On approximate majority and probabilistic time Emanuele Viola Institute for advanced study January 2007

BPP vs. POLY-TIME HIERARCHY Probabilistic Polynomial Time (BPP): for every x, Pr [ M(x) errs ] · 1/3 Strong belief: BPP = P [NW,BFNW,IW,…] Still open: BPP µ NP ? Theorem [SG,L; ‘83]: BPP µ S2 P Recall NP = S1 P ! 9 y M(x,y) S2 P ! 9 y 8 z M(x,y,z)

The problem we study More precisely [SG,L] give BPTime(t) µ S2Time( t2 ) Question[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 Time( t2 ) [DvM]

Approximate Majority Input: R = 101111011011101011 Task: Tell Pri [ Ri = 1] ¸ 2/3 from Pri [ Ri = 1] · 1/3 Approximate: Do not care if Pri [ Ri = 1] ~ 1/2 Model: Depth-3 circuit V Æ Æ Æ Æ Æ Æ Depth V V V V V V V V R = 101111011011101011

The connection [FSS] M(x;u) 2 BPTime(t) R = 11011011101011 Compute M(x): Tell Pru[M(x) = 1] ¸ 2/3 Compute Appr-Maj from Pru[M(x) = 1] · 1/3 BPTime(t) µ S2 Time(t’) = 9 8 Time(t’) Running time t’ Bottom fan-in f = t’ / t run M at most t’/t times |R| = 2t Ri = M(x;i) V Æ Æ Æ Æ Æ Æ V V V V V V V V L f L 101111011011101011

Our Negative Result Theorem[V] : Small depth-3 circuits for Approximate Majority on N bits have bottom fan-in W(log N) Corollary: Quadratic slow-down necessary for relativizing techniques: BPTime A (t) µ S2Time A (t1.99) Proof of Corollary: BPTime (t) µ S2 Time (t’) ) [FSS] Appr-Maj on N = 2t bits 2 depth-3, bottom fan-in t’ / t. By Theorem: t’ / t = (t). Q.E.D.

Quasilinear-time simulation? Question: BPTime(t) µ S3 Time(t ¢ polylog t) ? Related: Appr-Maj 2 depth-3 poly-size ? arbitrary bottom fan-in Previous results & problems: [SG,L] Appr-Maj 2 depth-3 size Nlog N [A] Appr-Maj 2 depth-3 size poly(N) nonuniform [A] Appr-Maj 2 depth-O(1) size poly(N)

Our Positive Results Theorem[V] : There are uniform depth-3 poly(N)-size circuits for Approximate Majority on N bits Uniform version of Ajtai’s result Theorem[DvM,V]: BPTime (t) µ S3Time (t ¢ log5 t)

Summary Appr-Maj on N bits BPTime(t) [SG,L] 2 size Nlog N depth 3 µ S2Time( t2 ) [A] 2 size poly(N) depth 3 non-uniform ---------- 2 size poly(N) depth O(1) µ SO(1)Time( t ) [V] 2 size 2N depth 3 bottom fan-in ¢log N µ S2Time (t1.99) w.r.t. oracle [DvM,V] µ S3Time (t¢log5 t)

Rest of slides Proof of bottom fan-in lower bound Other result 3Time (t) µ BPTime (t1+o(1)) on restricted models

Our negative result Theorem[V]: 2N-size depth-3 circuits for Approximate Majority on N bits have bottom fan-in W(log N) Switching lemmas fail Cannot use [H] for Approximate-Majority [SBI] ) bottom fan-in ¸ (log N)1/2 Independently: [R] improves [SBI] alternative proof of theorem Note: No 2W(N) bound for depth-3 w/ bottom fan-in w(1)

Our Negative Result Theorem[V]: 2N-size depth-3 circuits for Approximate Majority on N bits have bottom fan-in f = W(log N) Recall: Tells R 2 YES := { R : Pri [ Ri = 1] ¸ 2/3 } from R 2 NO := { R : Pri [ Ri = 1] · 1/3 } V Æ Æ Æ Æ Æ Æ V V V V V V V V L f L R = 101111011011101011 |R| = N

