1. On Probabilistic Time versus Alternating 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 µ S2 P
Recall
NP = S1 P ! 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) 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]

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 =

5. The connection [FSS] M(x;u) 2 BPTime(t) R = 11011011101011
Compute M(x):
Tell Pru[M(x) = 1] ¸ 2/ 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

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) 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.

7. 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)

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) µ S3Time (t ¢ log5 t)

9. 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)

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 = |R| = N

13. 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 (x) = 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

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

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

16. Case |S| BIG PrR 2 YES [C(R) = 1] · Pr [ 8 i, Gi(R) = 1 ]
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. Case |S| tiny Either PrR 2 YES [C(x)=1]·1/2N or 9 R 2 NO : C(x) = 1
|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.
x3 Ã 0
x4 Ã 1

18. Finish up Recall: Repeat ) shrink clauses
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
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.
Work tape
Input

21. Our BPTime Lower Bound for 3
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
Work tape
Input

22. Conclusion 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)