On approximate majority and

probabilistic time

Emanuele ViolaInstitute for advanced study

January 2007

• 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 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

• More precisely [SG,L] give

BPTime(t) µ 2Time( t2 )

• Question[This Talk]:Is quadratic slow-down necessary?

• Motivation: Lower boundsKnow NTime ≠ Time on some models [P+,F,…]

Technique: speed-up computation with quantifiersTo prove NTime ≠ BPTime cannot afford Time( t2 ) [DvM]

The problem we study

• 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

Approximate Majority

R = 101111011011101011

V V V V V V V V

Æ Æ Æ Æ Æ Æ

V

Depth

M(x;u) 2 BPTime(t) R = 11011011101011

Compute M(x):Tell Pru[M(x) = 1] ¸ 2/3 Compute Appr-Majfrom Pru[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 [FSS]

V V V V V V V V

Æ Æ Æ Æ Æ Æ

V

f

|R| = 2t Ri = M(x;i)

101111011011101011

• 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 2Time A (t1.99)

• Proof of Corollary:

BPTime (t) µ 2 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.

Our Negative Result

• Question: BPTime(t) µ 3 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)

Quasilinear-time simulation ?

• 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 3Time (t ¢ log5 t)

Our Positive Results

Summary

Appr-Maj on N bits BPTime(t)

[SG,L] 2 size Nlog N depth 3 µ2Time( t2 )

[A] 2 size poly(N) depth 3

non-uniform

----------

[A] 2 size poly(N) depth O(1) µO(1)Time( t )

[V] 2 size 2N depth 3 bottom fan-in ¢log N

µ 2Time (t1.99)

w.r.t. oracle

[DvM,V] 2 size poly(N) depth 3 µ 3Time (t¢log5 t)

• Proof of bottom fan-in lower bound

• Other result

3Time (t) µ BPTime (t1+o(1))

on restricted models

Rest of slides

• Theorem[V]: 2N-size depth-3 circuits for Approximate Majority on N bits have bottom fan-in (log N)

• Switching lemmas failCannot use [H] for Approximate-Majority[SBI] ) bottom fan-in ¸ (log N)1/2

• Independently: [R] improves [SBI]alternative proof of theorem

• Note: No 2(N) bound for depth-3 w/ bottom fan-in (1)

Our negative result

• Theorem[V]: 2N-size depth-3 circuits for Approximate Majority on N bits have bottom fan-in f = (log N)

• Recall:

Tells R 2 YES := { R : Pri [ Ri = 1] ¸ 2/3 }

from R 2 NO := { R : Pri [ Ri = 1] · 1/3 }

Our Negative Result

V V V V V V V V

Æ Æ Æ Æ Æ Æ

V

f R = 101111011011101011 |R| = N

• 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/s8 R 2 NO ) C (R) = 0

• Claim: Impossible if C has bottom fan-in · log N

ProofV

C1 C2 C3 Cs

• 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

CNF Claim

V V V V V

Æ

x1 x2 x3 … xN

• 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

Proof Outline

Either PrR 2 YES [C(R)=1]·1/2Nor 9 R 2 NO : C(R) = 1

• |S| ¸ N ) have N /(¢log N) disjoint clauses i

– Can find i greedily

• PrR 2 YES [C(R) = 1] · Pr [ 8 i, i(R) = 1 ]

= i Pr[ i(R) = 1] (independence)

· i (1 – 1/3log N ) = i (1 – 1/NO())

= (1 – 1/NO()) |S| · e-N(1)

Case |S| BIG

Either PrR 2 YES [C(R)=1]·1/2Nor 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)

• RepeatConsider smallest covering S’, etc.

Case |S| tiny

x3 Ã 0

x4 Ã 1

Either PrR 2 YES [C(R)=1]·1/2Nor 9 R 2 NO : C(R) = 1

• Recall: Repeat ) shrink clausesSo 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 PrR 2 YES [C(R)=1]·1/2Nor 9 R 2 NO : C(R) = 1

• Proof of bottom fan-in lower bound

• Other result

3Time (t) µ BPTime (t1+o(1))

on restricted models

Rest of slides

The model Time1

0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

(x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Æ (x1Vx6V:x3 )

Sequential access work tape

Input

Random access

• 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.

Time Lower Bound for SAT

The model BPTime1

(x1Vx2V:x3 ) Æ (:x4) Æ (x5Vx3) Æ (x1Vx6V:x3 )

Input

Random access

0 1Sequential access work tape with coins

• Theorem [V] : 3Time (n) µ BPTime1 (n1+o(1))

• Proof by contradiction (Inspired by [DvM]):

Suppose 3Time(n) µ BPTime1 (n1+o(1))

µ BPn 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

Our BPTime Lower Bound for 3

• Theorem[SG,L]: BPTime(t) µ 2Time( t2 )– Related to Approximate Majority

• Theorem [V] : 3Time (n) µ BPTime1 (n1+o(1))

Seen so far

Appr-Maj on N bits BPTime(t)

[V] 2 size 2N depth 3 bottom fan-in ¢log N

µ 2Time (t1.99)

w.r.t. oracle

[DvM,V] 2 size poly(N) depth 3

uniform

µ 3Time (t¢log5 t)