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Interactive Proofs Lecture 16 What the all-powerful can convince - PowerPoint PPT Presentation

Interactive Proofs Lecture 16 What the all-powerful can convince mere mortals of 1 Recap 2 Recap Non-deterministic Computation 2 Recap Non-deterministic Computation Polynomial Hierarchy 2 Recap Non-deterministic Computation


  1. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier 10

  2. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier 10

  3. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover 10

  4. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover 10

  5. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon 10

  6. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon 10

  7. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system 10

  8. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system Arthur sends no messages nor flips any coins 10

  9. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system Arthur sends no messages nor flips any coins 10

  10. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system Arthur sends no messages nor flips any coins 10

  11. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system Arthur sends no messages nor flips any coins 10

  12. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system Arthur sends no messages nor flips any coins 10

  13. Arthur Merlin Proofs Arthur-Merlin proof-systems Arthur: polynomial time verifier Merlin: unbounded prover Random coins come from a beacon Public coin proof-system Arthur sends no messages nor flips any coins 10

  14. MA and AM 11

  15. MA and AM Class of languages with two message Arthur-Merlin protocols 11

  16. MA and AM Class of languages with two message Arthur-Merlin protocols AM (or AM[2]): One message from beacon, followed by one message from Merlin 11

  17. MA and AM Class of languages with two message Arthur-Merlin protocols AM (or AM[2]): One message from beacon, followed by one message from Merlin MA (or MA[2]): One message from Merlin followed by one message from beacon 11

  18. MA and AM Class of languages with two message Arthur-Merlin protocols AM (or AM[2]): One message from beacon, followed by one message from Merlin MA (or MA[2]): One message from Merlin followed by one message from beacon Contain NP and BPP 11

  19. Multiple-message proofs 12

  20. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages 12

  21. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages Turns out IP[k] ⊆ AM[k+2]! 12

  22. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages Turns out IP[k] ⊆ AM[k+2]! Turns out IP[const] = AM[const] = AM[2]! 12

  23. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages Turns out IP[k] ⊆ AM[k+2]! Turns out IP[const] = AM[const] = AM[2]! Called AM 12

  24. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages Turns out IP[k] ⊆ AM[k+2]! Turns out IP[const] = AM[const] = AM[2]! Called AM Turns out IP[poly] = AM[poly] = PSPACE! 12

  25. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages Turns out IP[k] ⊆ AM[k+2]! Turns out IP[const] = AM[const] = AM[2]! Called AM Turns out IP[poly] = AM[poly] = PSPACE! Called IP (= PSPACE) 12

  26. Multiple-message proofs AM[k], MA[k], IP[k]: k(n) messages Turns out IP[k] ⊆ AM[k+2]! Turns out IP[const] = AM[const] = AM[2]! Called AM Turns out IP[poly] = AM[poly] = PSPACE! Called IP (= PSPACE) Later. 12

  27. How can private coins be avoided? 13

  28. How can private coins be avoided? Example: GNI 13

  29. How can private coins be avoided? Example: GNI Recall GNI protocol used private coins 13

  30. How can private coins be avoided? Example: GNI Recall GNI protocol used private coins An alternate view of GNI 13

  31. How can private coins be avoided? Example: GNI Recall GNI protocol used private coins An alternate view of GNI Each of G 0 and G 1 has n! isomorphic graphs 13

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