On approximate majority
and
probabilistic time
Emanuele Viola
Institute for advanced study
Work done during Ph.D. at Harvard University
June 2007
BPP vs. POLY-TIME HIERARCHY
• 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 S2 P
• Recall
NP = S1 P !
S2 P
!
9 y M(x,y)
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 )
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
V V V V V V V V
R = 101111011011101011
Depth
The connection [Furst Saxe Sipser]
M(x;u) 2 BPTime(t)
R = 11011011101011
|R| = 2t
Ri = M(x;i)
Compute M(x):
Tell Pru[M(x) = 1] ¸ 2/3
from Pru[M(x) = 1] · 1/3
BPTime(t) µ S2 Time(t’)
= 9 8 Time(t’)
Compute Appr-Maj
V
Æ Æ Æ Æ Æ Æ
V V V V V V V V
LfL
101111011011101011
Running time t’
– run M at most t’/t times
Bottom fan-in f = t’ / t
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 = W(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:
[Sipser & Gacs, Lautemann] Appr-Maj 2 depth-3 size Nlog N
[Ajtai] Appr-Maj 2 depth-3 size poly(N) nonuniform
[Ajtai] 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[Diehl & van Melkebeek,V]:
BPTime (t) µ S3Time (t ¢ log5 t)
Summary
Appr-Maj on N bits
[SG,L] 2 size Nlog N depth 3
[A]
[A]
[V]
2 size poly(N) depth 3
BPTime(t)
µ S2Time( t2 )
----------
non-uniform
2 size poly(N) depth O(1) µ SO(1)Time( t )
0.1
N
2
2 size
depth 3
bottom fan-in ¢log N
[DvM,V] 2 size poly(N) depth 3
µ S2Time (t1.99)
w.r.t. oracle
µ S3Time (t¢log5 t)
Rest of slides
• Proof of bottom fan-in lower bound
• Other result
S3Time (t) µ BPTime (t1+o(1))
on restricted models
Our negative result
0.1
• Theorem[V]: 2N -size depth-3 circuits for N-bit
Approximate Majority have bottom fan-in W(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 2W(N) bound for depth-3 w/ bottom fan-in w(1)
Our negative result
0.1
• Theorem[V]: 2N -size depth-3 circuits for N-bit
Approximate Majority have bottom fan-in f = W(log N)
• Recall:
V
Æ Æ Æ Æ Æ Æ
V V V V V V V V
LfL
R = 101111011011101011
|R| = N
Tells R 2 YES := { R : Pri [ Ri = 1] ¸ 2/3 }
from R 2 NO := { R : Pri [ Ri = 1] · 1/3 }
Proof
0.1
N
2
V
• Circuit is OR of s =
CNF
E.g. Ci=(x1Vx2V:x3)Æ(:x4)Æ(x5Vx3) C1 C2 C3 LL Cs
Bottom fan-in ) clause size
• By definition of OR : R 2 YES ) some Ci (R) = 1
R 2 NO ) all Ci (R) = 0
• By averaging, fix CNF C = Ci s.t.
PrR 2 YES [C (R) = 1 ] ¸ 1/s = 1/2N0.1
8 R 2 NO ) C (R) = 0
• Claim: Impossible if C has clauses of size ¢log N
Either PrR 2 YES [C(R)=1]·1/2N0.1 or 9 R 2 NO : C(R)=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 (R) = 1 ] · 1 / 2N0.1
Case |S| tiny : Fix few variables and repeat
Either PrR 2 YES [C(R)=1]·1/2N0.1 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 ]
= i Pr[ Gi(R) = 1]
(independence)
· i (1 – 1/3log N ) = i (1 – 1/NO())
= (1 –
) ·
1/NO() |S|
W(1)
-N
e
Either PrR 2 YES [C(R)=1]·1/2N0.1 or 9 R 2 NO : C(R)=1
Case |S| tiny
• |S| < N
)
Fix variables in S
– Maximize PrR 2 YES [C(R)=1]
• Note: S covering ) clauses shrink
Example
x3 Ã 0
(x1Vx2Vx3 ) Æ (:x3) Æ (x5V:x4) x à 1
4
• Repeat
Consider smallest covering S’, etc.
(x1Vx2 ) Æ (x5)
Either PrR 2 YES [C(R)=1]·1/2N0.1 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
S3Time (t) µ BPTime (t1+o(1))
on restricted models
Lower bounds on probabilistic models
• Space-bounded, probabilistic models
• [Ajtai, Beame Saks Sun Vee] n log0.5n time lower bound
– branching programs
• [Allender Koucký Ronneburger Roy Vinay, Diehl & van Melkebeek]
n1+W(1) time lower bounds
one-way randomness
• Theor.[V]: S3Time (n) µ BPTime(n1+o(1))
two-way randomness (sequential)
– Proof uses our positive result
– Our negative result is obstacle for S2
Conclusion
• [SG,L]: BPTime(t) µ S2Time( t2 )
– Related to Approximate Majority
[V]
Appr-Maj on N bits
0.1
N
2 size 2
depth 3
bottom fan-in ¢log N
[DvM,V] 2 size poly(N) depth 3
uniform
BPTime(t)
µ S2Time (t1.99)
w.r.t. oracle
µ S3Time (t¢log5 t)
• Theorem[V] : S3Time (n) µ BPTime(n1+o(1))
two-way access to randomness
Thank you!