Derandomization: New Results and Applications Emanuele Viola Harvard University March 2006

Useful throughout Computer Science –Algorithms –Learning Theory –Complexity Theory Question: Is Randomness necessary? Randomness in Computation InputAnswer (error probability 1%)

Goal: remove randomness Why study derandomization? Breakthrough [R ‘04]: Connectivity in logarithmic space (SL = L) Breakthrough [AKS ‘02]: Primality in polynomial time (PRIMES 2 P) Derandomization

Goal: simulate randomized computation deterministically Trivial Derandomization: If A uses n random bits, enumerate all 2 n possibilities Probabilistic polynomial-time µ exponential time BPP µ Time(2 poly(n) ) Strong Belief: BPP = P ( Time(poly(n)) ) Complexity Assumptions ) BPP = P [BFNW,NW,IW,…] Randomness vs. Time

Outline Overview of derandomization Derandomization of restricted models –Application: Hardness Amplification in NP –New derandomization Derandomization of general models –BPP vs. PH –Proof of Lower Bound

Constant-Depth Circuits Probabilistic constant-depth circuit (BP AC 0 ) Theorem [N ‘91]: BP AC  µ Time(n polylog n ) –Compare to BP P µ Time(2 poly(n) ) Input Depth Random Bits VVVVVVVV ÆÆÆÆÆÆ V

Application: Avg-Case Hardness of NP Study hardness of NP on random instances –Natural question, essential for cryptography Currently cannot relate to P  NP [FF,BT,V] Hardness amplification Definition: f : {0,1} n ! {0,1} is  -hard if for every efficient algorithm M : Pr x [M(x)  f(x)] ¸   Hardness Amplification f.01-hard f ’ (1/2 -  )-hard

Previous Results Yao’s XOR Lemma: f 0 (x 1,…, x n ) := f(x 1 ) ©  © f(x n ) f 0 ¼ (1/2 – 2 -n )-hard, almost optimal Cannot use XOR in NP: f 2 NP ; f 0 2 NP Idea: f´(x 1,…, x n ) = C( f(x 1 ),…, f(x n ) ), C monotone –e.g. f(x 1 ) Æ ( f(x 2 ) Ç f(x 3 ) ). f 2 NP ) f 0 2 NP Theorem [O’D]: There is C s.t. f 0 ¼ (1/2 – 1/n)-hard Barrier: No monotone C can do better!

Theorem [HVV]: Amplification in NP up to ¼ 1/ n –Matches the XOR Lemma Technique: Derandomize! Intuitively, f´ := C( f(x 1 ),…, f(x n ), … … f(x 2 n ) ) f´ (1/2 – 1/2 n )-hard by previous result Problem: Input length = 2 n Note C is constant-depth Derandomize: input length ! n, keep hardness Our Result on Hardness Amplification C f(x 1 ),…, f(x n ), … … f(x 2 n ) f ’

Outline Overview of derandomization Derandomization of restricted models –Application: Hardness Amplification in NP –New derandomization Derandomization of general models –BPP vs. PH –Proof of Lower Bound

Previous Results Recall Theorem [N]: BP AC  µ Time(n polylog n ) But AC 0 is weak: Majority  AC 0 –Majority(x 1,…,x n ) :=  i x i > n/2 ? Theorem [LVW]: BP Maj AND µ Time(2 n  ) Derandomize incomparable classes ÆÆÆÆÆÆ Maj VVVVVVVV ÆÆÆÆÆÆ V

Theorem [V] : BP Maj AC  µ Time(2 n  ) Derandomize constant-depth circuits with few Majority gates = Improves on [LVW]. Slower than [N] but richer richest probabilistic circuit class in Time(2 n  ) Techniques: Communication complexity + switching lemma [BNS,HG,H,HM,CH] Our New Derandomization ÆÆÆÆÆÆ ÇÇÇÇ Maj InputRandom Bits

Outline Overview of derandomization Derandomization of restricted models –Application: Hardness Amplification in NP –New derandomization Derandomization of general models –BPP vs. PH –Proof of Lower Bound

Probabilistic Polynomial Time (BPP): for every x, Pr [ M(x) errs ] · 1% Strong belief: BPP = P [NW,BFNW,IW,…] Still open: BPP µ NP ? Theorem [SG,L; ‘83]: BPP µ  2 P Recall NP =   P ! 9 y M(x,y)  2 P ! 9 y 8 z M(x,y,z) BPP vs. POLY-TIME HIERARCHY

More precisely [SG,L] give BPTime(t) µ  2 Time( t 2 ) Question[Rest of this Talk]: Is quadratic slow-down necessary? Motivation: Lower bounds Know NTime ≠ Time on some models [P+,F+,…] Technique: speed-up computation with quantifiers To prove NTime ≠ BPTime cannot afford  2 Time( t 2 ) The Problem we Study

Input: R = Task: Tell Pr i [ R i = 1] ¸ 99% from Pr i [ R i = 1] · 1% Do not care if Pr i [ R i = 1] ~ 50% (approximate) Model: Depth-3 circuit Approximate Majority R = VVVVVVVV ÆÆÆÆÆÆ V Depth

