ps7 - CS 151 Complexity Theory Spring 2011 Problem Set 7...

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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 write-up 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. 2-sided versus 1-sided error for MA and AM . For this problem you may want to recall “strong error-reduction 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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ps7 - CS 151 Complexity Theory Spring 2011 Problem Set 7...

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