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lecture9 - Notes on Complexity Theory Last updated October...

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Notes on Complexity Theory Last updated: October, 2011 Lecture 9 Jonathan Katz 1 The Polynomial Hierarchy 1.1 Defining the Polynomial Hierarchy via Oracle Machines Here we show a third definition of the levels of the polynomial hierarchy in terms of oracle machines. Definition 1 Define Σ i , Π i inductively as follows: Σ 0 def = P . Σ i +1 def = NP Σ i and Π i +1 = co NP Σ i . (Note that even though we believe Σ i 6 = Π i , oracle access to Σ i gives the same power as oracle access to Π i . Do you see why?) We show that this leads to an equivalent definition. For this section only , let Σ O i refer to the definition in terms of oracles. We prove by induction that Σ i = Σ O i . (Since Π O i = co Σ O i , this proves it for Π i , Π O i as well.) For i = 1 this is immediate, as Σ 1 = NP = NP P = Σ O 1 . Assuming Σ i = Σ O i , we prove that Σ i +1 = Σ O i +1 . Let us first show that Σ i +1 Σ O i +1 . Let L Σ i +1 . Then there exists a polynomial-time Turing machine M such that x L ⇔ ∃ w 1 w 2 · · · Q i +1 w i +1 M ( x, w 1 , . . . , w i +1 ) = 1 . In other words, there exists a language L 0 Π i such that x L ⇔ ∃ w 1 ( x, w 1 ) L 0 . By our inductive assumption, Π i = Π O i ; thus, L ∈ NP Π O i = NP Σ O i = Σ O i +1 and so Σ i +1 Σ O i +1 . It remains to show that Σ O i +1 Σ i +1 (assuming Σ O i = Σ i ). Let L Σ O i +1 . This means there exists a non-deterministic polynomial-time machine M and a language L 0 Σ O i such that M , given oracle access to L i , decides L . In other words, x L iff y, q 1 , a 1 , . . . , q n , a n (here, y represents the non-deterministic choices of M , while q j , a j represent the queries/answers of M to/from its oracle) such that: 1. M , on input x , non-deterministic choices y , and oracle answers a 1 , . . . , a n , makes queries q 1 , . . . , q n and accepts. 2. For all j , we have a j = 1 iff q j L 0 . Since L 0 Σ O i = Σ i (by our inductive assumption) we can express the second condition, above, as: a j = 1 ⇔ ∃ y j 1 y j 2 · · · Q i y j i M 0 ( q j , y j 1 , . . . , y j i ) = 1 a j = 0 ⇔ ∀ y j 1 y j 2 · · · Q 0 i y j i M 0 ( q j , y j 1 , . . . , y j i ) = 0 9-1
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for some (deterministic) polynomial-time machine M 0 . The above leads to the following specifica- tion of L as a Σ i +1 language: x L iff y, q 1 , a 1 , . . . , q n , a n , { y j 1 } n j =1 · { y j 2 } n j =1 · · · · Q i +1 { y j i +1 } n j =1 · : M , on input x , non-deterministic choices y , and oracle answers a 1 , . . . , a n , makes queries q 1 , . . . , q n and accepts, and Let Y be the set of j ’s such that a j = 1, and let N be the set of j
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