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Unformatted text preview: CS 151 Complexity Theory Spring 2011 Problem Set 7 Out: May 19 Due: May 26 Reminder: you are encouraged to work in groups of two or three; however you must turn in your own writeup and note with whom you worked. You may consult the course notes and the text (Papadimitriou). Please attempt all problems. To facilitate grading, please turn in each problem on a separate sheet of paper and put your name on each sheet. Do not staple the separate sheets. 1. 2sided versus 1sided error for MA and AM . For this problem you may want to recall “strong errorreduction via extractors” from Lecture 11, and the proof of BPP ⊆ Σ P 2 ∩ Π P 2 from Lecture 13. In the characterizations below, y,z,y ′ ,z ′ are all strings whose length is polynomial in  x  . (a) Recall that a language L is in MA if there is a language R in P for which: x ∈ L ⇒ ∃ y for which Pr z [( x,y,z ) ∈ R ] ≥ 2 / 3, and x ̸∈ L ⇒ ∀ y Pr z [( x,y,z ) ∈ R ] ≤ 1 / 3 . Prove that for every such language L , there is a language R ′ in P for which: x ∈ L ⇒ ∃ y ′ for which Pr z ′ [( x,y ′ ,z ′ ) ∈ R ′ ] = 1, and x ̸∈ L ⇒ ∀ y ′ Pr z ′ [( x,y ′ ,z ′ ) ∈ R ′ ] ≤ 1 / 3 . (b) Recall that a language L is in AM if there is a language R in P for which: x ∈ L ⇒ Pr y [ ∃ z for which ( x,y,z ) ∈ R ] ≥ 2 / 3, and x ̸∈ L ⇒ Pr y [ ∃ z ( x,y,z ) ∈ R ] ≤ 1 / 3 . Prove that for every such language L , there is a language R ′ in P for which: x ∈ L ⇒ Pr y ′ [ ∃ z ′ for which ( x,y ′ ,z ′ ) ∈ R ′ ] = 1, and x ̸∈ L ⇒ Pr y ′ [ ∃ z ′ ( x,y ′ ,z ′ ) ∈ R ′ ] ≤ 1 / 3 . Hint: for both parts, use strong error reduction, and then allow Merlin to pick half of Arthur’s random string....
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This document was uploaded on 01/05/2012.
 Fall '09

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