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On Probabilistic Time versus Alternating Time

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This talk by Emanuele Viola from Harvard University delves into the complexities of Probabilistic Polynomial Time (BPP) and its relations to various computational classes, particularly concerning the necessity of quadratic slow-downs in approximating majority functions. The discussion builds upon foundational theorems, positing questions about the lower bounds of computation within these frameworks, and examines the implications for circuit depth in theoretical computer science. Key results and their significance suggest potential breakthroughs in understanding BPP and its limitations against polynomial-time constraints.

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On Probabilistic Time versus Alternating Time

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  1. On Probabilistic TimeversusAlternating Time Emanuele Viola Harvard University March 2006

  2. 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 µS2 P • Recall NP = S1P!9 y M(x,y) S2 P !9 y 8 z M(x,y,z)

  3. The Problem we Study • More precisely [SG,L] give BPTime(t) µS2Time( 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]

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

  5. The connection [FSS] |R| = 2t Ri = M(x;i) 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 8Time(t’) Running time t’ Bottom fan-in f = t’ / t • run M at most t’/t times V Æ Æ Æ Æ Æ Æ V V V V V V V V L f L 101111011011101011

  6. 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) µS2Time A(t1.99) Proof of Corollary: BPTime(t) µS2 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.

  7. Quasilinear-time simulation? • Question: BPTime(t) µS3 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)

  8. 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) µS3Time (t ¢ log5 t)

  9. Summary

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

  11. 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 • Note: No known 2W(N) size lower bound for depth-3 circuits w/ bottom fan-in w(1)

  12. 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

  13. Proof V C1 C2 C3 LL Cs • 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 (x) = 1 ] ¸ 1/s 8 R 2 NO ) C (R) = 0 • Claim: Impossible if C has bottom fan-in ·  log N

  14. CNF Claim Æ • 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 (x) = 1 ] · 1 / 2N or there is R 2 NO : C(x) = 1 • Note: Claim ) Theorem V V V V V x1 x2 x3 … xN

  15. Either PrR 2 YES [C(x)=1]·1/2Nor 9 R 2 NO : C(x) = 1 Proof Outline • 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 (x) = 1 ] · 1 / 2N Case |S| tiny : Fix few variables and repeat

  16. Either PrR 2 YES [C(x)=1]·1/2Nor 9 R 2 NO : C(x) = 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)

  17. x3Ã 0 x4Ã 1 Either PrR 2 YES [C(x)=1]·1/2Nor 9 R 2 NO : C(x) = 1 Case |S| tiny • |S| < N) Fix variables in S • Maximize PrR 2 YES [C(x)=1] • Note: S covering ) clauses shrink Example (x1Vx2Vx3 ) Æ (:x3) Æ (x5V:x4) (x1Vx2 ) Æ (x5) • Repeat Consider smallest covering S’, etc.

  18. Either PrR 2 YES [C(x)=1]·1/2Nor 9 R 2 NO : C(x) = 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.

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

  20. Time Lower Bound for SAT Work tape Input 0 0 0 0 0 0 0 0 0 0 • Model: Time1 • Theorem [MS,vMR]: NTime (n) µ Time1 (n1.22) • Proof (by contradiction): NTime(n) µ Time1 (n1.22) µO(1) Time(n.1) (speed-up with quantifiers) µ NTime(n.9) (collapse by assumption) Contradiction with NTime hierarchy Q.E.D.

  21. Our BPTime Lower Bound for 3 Work tape Input • Model: BPTime1 • Theorem [V]:3Time (n) µ BPTime1 (n1+o(1)) • Proof [Inspired by DvM]: 3Time(n) µ BPTime1 (n1+o(1)) µ BPn Time1(n1+o(1)) (derandomize [INW]) µ3 Time(n.9) (collapse [DvM,V] ) Q.E.D. • Note: Quadratic slow-down in 2) idea won’t work for 2

  22. Conclusion • Theorem[SG,L]: BPTime(t) µS2Time( t2 ) • Related to Approximate Majority • Theorem [V]: 3Time (n) µ BPTime1 (n1+o(1))

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