On approximate majority and probabilistic time Emanuele Viola
On approximate majority and probabilistic time Emanuele Viola Institute for advanced study January 2007
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 2 P • Recall NP = 1 P ! 2 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 2 Time(](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-3.jpg "The problem we study • More precisely [SG, L] give BPTime(t) µ 2 Time(")
The problem we study • 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 ) [Dv. M]
Approximate Majority • Input: R = 10111101101011 • 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 V V V V R = 10111101101011 Depth
![The connection [FSS] M(x; u) 2 BPTime(t) Compute M(x): Tell Pru[M(x) = 1] ¸](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-5.jpg "The connection [FSS] M(x; u) 2 BPTime(t) Compute M(x): Tell Pru[M(x) = 1] ¸")
The connection [FSS] M(x; u) 2 BPTime(t) Compute M(x): Tell Pru[M(x) = 1] ¸ 2/3 from Pru[M(x) = 1] · 1/3 BPTime(t) µ 2 Time(t’) = 9 8 Time(t’) R = 1101101011 |R| = 2 t Ri = M(x; i) Compute Appr-Maj V Æ Æ Æ V V V V Lf. L 10111101101011 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](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-6.jpg "Our Negative Result • Theorem[V] : Small depth-3 circuits for Approximate Majority on N")
Our Negative Result • 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.
Quasilinear-time simulation? • 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)
![Our Positive Results • Theorem[V] : There are uniform depth-3 poly(N)-size circuits for Approximate](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-8.jpg "Our Positive Results • Theorem[V] : There are uniform depth-3 poly(N)-size circuits for Approximate")
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[Dv. M, V]: BPTime (t) µ 3 Time (t ¢ log 5 t)
![Summary Appr-Maj on N bits [SG, L] 2 size Nlog N depth 3 BPTime(t)](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-9.jpg "Summary Appr-Maj on N bits [SG, L] 2 size Nlog N depth 3 BPTime(t)")
Summary Appr-Maj on N bits [SG, L] 2 size Nlog N depth 3 BPTime(t) µ 2 Time( t 2 ) [A] 2 size poly(N) depth 3 [A] non-uniform 2 size poly(N) depth O(1) µ O(1)Time( t ) [V] N 2 2 size depth 3 bottom fan-in ¢log N [Dv. M, V] 2 size poly(N) depth 3 ----- µ 2 Time (t 1. 99) w. r. t. oracle µ 3 Time (t¢log 5 t)
Rest of slides • Proof of bottom fan-in lower bound • Other result 3 Time (t) µ BPTime (t 1+o(1)) on restricted models
![Our negative result N 2 -size • Theorem[V]: depth-3 circuits for Approximate Majority on](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-11.jpg "Our negative result N 2 -size • Theorem[V]: depth-3 circuits for Approximate Majority on")
Our negative result N 2 -size • Theorem[V]: depth-3 circuits for Approximate Majority on N bits have bottom fan-in (log N) • Switching lemmas fail Cannot 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 w(1)
![Our Negative Result • Theorem[V]: 2 N -size depth-3 circuits for Approximate Majority on](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-12.jpg "Our Negative Result • Theorem[V]: 2 N -size depth-3 circuits for Approximate Majority on")
Our Negative Result • Theorem[V]: 2 N -size depth-3 circuits for Approximate Majority on N bits have bottom fan-in f = (log N) • Recall: V Æ Æ Æ V V V V Lf. L R = 10111101101011 |R| = N Tells R 2 YES : = { R : Pri [ Ri = 1] ¸ 2/3 } from R 2 NO : = { R : Pri [ Ri = 1] · 1/3 }
Proof • Circuit is OR of s depth-2 circuits V C 1 C 2 C 3 L L C s • 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. Pr. R 2 YES [C (R) = 1 ] ¸ 1/s 8 R 2 NO ) C (R) = 0 • Claim: Impossible if C has bottom fan-in · log N
