lecture8 - Notes on Complexity Theory Last updated:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 IND-SET: IND-SET def = { ( G,k ) : G has an independent set of size ≥ k } . We know that IND-SET ∈ 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? MAX-IND-SET 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∈ MAX-IND-SET if G happened to have an independent set of size larger than k , there is no easy way to prove that ( G,k ) 6∈ MAX-IND-SET in case its largest independent set has size smaller than k . As another example, consider the problem of CNF-formula 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 MIN-CNF 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 . 8-1 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....
View Full Document

This note was uploaded on 01/13/2012 for the course CMSC 652 taught by Professor Staff during the Fall '08 term at Maryland.

Page1 / 5

lecture8 - Notes on Complexity Theory Last updated:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online