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3. 2 Probabilistic Computations  Extend the notion of “efficient computation” beyond polynomial-time- Turing machines.  We will still consider only machines that are allowed to run no more than a fixed polynomial in the input length. 7.1

4. 3 Random vs. Non-deterministic computations  In an NDTM, even a single wizard’s witness is enough to cause the input be accepted.  In the randomized model - we require the probability that such a witness is detected to be larger than a certain threshold.  There are two approaches to randomness, online & offline, which we will show to be identical.

5. 4 Online approach  NDTM  : (,, )  (,, )  Randomized TM  : (,, )  (,, )  NDTM accepts if a “good” guess exists - a single accepting computation is enough.  A Random TM accepts if sufficient accepting computations exist. 7.2

6. 5 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

7. 6 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…

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11. 10 Online approach  A random online TM has a creates a probability space on possible computations.  The computations can be described as a tree.  We usually regard only binary tree, because we can simulate any other tree as close as needed.

12. 11 Offline approach  A language L  NP iff there exists a DTM and a polynomial p(  ) and a TM M, s.t. for every x  L  y |y|  p(|x|) and TM(x,y) accepts.  The DTM has access to a “witness” string of polynomial length.  Similarly, a random TM has access to a “guess” input string. BUT...  There must be sufficient such witnesses 7.3

13. 12 The Random Complexity classes  A language that can be recognized by a polynomial time probabilistic TM with good probability, we consider it relatively easy.  We first consider a one-sided error complexity class.

14. 13  A language L  RP, iff a probabilistic TM M exists, such that x  L  Prob[M(x)=1]  ½ x  L  Prob[M(x)=1] = 0  A language L  coRP, iff a probabilistic TM exists M, such that x  L  Prob[M(x)=1] = 1 x  L  Prob[M(x)=0]  ½  These classes complement each other: coRP = { L: L  RP } The classes RP and coRP 7.4

15. 14 Comparing RP with NP  Let L be a language in NP or RP.  Let R L be the relation defining the witness/guess for L for a certain TM.  Then: NP RP x  L   y s.t. (x,y)  R L x  L  Prob[(x,r)  R L ]  ½ x  L   y (x,y)  R L x  L   r (x,r)  R L  Obviously, RP  NP 7.5

16. 15 Amplification  The constant ½ in the definition of RP is arbitrary.  If we have a probabilistic TM that accepts x  L with probability p<½, we can run this TM several times to “amplify” the probability.  If x  L, all runs will return 0.  If x  L, and we run it n times than the probability that none of these accepts is Prob[M n (x)=1] = 1-Prob[M n (x)  1] = = 1-Prob[M(x)  1] n = = 1-(1-Prob[M(x)=1]) n = 1-(1-p) n 7.6

17. 16 Alternative Definitions for RP  RP1 will mistake quite often, while RP2 will almost never mistake:  Define: L  RP1 iff  probabilistic Poly-time TM M and a polynomial p(  ), s.t. x  L  Prob[M(x)=1]  1/p(|x|) x  L  Prob[M(x)=1] = 0  Define: L  RP2 iff  probabilistic Poly-time TM M and a polynomial p(  ), s.t. x  L  Prob[M(x)=1]  1-2 -p(|x|) x  L  Prob[M(x)=1] = 0

18. 17 Claim: RP1=RP2  Trivially, RP2  RP1: if |x| is big enough, then 1-2 - p(|x|)  1/p(|x|)  Suppose L  RP1: there exists M 1 s.t.  x  L: Prob[M 1 (x,r)=1]  1/p(|x|)  M 2 runs M 1 t(|x|) times, to find that Prob[M 2 (x,r)=0]  (1-1/p(|x|)) t(|x|)  Solving (1-1/p(|x|)) t(|x|) = 1-2 -p(|x|) gives t(|x|)  p(|x|) 2 /log 2 e  Hence, M 2 runs in polynomial time and thus L  RP2  RP1  RP2 7.7

19. 18 The class BPP  Define: L  BPP iff there exists a polynomial-time probabilistic TM M, such that  x  L: Prob[M(x)=  L (x)]  2/3  where  L (x)=1 if x  L, and  L (x)=0 if x  L.  The BPP machine success probability is bounded away from failure probability 7.8

20. 19 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’

21. 20 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...

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24. 23 Relations to P and NP P  BPP  NP ignore the random input ?

25. 24 Does BPP  NP? We may have considered saying: “Use the random string as a witness” Why is that wrong?

