{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture18 - Notes on Complexity Theory Last updated October...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Notes on Complexity Theory Last updated: October, 2011 Lecture 18 Jonathan Katz 1 The Power of IP We have seen a (surprising!) interactive proof for graph non-isomorphism. This begs the question: how powerful is IP ? 1.1 co NP ⊆ IP As a “warm-up” we show that co NP ⊆ IP . We have seen last time that co NP is unlikely to have a constant-round interactive proof system (since this would imply 1 that the polynomial hierarchy collapses). For this reason it was conjectured at one point that IP was not “too much more powerful” than NP . Here, however, we show this intuition wrong: any language in co NP has a proof system using a linear number of rounds. We begin by arithmetizing a 3CNF formula φ to obtain a polynomial expression that evaluates to 0 iff φ has no satisfying assignments. (This powerful technique, by which a “combinatorial” statement about satisfiability of a formula is mapped to an algebraic statement about a polynomial, will come up again later in the course.) We then show how to give an interactive proof demonstrating that the expression indeed evaluates to 0. To arithmetize φ , the prover and verifier proceed as follows: identify 0 with “false” and positive integers with “true.” The literal x i becomes the variable x i , and the literal ¯ x i becomes (1 - x i ). We replace “ ” by multiplication, and “ ” by addition. Let Φ denote the polynomial that results from this arithmetization; note that this is an n -variate polynomial in the variables x 1 , . . . , x n , whose total degree is at most the number of clauses in φ . Now consider what happens when the { x i } are assigned boolean values: all literals take the value 1 if they evaluate to “true,” and 0 if they evaluate to “false.” Any clause (which is a disjunction of literals) takes a positive value iff at least one of its literals is true; thus, a clause takes a positive value iff it evaluates to “true.” Finally, note that Φ itself (which is a conjunction of clauses) takes on a positive value iff all of its constituent clauses are positive. We can summarize this as: Φ( x 1 , . . . , x n ) > 0 if φ ( x 1 , . . . , x n ) = true , and Φ( x 1 , . . . , x n ) = 0 if φ ( x 1 , . . . , x n ) = false . Summing over all possible (boolean) settings to the variables, we see that φ SAT X x 1 ∈{ 0 , 1 } · · · X x n ∈{ 0 , 1 } Φ( x 1 , . . . , x n ) = 0 . If φ has m clauses, then Φ has degree (at most) m (where the [total] degree of a polynomial is the maximum degree on any of its monomials, and the degree of a monomial is the sum of the degrees of its constituent variables). Furthermore, the sum above is at most 2 n · 3 m . So, if we work
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern