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Unformatted text preview: Notes on Complexity Theory Last updated: September, 2011 Lecture 8 Jonathan Katz 1 The Polynomial Hierarchy We have seen the classes NP and co NP , which are defined as follows: L ∈ NP if there is a (deterministic) Turing machine M running in time polynomial in its first input, such that x ∈ L ⇔ ∃ w M ( x,w ) = 1 . L ∈ co NP if there is a (deterministic) Turing machine M running in time polynomial in its first input, such that x ∈ L ⇔ ∀ w M ( x,w ) = 1 . It is natural to generalize the above; doing so allows us to capture problems that are “more difficult” than NP ∪ co NP . As an example, consider the language INDSET: INDSET def = { ( G,k ) : G has an independent set of size ≥ k } . We know that INDSET ∈ NP ; a certificate is just an independent set of vertices of size at least k (that the vertices are independent can easily be verified). How about the following language? MAXINDSET def = { ( G,k ) : the largest independent set in G has size exactly k } . This language does not appear to be in NP : we can certify that some graph G has an independent set of size k , but how do we certify that this is the largest independent set in G ? The language does not appear to be in co NP , either: although we could prove that ( G,k ) 6∈ MAXINDSET if G happened to have an independent set of size larger than k , there is no easy way to prove that ( G,k ) 6∈ MAXINDSET in case its largest independent set has size smaller than k . As another example, consider the problem of CNFformula minimization. A CNF formula φ on n variables naturally defines a function f φ : { , 1 } n → { , 1 } , where f φ ( x ) = 1 iff the given assignment x satisfies φ . Can we tell when a given formula is minimal? Consider the language MINCNF def = { φ : no formula with fewer clauses than φ computes f φ } . This language does not appear to be in NP or co NP , either. (Even if I give you a smaller formula φ that computes f φ , there is no obvious way for you to verify that fact efficiently.) The above examples motivate the following definition: Definition 1 Let i be a positive integer. L ∈ Σ i if there is a (deterministic) Turing machine M running in time polynomial in its first input, such that x ∈ L ⇔ ∃ w 1 ∀ w 2 ··· Q i w i  {z } i times M ( x,w 1 ,...,w i ) = 1 . 81 where Q i = ∀ if i is even, and Q i = ∃ if i is odd. L ∈ Π i if there is a (deterministic) Turing machine M running in time polynomial in its first input, such that x ∈ L ⇔ ∀ w 1 ∃ w 2 ··· Q i w i  {z } i times M ( x,w 1 ,...,w i ) = 1 . where Q i = ∀ if i is odd, and Q i = ∃ if i is even....
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This note was uploaded on 01/13/2012 for the course CMSC 652 taught by Professor Staff during the Fall '08 term at Maryland.
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