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lect13

# lect13 - 6.841 Advanced Complexity Theory Lecture 13...

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6.841 Advanced Complexity Theory March 18, 2009 Lecture 13 Lecturer: Madhu Sudan Scribe: Alex Cornejo 1 Overview of today’s lecture Toda’s Theorem: PH := S k N P k P #P , steps: Prove some properties concerning C , C , ⊕ · C , BP · C Do some operator calculus to prove PH BP · ⊕ · P . Prove that BP · ⊕ · P P #P 2 Review: Operator definitions Let L be a language, C a complexity class and q ( n ) some growing function of n . BP q ( n ) · L = n Π q ( n ) yes ( L ) , Π q ( n ) no ( L ) o Π q ( n ) yes ( L ) = n x | Pr [( x, y ) L ] 1 - 2 - q ( n ) o Π q ( n ) no ( L ) = n x | Pr [( x, y ) L ] 2 - q ( n ) o When we omit the q ( n ) subscript we assume that q ( n ) P . BP · L = { Π yes ( L ) , Π no ( L ) } BP · C = { BP · L | L C } ⊕ · L = { x | #( y ) s.t. ( x, y ) L is even } ⊕ · C = {⊕ · L | L C } ⊕ · L = { x | #( y ) s.t. ( x, y ) L is odd } ⊕ · C = { ⊕ · L | L C } ∃ · L = { x | ∃ y s.t. ( x, y ) L } ∃ · C = {∃ · L | L C } ∀ · L = { x | ∀ y s.t. ( x, y ) L } ∀ · C = {∀ · L | L C } For this lecture, the correct way to think about an expression involving Toda’s complexity operators is visualizing the execution tree that represents the expression. For example consider a language L BP ·⊕ · P , then for x L we have the following tree: BP + M ( x, y, z ) . . . M ( x, y, z ) . . . + M ( x, y, z ) . . . M ( x, y, z ) y z z 13-1

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3 Operator properties To prove Toda’s theorem we need to prove the following properties: Property 1. ⊕ · ⊕ · C = ⊕ · C Property 2. BP · BP · C = BP · C Property 3. ⊕ · BP · C = BP · ⊕ · C Property 4. ∃ · C , ∀ · C BP · ⊕ · C Observe that for our purposes it would suffice to prove them for C ∈ { P , ⊕ · P , BP · ⊕ · P } . Lets warm up by proving ⊕ · C = ⊕ · C .
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lect13 - 6.841 Advanced Complexity Theory Lecture 13...

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