof: by cases CS151 Lecture 13

  16. IRR is Σ2-complete S = {φ1, φ2, φ3,..., φm = t1  …  tn }; S’  S φ=i=1…n(z1…zi-1 zi+1…zn ti)  j=1…m-1(z1…znφj) φ’=i=1…n(z1…zi-1 zi+1…zn ti)  j<m,φj S’(z1z2…znφj) • more than one z variable 0: both φ’, φ are 0 • zi 0, other z’s 1: φ’, φ equivalent to ti • all z’s 1: • φ’ equivalent to φj  S’(z1z2…znφj) • φ’ equivalent to φj  S(z1…znφj) • S’ is a cover implies both equivalent to 1 CS151 Lecture 13

  17. IRR is Σ2-complete S = {φ1, φ2, φ3,..., φm = t1  …  tn } φ=i=1…n(z1…zi-1 zi+1…zn ti)  j=1…m-1(z1…znφj) • Proof (continued): Claim: if φ’  φ uses k+n-1 terms of φ, then there exists a cover S’ of size k Proof: • each “ti term” of φ must be present CS151 Lecture 13

  18. IRR is Σ2-complete S = {φ1, φ2, φ3,..., φm = t1  …  tn } φ=i=1…n(z1…zi-1 zi+1…zn ti)  j=1…m-1(z1…znφj) φ’=i=1…n(z1…zi-1 zi+1…zn ti)  ??? (k+n-1 termstotal) • other k-1 terms all involve some φj • let S’ be these φj together with φm (φjS’ φj) φ’z11…1 φz11…1  (φjS φj) 1 • conclude S’ is a cover of size k CS151 Lecture 13

  19. PSPACE • General phenomenon: many 2-player games are PSPACE-complete. • 2 players I, II • alternate pick- ing edges • lose when no unvisited choice pasadena auckland athens davis san francisco oakland • GEOGRAPHY = {(G, s) : G is a directed graph and player I can win from node s} CS151 Lecture 13

  20. PSPACE Theorem: GEOGRAPHY is PSPACE-complete. Proof: • in PSPACE • easily expressed with alternating quantifiers • PSPACE-hard • reduction from QSAT CS151 Lecture 13

  21. PSPACE x1x2x3 … xn φ(x1, x2, …, xn)? clause choice gadget true false variable gadget for xi … C1 C2 Cm CS151 Lecture 13

  22. PSPACE x1x2x3…(x1x2x3)(x3x1)…(x1x2) I false alternately pick truth assignment true x1 II I II x2 true false I II I I x3 true false pick a clause II II … I I pick a true literal? CS151 Lecture 13

  23. Proof systems L = { (A, 1k) : A is a true mathematical assertion with a proof of length k} • New topic: What is a “proof”? complexity insight: meaningless unless can be efficiently verified CS151 Lecture 13

  24. Proof systems • given language L, goal is to prove x  L • proof system for L is a verification algorithm V • completeness: x  L   proof, V accepts (x, proof) “true assertions have proofs” • soundness: x  L   proof*, V rejects (x, proof*) “false assertions have no proofs” • efficiency:  x, proof, V(x, proof) runs in polynomial time in |x| CS151 Lecture 13

  25. Classical Proofs • previous definition: “classical” proof system • recall: L  NP iff expressible as L = { x |  y, |y| < |x|k, (x, y)  R } and R  P. • NP is the set of languages with classical proof systems (R is the verifier) CS151 Lecture 13

  26. Interactive Proofs • Two new ingredients: • randomness: verifier tosses coins, errs with some small probability • interaction: rather than “reading” proof, verifier interacts with computationally unbounded prover • NP proof systems lie in this framework: prover sends proof, verifier does not use randomness CS151 Lecture 13

  27. 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 13

  28. 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 13

  29. 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 13

  30. 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 13

  31. 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 13

  32. 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 13

  33. 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 13

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