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

2. 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

3. Complexity ©D. Moshkovitz 3 Deciding Satifiability We’ve already seen, that deciding whether a formula is satisfiable…  x 1  x 2  x 3 … [(x 1  x 2  x 8 )  …  (  x 6  x 3 )]  x 1 …x n (x 1  x 2  x 8 )  …  (  x 6  x 3 ) only existential quantifierexistential & universal quantifiers

4. Complexity ©D. Moshkovitz 4 Technical Note  x 1  x 2 …  x k is the same as  x=  x 1  x 2 …  x k is the same as  x= Thus, allowing several adjacent quantifiers of the same type does not change the problem.

5. Complexity ©D. Moshkovitz 5 The Hierarchy Definition (  i ):  i is the class of all languages reducible to deciding the sat. of a formula of type:  x 1  x 2  x 3 … R(x 1,x 2,x 3,…) i alternating quantifiers

6. Complexity ©D. Moshkovitz 6 The Hierarchy Definition (  i ):  i is the class of all languages reducible to deciding the sat. of a formula of type:  x 1  x 2  x 3 … R(x 1,x 2,x 3,…) i alternating quantifiers

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

8. 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

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

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

11. 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…

12. 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 Pr r [M(x,r) =  L (x)]  2/3  L (x)=1  x  L such TMs are called ‘Atlantic City’

13. Complexity ©D. Moshkovitz 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. Complexity ©D. Moshkovitz 14 Amplification Claim: If L  BPP, then there exists a probabilistic polynomial TM M’, and a polynomial p(n) s.t  x  {0,1} n Pr r  {0,1} p(n) [M’(x,r)  L (x)] < 1/(3p(n)) We can get better amplifications, but this will suffice here...

15. Complexity ©D. Moshkovitz 15 Proof Idea Repeat –Pick r uniformly at random –Simulate M(x,r) Output the majority answer rM(x,r) 0111001Yes 1011100Yes 0001001No 1100000Yes 0010011No 0110001Yes

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

17. 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

18. Complexity ©D. Moshkovitz 18 “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   s 1,…,s m  {0,1} m  r  {0,1} m  1  i  m M(x,r  s i )=1 Make sure you understand why the theorem follows

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

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

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

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

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

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

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

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

27. 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   s 1,…,s m  r  1  i  m M(x,r  s i )=1 Thus, L  2  BPP  2

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

29. 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 

30. Complexity ©D. Moshkovitz 30 Summary We also presented a new paradigm for proving existence utilizing the algebraic tools of probability theory  Pr r [r has property P] > 0   r with property P The probabilistic method