M(x;u) 2 BPTime(t) R = Compute M(x): Tell Pr u [M(x) = 1] ¸ 99% Compute Appr-Maj from Pr u [M(x) = 1] · 1% BPTime(t) µ  2 Time(t’) = 9 8 Time(t’) Running time t’ Bottom fan-in f = t’ / t –run M at most t’/t times The connection [FSS] VVVVVVVV ÆÆÆÆÆÆ V  f  |R| = 2 t R i = M(x;i)

Theorem[V] : Small depth-3 circuits for Approximate Majority on N bits have bottom fan-in  (log N) Corollary: Quadratic slow-down necessary for relativizing techniques: BPTime A (t) µ  2 Time A (t 1.99 ) Theorem[DvM,V]: BPTime (t) µ  3 Time (t ¢ log 5 t) –Previous result [A]: BPTime (t) µ  O(1) Time( t ) For time, the level is the third! Our Results

Outline Overview of derandomization Derandomization of restricted models –Application: Hardness Amplification in NP –New derandomization Derandomization of general models –BPP vs. PH –Proof of Lower Bound

Theorem[V]: 2 N  -size depth-3 circuits for Approximate Majority on N bits have bottom fan-in f =  (log N) Recall: Tells R 2 YES := { R : Pr i [ R i = 1] ¸ 99% } from R 2 NO := { R : Pr i [ R i = 1] · 1% } Our Negative Result VVVVVVVV ÆÆÆÆÆÆ V  f  R = |R| = N

Circuit is OR of s depth-2 circuits By definition of OR : R 2 YES ) some C i (R) = 1 R 2 NO ) all C i (R) = 0 By averaging, fix C = C i s.t. Pr R 2 YES [C (x) = 1 ] ¸ 1/s 8 R 2 NO ) C (R) = 0 Claim: Impossible if C has bottom fan-in ·  log N Proof V C 1 C 2 C 3   C s

Depth-2 circuit ) CNF (x 1 Vx 2 V : x 3 ) Æ ( : x 4 ) Æ (x 5 Vx 3 ) bottom fan-in ) clause size Claim: All CNF C with clauses of size  ¢ log N Either Pr R 2 YES [C (x) = 1 ] · 1 / 2 N  or there is R 2 NO : C(x) = 1 Note: Claim ) Theorem CNF Claim VVVVV Æ x 1 x 2 x 3 … x N

Definition: S µ {x 1,x 2,…,x N } is a covering if every clause has a variable in S E.g.: S = {x 3,x 4 } C = (x 1 Vx 2 V : x 3 ) Æ ( : x 4 ) Æ (x 5 Vx 3 ) Proof idea: Consider smallest covering S Case |S| BIG : Pr R 2 YES [C (x) = 1 ] · 1 / 2 N  Case |S| tiny : Fix few variables and repeat Proof Outline Either Pr R 2 YES [C(x)=1] · 1/2 N   or 9 R 2 NO : C(x) = 1

|S| ¸ N  ) have N  /(  ¢ log N) disjoint clauses  i –Can find  i greedily Pr R 2 YES [ C(R) = 1 ] · Pr [ 8 i,  i (R) = 1 ] =  i Pr[  i (R) = 1] (independence) ·  i (1 – 1/100  log N ) =  i ( 1 – 1/N O(  ) ) = ( 1 – 1/N O(  ) ) |S| · e -N  (1) Case |S| BIG Either Pr R 2 YES [C(x)=1] · 1/2 N   or 9 R 2 NO : C(x) = 1

|S| < N  ) Fix variables in S –Maximize Pr R 2 YES [C(x)=1] Note: S covering ) clauses shrink Example (x 1 Vx 2 Vx 3 ) Æ ( : x 3 ) Æ (x 5 V : x 4 ) (x 1 Vx 2 ) Æ (x 5 ) Repeat Consider smallest covering S’, etc. Case |S| tiny Either Pr R 2 YES [C(x)=1] · 1/2 N   or 9 R 2 NO : C(x) = 1 x 3 Ã 0 x 4 Ã 1

Recall: Repeat ) shrink clauses So repeat at most  ¢ log N times When you stop: Either smallest covering size ¸ N  Or C = 1 Fixed · (  ¢ log N) N  ¿ N vars. Set rest to 0 ) R 2 NO : C(R) = 1 Q.E.D. Finish up Either Pr R 2 YES [C(x)=1] · 1/2 N   or 9 R 2 NO : C(x) = 1

Derandomization: powerful technique Restricted models: Constant-depth circuits (AC 0 ) –Derandomization of AC 0 [N] –Application: Hardness Amplification in NP [HVV] –Derandomization of AC 0 with few Maj gates [V] General models: BPP vs. PH –BPTime(t) µ  2 Time( t 2 ) [SG,L] –BPTime (t) µ  2 Time (t 1.99 ) (w.r.t. oracle) [V] Lower Bound for Approximate Majority Conclusion

Thank you!