Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.

Complexity©D. Moshkovitz

1

And Randomized Computations

The Polynomial Hierarchy


2

Introduction

• Objectives:– To introduce the polynomial-time

hierarchy (PH)– To introduce BPP– To show the relationship between the two

• Overview:– satisfiability and PH– probabilistic TMs and BPP

– BPP2


3

Deciding Satifiability

We’ve already seen, that deciding whether a formula is satisfiable…

x1x2x3… [(x1x2x8)…(x6x3)]x1 …xn(x1x2x8)… (x6x3)

only existential quantifier existential & universal quantifiers


4

Technical Note

x1x2…xk is the same as x=<x1,x2,…,xk>

x1x2…xk is the same as x=<x1,x2,…,xk>

• Thus, allowing several adjacent quantifiers of the same type does not change the problem.


5

The Hierarchy

Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:

x1x2 x3… R(x1,x2,x3,…)

i alternating quantifiers


6

The Hierarchy

Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:

x1x2x3… R(x1,x2,x3,…)

i alternating quantifiers


7

PH (Polynomial-time Hierarchy)

Definition:

PH = i i


8

Simple Observations

• “base”: 1=NP

• “connection between and ”: i=coi

• “hierarchy”: ii+1 and ii+1

• “upper bound”: PHPSPACE


9

Can the Hierarchy Collapse?

Proposition: If NP=coNP, then PH=NP.

Proof Idea: By induction on i, i=NP.


10

Probabilistic Turing Machines

• Probabilistic TMs have an “extra” tape: the random tape

M(x) Prr[M(x,r)]content of input

tape

“standard” TMs probabilistic TMs

content of random

tape


11

Does It Really Capture The Notion of Randomized Algorithms?

It doesn’t matter if you toss all your coins in

advance or throughout the computation…


12

BPP (Bounded-Probability Polynomial-Time)

Definition: BPP is the class of all languages L which have a probabilistic polynomial time TM M, s.t

x Prr[M(x,r) = L(x)] 2/3

L(x)=1 xL

such TMs are called ‘Atlantic City’


13

BPP Illustrated

For any input x,

all random strings

random strings for which M is

right

Note: TMs which are right for most x’s (e.g for PRIMES: always say

‘NO’) are NOT acceptable!


14

Amplification

Claim: If LBPP, then there exists a probabilistic polynomial TM M’, and a polynomial p(n) s.t

x{0,1}n Prr{0,1}p(n)[M’(x,r)L(x)] <

1/(3p(n))We can get better

amplifications, but this will suffice here...


15

Proof Idea

• Repeat– Pick r uniformly at

random– Simulate M(x,r)

• Output the majority answer

r M(x,r)

0111001 Yes

1011100 Yes

0001001 No

1100000 Yes

0010011 No

0110001 Yes


16

Relations to P and NP

P BPP NP

ignore the random input

?


17

Does BPPNP?

We may have considered saying:“Use the random string as a

witness”

Why is that wrong?Because non-members may be recognized as members


18

“Some Comfort”

Theorem (Sipser,Lautemann): BPP2

Underlying observation:LBPP there exists a poly. probabilistic TM M,

s.t for any n and x{0,1}n let m=p(n) s.t

xL s1,…,sm{0,1}m r{0,1}m 1imM(x,rsi)=1

Make sure you understand why the theorem follows


19

{0, 1}m

Yes-instance


20

No-instance

{0, 1}m


21

Our Starting Point

• LBPP• By amplification, there’s a

poly-time machine M which– uses m random coins – errs w.p < 1/3m

M

x r

xL?

n bits

m bits

false for less than 1/3m of the

r’s


22

Proving the Underlying Observation

We will follow the Probabilistic Method

Prr[r has property P] > 0 r with property P


23

Yes-Instances Accepted

• Let xL.

• We want s1,…,sm{0,1}m s.t

r{0,1}m 1imM(x,rsi)=1

• So we’ll bound the probability over si’s that it doesn’t hold.


24

Bounding The Probability Random si’s Do Not Satisfy This

m1 m R

mm

is ,...,s {0,1}i 1

Pr r 0,1 , M x,r s 0

m1 m R

mm

is ,...,s {0,1}i 1

Pr r 0,1 , M x,r s 0

union-bound m

1 m Rm

m

is ,...,s {0,1}i 1r {0,1}

Pr M x,r s 0

m

1 m Rm

m

is ,...,s {0,1}i 1r {0,1}

Pr M x,r s 0

m1 m Rm

m

is ,...,s {0,1}i 1r {0,1}

Pr M(x,r s ) 0

m1 m Rm

m

is ,...,s {0,1}i 1r {0,1}

Pr M(x,r s ) 0

si’s

independent

mR

mm

s {0,1}i 1

2 Pr M x,s 0

mR

mm

s {0,1}i 1

2 Pr M x,s 0

r: s is random rs is random

mm 1

2 13m

mm 1

2 13m

xL


25

No-Instances Rejected

• Let xL.

• Let s1,…,sm{0,1}m .

• We want r{0,1}m s.t

1imM(x,rsi)=0

• So we’ll bound the probability over r that it doesn’t hold.


26

Bounding The Probability Random r Does Not Satisfy This

mR

m

ir {0,1}i 1

Pr M x,r s 1

mR

m

ir {0,1}i 1

Pr M x,r s 1

union-bound m

m

ir {0,1}i 1

Pr M x,r s 1

m

m

ir {0,1}i 1

Pr M x,r s 1

1

m 13m

1

m 13m

xL


27

Q.E.D!

It follows that:LBPP there’s a poly. prob. TM M,

s.t for any x there is m s.t

xL s1,…,sm r

1imM(x,rsi)=1

Thus, L2

BPP2


28

Summary

• We defined the polynomial-time hierarchy – Saw NP PH PSPACE– NP=coNP PH=NP (“the

hierarchy collapses”)


29

Summary

• We presented probabilistic TMs– We defined the complexity class BPP– We saw how to amplify randomized

computations

– We proved P BPP 2


30

Summary

• We also presented a new paradigm for proving existence utilizing the algebraic tools of probability theory

Prr[r has property P] > 0 r with property P

The probabilistic method

Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.

Documents

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