Proof Circuit is OR of s depth-2 circuits By definition of OR : R 2 YES ) some Ci (R) = 1 R 2 NO ) all Ci (R) = 0 By averaging, fix C = Ci s.t. PrR 2 YES [C (R) = 1 ] ¸ 1/s 8 R 2 NO ) C (R) = 0 Claim: Impossible if C has bottom fan-in ·  log N V C1 C2 C3 L L Cs

CNF Claim Depth-2 circuit ) CNF (x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) bottom fan-in ) clause size Claim: All CNF C with clauses of size ¢log N Either PrR 2 YES [C (R) = 1 ] · 1 / 2N or there is R 2 NO : C(R) = 1 Note: Claim ) Theorem Æ V V V V V x1 x2 x3 … xN

Proof Outline Either PrR 2 YES [C(R)=1]·1/2N or 9 R 2 NO : C(R) = 1 Definition: S µ {x1,x2,…,xN} is a covering if every clause has a variable in S E.g.: S = {x3,x4} C = (x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Proof idea: Consider smallest covering S Case |S| BIG : PrR 2 YES [C (R) = 1 ] · 1 / 2N Case |S| tiny : Fix few variables and repeat

Case |S| BIG PrR 2 YES [C(R) = 1] · Pr [ 8 i, Gi(R) = 1 ] Either PrR 2 YES [C(R)=1]·1/2N or 9 R 2 NO : C(R) = 1 Case |S| BIG |S| ¸ N ) have N /(¢log N) disjoint clauses Gi Can find Gi greedily PrR 2 YES [C(R) = 1] · Pr [ 8 i, Gi(R) = 1 ] = Õi Pr[ Gi(R) = 1] (independence) · Õi (1 – 1/3log N ) = Õi (1 – 1/NO()) = (1 – 1/NO()) |S| · e-N(1)

Case |S| tiny Either PrR 2 YES [C(R)=1]·1/2N or 9 R 2 NO : C(R) = 1 |S| < N ) Fix variables in S Maximize PrR 2 YES [C(R)=1] Note: S covering ) clauses shrink Example (x1Vx2Vx3 ) Æ (:x3) Æ (x5V:x4) (x1Vx2 ) Æ (x5) Repeat Consider smallest covering S’, etc. x3 Ã 0 x4 Ã 1

Finish up Recall: Repeat ) shrink clauses Either PrR 2 YES [C(R)=1]·1/2N or 9 R 2 NO : C(R) = 1 Finish up 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.

Rest of slides Proof of bottom fan-in lower bound Other result 3Time (t) µ BPTime (t1+o(1)) on restricted models

(x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Æ (x1Vx6V:x3 ) The model Time1 Input (x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Æ (x1Vx6V:x3 ) Random access 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 Sequential access work tape

Time Lower Bound for SAT Theorem [MS,vMR]: NTime (n) µ Time1 (n1.22) Note: ``combinatorics’’ does not work Palindromes 2 Time1 (n1+o(1)) Proof by contradiction: Suppose NTime(n) µ Time1 (n1.22) µ O(1) Time(n0.1) (speed-up with quantifiers) µ NTime(n0.9) (collapse by assumption) Contradicts NTime hierarchy Q.E.D.

(x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Æ (x1Vx6V:x3 ) The model BPTime1 Input (x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Æ (x1Vx6V:x3 ) Random access 0 1 Sequential access work tape with coins

Our BPTime Lower Bound for 3 Theorem [V] : 3Time (n) µ BPTime1 (n1+o(1)) Proof by contradiction (Inspired by [DvM]): Suppose 3Time(n) µ BPTime1 (n1+o(1)) µ BPn0.9 Time1(n1+o(1)) (derandomize [INW]) µ 3 Time(n.99) ([V] + [MS]) Contradicts 3 Time hierarchy Q.E.D. Note: Quadratic slow-down ) won’t work for 2

Seen so far Theorem[SG,L]: BPTime(t) µ S2Time( t2 ) Related to Approximate Majority Theorem [V] : 3Time (n) µ BPTime1 (n1+o(1)) Appr-Maj on N bits BPTime(t) [V] 2 size 2N depth 3 bottom fan-in ¢log N µ S2Time (t1.99) w.r.t. oracle [DvM,V] 2 size poly(N) depth 3 uniform µ S3Time (t¢log5 t)