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BPP Contained in PH

BPP Contained in PH. By Michael Sipser; Clemens Lautemann. Presenter: Jie Meng. M. Sipser. A complexity theoretic approach to randomness , In Proceedings of the 15th ACM STOC, 1983 C. Lautemann, BPP and the polynomial hierarchy , Information Process Letter 14 215-217, 1983.

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BPP Contained in PH

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  1. BPP Contained in PH By Michael Sipser; Clemens Lautemann Presenter: Jie Meng

  2. M. Sipser. A complexity theoretic approach to randomness, In Proceedings of the 15th ACM STOC, 1983 • C. Lautemann, BPP and the polynomial hierarchy, Information Process Letter 14 215-217, 1983

  3. Outline • Definition and Background • Techniques • Proof • Questions • Homework

  4. BPP: A language L is in BPP if and only if there exists a randomized Turing Machine M, s.t.

  5. Trivially

  6. Main Theorem:

  7. Let C be a language, NPC be the class that L is in NPC if there is a non-deterministic Turing Machine M, which can accept L, with the power that M can query an oracle such questions like “if y is in C” and get the correct answer in one step. • This can be generalized to NPA, A is a language class.

  8. NP: L is in NP if there exists a deterministic polynomial Turing Machine M, s.t. xL y M(x,y)=1

  9. L in NP, for any x in L                Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Acc Acc

  10. L in NP, for any x not in L                Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej

  11. L in , x in L, L’ in NP  y in L’ ? Yes/No               Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Acc Acc

  12. L in , x in L,   y in L’               Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Acc

  13. L in , x in L,   y not in L’               Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Acc

  14.               Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

  15. Equivalent definition • NP: L is in NP, if and only if there exists a deterministic poly-time TM M, s.t. xL y M(x,y)=1 • : L is in , if and only if there exists a deterministic poly-time TM M’, s.t.x L y z M’(x,y,z)=1 xL yz M’(x,y,z)=0

  16. Technique fat 325 lbs 7’ 1’’ VS Thin

  17. Technique

  18. Technique

  19. PROOF • L in BPP: • By amplifying method and Chernoff Bound

  20. PROOF • Wx={ y | M(x, y)=1} • x in L, |Wx|>2m (1-1/m), Wx is very fat; • x not in L, |Wx|<2m 1/m Wx is very thin; • {0,1}m is the whole space;

  21. PROOF • Shifting: • If Wx is fat, |Wx|>2m (1-1/m), • There exists a set of strings y1, y2, … yr, r=m/2 s.t. • If Wx is thin, |Wx|<2m 1/m • There is no such set of strings

  22. PROOF X in L, Wx is fat, there exists a set of string y1, y2, … yr Then for all z in {0, 1}m , That is, there exists i, s.t.

  23. PROOF • x in L, Wx is fat, • There exists y1, y2, … yr, • For all z in {0,1}m, M(x, z yi)=1, for some i; • x in L,

  24. Question?

  25. HOMEWORK • Finish the proof in case x is not in L, which is to say, fill out the blank in the following statement: • L in BPP, xL […][…] M’(x,y,z)=[.] • Give all necessary explanations about your statement.

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