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

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
BPP2
Complexity ©D. Moshkovitz

3. Deciding Satifiability
We’ve already seen, that deciding whether a formula is satisfiable…
x1 …xn(x1x2x8)… (x6x3)
x1x2x3… [(x1x2x8)…(x6x3)]
only existential quantifier
existential & universal quantifiers
PSPACE-complete
NP-complete
Complexity ©D. Moshkovitz

4. Technical Note x1x2…xk is the same as x=<x1,x2,…,xk>
Thus, allowing several adjacent quantifiers of the same type does not change the problem.
Complexity ©D. Moshkovitz

5. i alternating quantifiers
The Hierarchy
Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:
x1x2 x3… R(x1,x2,x3,…)
i alternating quantifiers
Complexity ©D. Moshkovitz

6. i alternating quantifiers
The Hierarchy
Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:
x1x2x3… R(x1,x2,x3,…)
i alternating quantifiers
Complexity ©D. Moshkovitz

7. PH (Polynomial-time Hierarchy)
Definition:
PH = i i
Complexity ©D. Moshkovitz

8. Simple Observations “base”: 1=NP
“connection between  and ”: i=coi
“hierarchy”: ii+1 and ii+1
“upper bound”: PHPSPACE
Complexity ©D. Moshkovitz

9. Can the Hierarchy Collapse?
Proposition:
If NP=coNP,
then PH=NP.
Proof Idea:
By induction on i,
i=NP.
Complexity ©D. Moshkovitz

10. Probabilistic Turing Machines
Probabilistic TMs have an “extra” tape: the random tape
“standard” TMs
probabilistic TMs
M(x)
Prr[M(x,r)]
content of input tape
content of random tape
Complexity ©D. Moshkovitz

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…
Complexity ©D. Moshkovitz

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  xL
such TMs are called ‘Atlantic City’
Complexity ©D. Moshkovitz

13. random strings for which M is right
BPP Illustrated
Note: TMs which are right for most x’s (e.g for PRIMES: always say ‘NO’) are NOT acceptable!
For any input x,
all random strings
random strings for which M is right
Complexity ©D. Moshkovitz

14. We can get better amplifications, but this will suffice here...
Claim: If LBPP, 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...
Complexity ©D. Moshkovitz

15. Proof Idea Repeat Output the majority answer
M(x,r)
Repeat
Pick r uniformly at random
Simulate M(x,r)
Output the majority answer
Yes
Yes No
Yes No
Yes
Complexity ©D. Moshkovitz

16. ignore the random input
Relations to P and NP ? P 
BPP  NP
ignore the random input
Complexity ©D. Moshkovitz

17. Does BPPNP? We may have considered saying:
“Use the random string as a witness”
Why is that wrong?
Because non-members may be recognized as members
Complexity ©D. Moshkovitz

18. Make sure you understand why the theorem follows
“Some Comfort”
Theorem (Sipser,Lautemann): BPP2
Underlying observation:
LBPP  there exists a poly. probabilistic TM M, s.t for any n and x{0,1}n let m=p(n) s.t
xL  s1,…,sm{0,1}m r{0,1}m 1imM(x,rsi)=1
Make sure you understand why the theorem follows
Complexity ©D. Moshkovitz

19. Yes-instance
{0, 1}m
Complexity ©D. Moshkovitz

20. No-instance
{0, 1}m
Complexity ©D. Moshkovitz

21. false for less than 1/3m of the r’s
Our Starting Point
m bits
n bits M x r
xL?
LBPP
By amplification, there’s a poly-time machine M which
uses m random coins
errs w.p < 1/3m
false for less than 1/3m of the r’s
Complexity ©D. Moshkovitz

22. Proving the Underlying Observation
We will follow the Probabilistic Method
Prr[r has property P] > 0   r with property P
Complexity ©D. Moshkovitz

23. Yes-Instances Accepted
Let xL.
We want s1,…,sm{0,1}m s.t
r{0,1}m 1imM(x,rsi)=1
So we’ll bound the probability over si’s that it doesn’t hold.
Complexity ©D. Moshkovitz

24. Bounding The Probability Random si’s Do Not Satisfy This
union-bound
si’s independent
r: s is random  rs is random
xL
Complexity ©D. Moshkovitz

25. No-Instances Rejected
Let xL.
Let s1,…,sm{0,1}m .
We want r{0,1}m s.t
1imM(x,rsi)=0
So we’ll bound the probability over r that it doesn’t hold.
Complexity ©D. Moshkovitz

26. Bounding The Probability Random r Does Not Satisfy This
union-bound
xL
Complexity ©D. Moshkovitz

27. Q.E.D!
It follows that:
LBPP  there’s a poly. prob. TM M, s.t for any x there is m s.t
xL  s1,…,sm r 1imM(x,rsi)=1
Thus, L2
 BPP2
Complexity ©D. Moshkovitz

28.  Summary We defined the polynomial-time hierarchy
Saw NP  PH  PSPACE
NP=coNP  PH=NP (“the hierarchy collapses”)
Complexity ©D. Moshkovitz

29.  Summary We presented probabilistic TMs
We defined the complexity class BPP
We saw how to amplify randomized computations
We proved P  BPP  2
Complexity ©D. Moshkovitz

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