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lec12 - CS151 Complexity Theory Lecture 12 May 5 2011...

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CS151 Complexity Theory Lecture 12 May 5, 2011
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May 5, 2011 2 Oracle Turing Machines Oracle Turing Machine (OTM): multitape TM M with special “query” tape – special states q ? , q yes , q no on input x, with oracle language A M A runs as usual, except… – when M A enters state q ? : y = contents of query tape • y A transition to q yes • y A transition to q no
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May 5, 2011 3 Oracle Turing Machines Shorthand #1: applying oracles to entire complexity classes: complexity class C language A C A = {L decided by OTM M with oracle A with M “in” C } example: P SAT
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May 5, 2011 4 Oracle Turing Machines Shorthand #2: using complexity classes as oracles: OTM M complexity class C M C decides language L if for some language A C , M A decides L Both together: C D = languages decided by OTM “in” C with oracle language from D exercise: show P SAT = P NP
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May 5, 2011 5 The Polynomial-Time Hierarchy Σ 0 = Π 0 = P Δ 1 =P P Σ 1 = NP Π 1 = coNP Δ 2 =P NP Σ 2 =NP NP Π 2 =coNP NP Δ i+1 =P Σ i Σ i+1 =NP Σ i Π i+1 =coNP Σ i Polynomial Hierarchy PH = i Σ i
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May 5, 2011 6 The PH PSPACE : generalized geography, 2-person games… 3rd level : V-C dimension… 2nd level : MIN CIRCUIT, BPP… 1st level : SAT, UNSAT, factoring, etc… P NP coNP Σ 3 Π 3 Δ 3 PSPACE EXP PH Σ 2 Π 2 Δ 2
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May 5, 2011 7 Useful characterization Recall: L NP iff expressible as L = { x | 9 y, |y| ≤ |x| k , (x, y) R } where R P . Corollary: L coNP iff expressible as L = { x | 8 y, |y| ≤ |x| k , (x, y) R } where R P .
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May 5, 2011 8 Useful characterization Theorem : L Σ i iff expressible as L = { x | 9 y, |y| ≤ |x| k , (x, y) R } where R Π i-1 . Corollary: L Π i iff expressible as L = { x | 8 y, |y| ≤ |x| k , (x, y) R } where R Σ i-1 .
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May 5, 2011 9 Useful characterization Proof of Theorem: induction on i base case (i =1) on previous slide ( ) – we know Σ i = NP Σ i-1 = NP Π i-1 guess y, ask oracle if (x, y) R Theorem : L Σ i iff expressible as L = { x | 9 y, |y| ≤ |x| k , (x, y) R }, where R Π i-1 .
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May 5, 2011 10 Useful characterization ( ) – given L Σ i = NP Σ i-1 decided by ONTM M running in time n k try: R = { (x, y) : y describes valid path of M’s computation leading to q accept } but how to recognize valid computation path when it depends on result of oracle queries? Theorem : L Σ i iff expressible as L = { x | 9 y, |y| ≤ |x| k , (x, y) R }, where R Π i-1 .
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May 5, 2011 11 Useful characterization try: R = { (x, y) : y describes valid path of M’s computation leading to q accept } valid path = step-by-step description including correct yes/no answer for each A-oracle query z j (A Σ i-1 ) verify “no” queries in Π i-1 : e.g: z 1 A z 3 A z 8 A for each “yes” query z j : 9 w j , |w j | ≤ |z j | k with (z j , w j ) R’ for some R’ Π i-2 by induction.
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