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Unformatted text preview: 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 i , i inductively as follows: def = P . i +1 def = NP i and i +1 = co NP i . (Note that even though we believe i 6 = i , oracle access to i gives the same power as oracle access to i . Do you see why?) We show that this leads to an equivalent definition. For this section only , let O i refer to the definition in terms of oracles. We prove by induction that i = O i . (Since O i = co O i , this proves it for i , O i as well.) For i = 1 this is immediate, as 1 = NP = NP P = O 1 . Assuming i = O i , we prove that i +1 = O i +1 . Let us first show that i +1 O i +1 . Let L i +1 . Then there exists a polynomialtime 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 i such that x L w 1 ( x,w 1 ) L . By our inductive assumption, i = O i ; thus, L NP O i = NP O i = O i +1 and so i +1 O i +1 . It remains to show that O i +1 i +1 (assuming O i = i ). Let L O i +1 . This means there exists a nondeterministic polynomialtime machine M and a language L 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 nondeterministic 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 , nondeterministic 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 . Since L O i = 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 ( q j ,y j 1 ,...,y j i ) = 1 a j = 0 y j 1 y j 2 Q i y j i M ( q j ,y j 1 ,...,y j i ) = 0 91 for some (deterministic) polynomialtime machine M . The above leads to the following specifica tion of L as a 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 , nondeterministic choices...
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