1. On approximate majority and probabilistic time Emanuele Viola Institute for advanced study Work done during Ph.D. at Harvard University June 2007

2. Probabilistic Polynomial Time (BPP): for every x, Pr [ M(x) errs ] · 1/3 Belief: BPP = P [Babai Fortnow Impagliazzo Nisan Wigderson …] Still open: BPP µ NP ? Theorem [Sipser & Gacs, Lautemann; ‘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

3. More precisely [SG,L] give BPTime(t) µ  2 Time( t 2 ) 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( t 2 ) The problem we study

4. Input: R = 101111011011101011 Task: Tell Pr i [ R i = 1] ¸ 2/3 from Pr i [ R i = 1] · 1/3 Approximate: Do not care if Pr i [ R i = 1] ~ 1/2 Model: Depth-3 circuit Approximate Majority R = 101111011011101011 VVVVVVVV ÆÆÆÆÆÆ V Depth

5. M(x;u) 2 BPTime(t) R = 11011011101011 Compute M(x): Tell Pr u [M(x) = 1] ¸ 2/3 Compute Appr-Maj from Pr u [M(x) = 1] · 1/3 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 [Furst Saxe Sipser] VVVVVVVV ÆÆÆÆÆÆ V  f  |R| = 2 t R i = M(x;i) 101111011011101011

6. 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 ) Proof of Corollary: BPTime (t) µ  2 Time (t’) ) [FSS] Appr-Maj on N = 2 t bits 2 depth-3, bottom fan-in t’ / t. By Theorem: t’ / t =  (t). Q.E.D. Our negative result

7. Question: BPTime(t) µ  3 Time(t ¢ polylog t) ? Related: Appr-Maj 2 depth-3 poly-size ? –arbitrary bottom fan-in Previous results & problems: [ Sipser & Gacs, Lautemann ] Appr-Maj 2 depth-3 size N log N [Ajtai] Appr-Maj 2 depth-3 size poly(N) nonuniform [Ajtai] Appr-Maj 2 depth-O(1) size poly(N) Quasilinear-time simulation?

8. Theorem[V] : There are uniform depth-3 poly(N)-size circuits for Approximate Majority on N bits –Uniform version of Ajtai’s result Theorem[Diehl & van Melkebeek,V]: BPTime (t) µ  3 Time (t ¢ log 5 t) Our positive results

9. Summary Appr-Maj on N bitsBPTime(t) [SG,L] 2 size N log N depth 3 µ  2 Time( t 2 ) [A] 2 size poly(N) depth 3 non-uniform ---------- [A] 2 size poly(N) depth O(1) µ  O(1) Time( t ) [V][V] 2 size 2 N 0.1 depth 3 bottom fan-in  ¢ log N µ  2 Time (t 1.99 ) w.r.t. oracle [DvM,V] 2 size poly(N) depth 3 µ  3 Time (t ¢ log 5 t)

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

11. Theorem[V]: 2 N 0.1 -size depth-3 circuits for N-bit Approximate Majority have bottom fan-in  (log N) Switching lemmas fail: Cannot use [Hastad] for Approximate-Majority [Segerlind Buss Impagliazzo] ) bottom fan-in ¸ (log N) 1/2 [Razborov] improves [SBI], alternative proof of theorem Note: No 2  (N) bound for depth-3 w/ bottom fan-in  (1) Our negative result

12. Theorem[V]: 2 N 0.1 -size depth-3 circuits for N-bit Approximate Majority have bottom fan-in f =  (log N) Recall: Tells R 2 YES := { R : Pr i [ R i = 1] ¸ 2/3 } from R 2 NO := { R : Pr i [ R i = 1] · 1/3 } Our negative result VVVVVVVV ÆÆÆÆÆÆ V  f  R = 101111011011101011 |R| = N

13. Circuit is OR of s = 2 N 0.1 CNF E.g. C i = (x 1 Vx 2 V : x 3 ) Æ ( : x 4 ) Æ (x 5 Vx 3 ) Bottom fan-in ) clause size By definition of OR :R 2 YES ) some C i (R) = 1 R 2 NO ) all C i (R) = 0 By averaging, fix CNF C = C i s.t. Pr R 2 YES [C (R) = 1 ] ¸ 1/s = 1/2 N 0.1 8 R 2 NO ) C (R) = 0 Claim: Impossible if C has clauses of size  ¢ log N Proof V C 1 C 2 C 3  C s

14. 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 (R) = 1 ] · 1 / 2 N 0.1 Case |S| tiny : Fix few variables and repeat Proof outline Either Pr R 2 YES [C(R)=1] · 1/2 N 0.1 or 9 R 2 NO : C(R)=1

15. |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 – 1/3  log N ) =  i ( 1 – 1/N O(  ) ) = ( 1 – 1/N O(  ) ) |S| · e -N  (1) Case |S| BIG Either Pr R 2 YES [C(R)=1] · 1/2 N 0.1 or 9 R 2 NO : C(R)=1

16. |S| < N  ) Fix variables in S –Maximize Pr R 2 YES [C(R)=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 x 3 Ã 0 x 4 Ã 1 Either Pr R 2 YES [C(R)=1] · 1/2 N 0.1 or 9 R 2 NO : C(R)=1

17. 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(R)=1] · 1/2 N 0.1 or 9 R 2 NO : C(R)=1

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

19. Space-bounded, probabilistic models [Ajtai, Beame Saks Sun Vee] n log 0.5 n time lower bound –branching programs [ Allender Koucký Ronneburger Roy Vinay, Diehl & van Melkebeek ] n 1+  (1) time lower bounds one-way randomness Theor.[V]:  3 Time (n) µ BPTime(n 1+o(1) ) two-way randomness (sequential) –Proof uses our positive result –Our negative result is obstacle for  2 Lower bounds on probabilistic models

20. [SG,L]: BPTime(t) µ  2 Time( t 2 ) –Related to Approximate Majority Theorem[V] :  3 Time (n) µ BPTime(n 1+o(1) ) two-way access to randomness Conclusion Appr-Maj on N bitsBPTime(t) [V][V] 2 size 2 N 0.1 depth 3 bottom fan-in  ¢ log N µ  2 Time (t 1.99 ) w.r.t. oracle [DvM,V] 2 size poly(N) depth 3 uniform µ  3 Time (t ¢ log 5 t)

21. Thank you!