CNF Claim • Depth-2 circuit ) CNF Æ (x 1 Vx 2 V: x 3 ) Æ (: x 4) Æ (x 5 Vx 3) V V V x 1 x 2 x 3 … bottom fan-in x. N ) clause size • Claim: All CNF C with clauses of size ¢log N N Either Pr. R 2 YES [C (R) = 1 ] · 1 / 2 or there is R 2 NO : C(R) = 1 • Note: Claim ) Theorem
![Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-15.jpg "Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :")
Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO : C(R) = 1 Proof Outline • 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 / N 2 Case |S| tiny : Fix few variables and repeat
![Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-16.jpg "Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :")
Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO : C(R) = 1 Case |S| BIG • |S| ¸ N ) have N /( ¢log N) disjoint clauses Gi – Can find Gi greedily • Pr. R 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| · (1) -N e
![Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-17.jpg "Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :")
Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO : C(R) = 1 Case |S| tiny • |S| < N ) Fix variables in S – Maximize Pr. R 2 YES [C(R)=1] • Note: S covering ) clauses shrink Example x 3 à 0 (x 1 Vx 2 Vx 3 ) Æ (: x 3) Æ (x 5 V: x 4) x à 1 4 • Repeat Consider smallest covering S’, etc. (x 1 Vx 2 ) Æ (x 5)
![Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-18.jpg "Either Pr. R 2 YES N [C(R)=1]· 1/2 or 9 R 2 NO :")
Either Pr. R 2 YES N [C(R)=1]· 1/2 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 3 Time (t) µ BPTime (t 1+o(1)) on restricted models
The model Time 1 Input (x 1 Vx 2 V: x 3 ) Æ (: x 4) Æ (x 5 Vx 3) Æ (x 1 Vx 6 V: x 3 ) Random access 0001010000000 Sequential access work tape
![Time Lower Bound for SAT • Theorem [MS, v. MR]: NTime (n) µ Time](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-21.jpg "Time Lower Bound for SAT • Theorem [MS, v. MR]: NTime (n) µ Time")
Time Lower Bound for SAT • Theorem [MS, v. MR]: NTime (n) µ Time 1 (n 1. 22) • Note: ``combinatorics’’ does not work – Palindromes 2 Time 1 (n 1+o(1)) • Proof by contradiction: Suppose NTime(n) µ Time 1 (n 1. 22) µ O(1) Time(n 0. 1) (speed-up with quantifiers) µ NTime(n 0. 9) (collapse by assumption) Contradicts NTime hierarchy Q. E. D.
The model BPTime 1 Input (x 1 Vx 2 V: x 3 ) Æ (: x 4) Æ (x 5 Vx 3) Æ (x 1 Vx 6 V: x 3 ) Random access 0 1 Sequential access work tape with coins
![Our BPTime Lower Bound for 3 • Theorem [V] : 3 Time (n) µ](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-23.jpg "Our BPTime Lower Bound for 3 • Theorem [V] : 3 Time (n) µ")
Our BPTime Lower Bound for 3 • Theorem [V] : 3 Time (n) µ BPTime 1 (n 1+o(1)) • Proof by contradiction (Inspired by [Dv. M]): Suppose 3 Time(n) µ BPTime 1 (n 1+o(1)) µ 0. 9 n BP Time 1(n 1+o(1)) µ 3 Time(n. 99) (derandomize [INW]) ([V] + [MS]) Contradicts 3 Time hierarchy Q. E. D. • Note: Quadratic slow-down ) won’t work for 2
![Seen so far • Theorem[SG, L]: BPTime(t) µ 2 Time( t 2 ) –](http://slidetodoc.com/presentation_image/6b2cd92d98f67abe3f6c2331c88fd284/image-24.jpg "Seen so far • Theorem[SG, L]: BPTime(t) µ 2 Time( t 2 ) –")
Seen so far • Theorem[SG, L]: BPTime(t) µ 2 Time( t 2 ) – Related to Approximate Majority [V] Appr-Maj on N bits N 2 size 2 depth 3 bottom fan-in ¢log N [Dv. M, V] 2 size poly(N) depth 3 uniform BPTime(t) µ 2 Time (t 1. 99) w. r. t. oracle µ 3 Time (t¢log 5 t) • Theorem [V] : 3 Time (n) µ BPTime 1 (n 1+o(1))
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