26. 25 Constant invariance  An alternative definition: x  L  Prob[M(x)=1]  p+  x  L  Prob[M(x)=1]  p-   We can build a machine M 2 that runs M n times, and return the majority of results of M. After a constant number of executions (depending on p and  but not on x) we would get the desired probability. 7.9

27. 26 The weakest possible BPP definition  Define: L  BPPW iff there exists a polynomial-time probabilistic TM M, and a polynomial-time computable function f:N  [0,1] and a polynomial p(  ), such that  x  L: x  L  Prob[M(x)=1]  f(|x|) + 1/p(|x|) x  L  Prob[M(x)=1]  f(|x|) - 1/p(|x|)  If we set f(|x|)=½ and p(|x|)=6 we get the original definition.  Hence, BPP  BPPW 7.10

28. 27 Claim: BPPW=BPP  Let L  BPPW and let M be its computing TM.  Define M’(x)  invoke t i =M(x)  i=1…n, compute p=  t i /n, if p>f(|x|) return YES else return NO.  Notice p is the mean of a sample of size n of the random variable M(x).  By setting  =1/p(|x|) and n=-ln(1/6)/2  2 we get the desired probability gap.

29. 28 The strongest possible BPP definition  Define: L  BPPS iff there exists a polynomial-time probabilistic TM M, and a polynomial p(  ), such that  x  L  Prob[M(x)=  L (x)]  1-2 -p(|x|)  If we set p(|x|)=2 we get the original definition, because 1-2 -2   Hence, BPPS  BPP 7.11

30. 29 BPP and other complexity classes  Clearly, RP  BPP.  BPP  NP? Unknown.  coBPP=BPP by definition. 7.12

31. 30 The class PP  The class BPP allowed for a small but not negligible gap for mistakes.  The class PP allows for even smaller gaps, even a single random toss is enough.  Running a PP machine polynomially many times, would not necessarily help.  Define: L  PP iff there exists a polynomial- time probabilistic TM M, such that  x  L: Prob[M(x)=  L (x)]  ½ 7.13

32. 31 Claim: PP Claim: PP  PSPACE  Let L  PP, M be the TM that recognizes L, and p(  ) be the time bound.  Define M’(x)  run M(x) using all p(|x|) possible coin tosses, and decides by majority.  M’ uses polynomial space for each of the (exponential number of) invocations, and only needs to count the number of successes. 7.14

33. 32 Other variants for PP  Define: L  PP1 iff a TM M exists s.t. x  L  Prob[M(x)=1]  ½ x  L  Prob[M(x)=0]  ½  Obviously, PP  PP1.

34. 33 Claim: PP=PP1  Let L  PP1, M be the TM that recognizes L, and p(  ) be the time bound.  M’(x,(a 1,a 2,...,a p(|x|),b 1,b 2,...,b p(|x|) ))  if a 1 =a 2 =...=a p(|x|) =1 then return NO, else return M(x,(b 1,b 2,...,b p(|x|) ))  M' is exactly the same as M, except for an additional probability of 2 -p(|x|)+1 for returning NO. 7.15

35. 34 PP=PP1 continued  If x  L, Prob[M(x)=1] > ½  Prob[M(x)=1]  ½+2 -p(|x|)+1  Prob[M’(x)=1]  (½+2 -p(|x|) )(1-2 -p(|x|)+1 ) > ½  If x  L, Prob[M(x)=0]  ½  Prob[M’(x)=0]  ½(1-2 -p(|x|)+1 )+2 -p(|x|)+1 > ½  In any case Prob[M(x)=  L (x)] > ½

36. 35 Claim: Claim: NP  PP  Let L  NP, M be the NDTM that recognizes L, and p(  ) be the time bound.  M’(x,(b 1,b 2,...,b p(|x|) ))  if b 1 =1 then return M(x,(b 2,...,b p(|x|) )), else return YES.  If x  L, then Prob[M’(x)=1]=½ (b 1 =1 only).  If x  L, then Prob[M’(x)=1]>½ (a witness exists)   L  PP1=PP  NP  PP  Also: coNP  PP because of symmetry. 7.16

37. 36 The class ZPP  Define: L  ZPP iff there exists a polynomial-time probabilistic TM M, such that  x  L: M(x)={0,1,  }, Prob[M(x)=  ]  ½, and Prob[M(x)=  L (x) or M(x)=  ] = 1  Prob[M(x)=  L (x)]>½  The symbol  is “I don’t know”.  The value ½ is arbitrary and can be replaced by 2 -p(|x|) or 1-1/p(|x|). 7.17

38. 37 Claim: ZPP=RP  coRP  Let L  ZPP, M be the NDTM that recognizes L.  Define: M’(x)  – let b=M(x) – if b=  then return 0, else return b.  If x  L, M’(x) will never return 1.  If x  L, Prob[M’(x)=1]>½, as required.  ZPP  RP.  The same way, ZPP  coRP.

39. 38  Let L  RP  coRP, M RP and M coRP be the NDTMs that recognize L according to RP and coRP.  Define: – M’(x)  – if M RP (x)=YES return 1 – if M coRP =NO return 0, else return .  M RP (x) never returns YES if x  L, and M coRP (x) never returns NO if x  L. Therefore, M’(x) never returns the opposite of  L (x).  The probability that M RP and M coRP are both wrong < ½  Prob[M’(x)=  ] < ½.  RP  coRP  ZPP RP  coRP RP  coRP  ZPP

40. 39 The random classes hierarchy  P  ZPP  RP  BPP  It is believed that BPP=P

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