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Unformatted text preview: 6.841 Advanced Complexity Theory Mar 04, 2009 Lecture 09 Lecturer: Madhu Sudan Scribe: Jeremy Hurwitz In this lecture, we introduce a new model of computation and a set of corresponding complexity classes which sit between NP and PSPACE . This model is based on considering alternation as an interesting phenomenon in its own right. This will lead to the complexity classes comprising the Polynomial Hierarchy ( PH ) and the Infinite Hierarchy Assumption ( IHA ), which informally says that having more alternations gives you more power. This will then result in the Karp-Lipton Theorem, which states Theorem 1 (Karp-Lipton Theorem) IHA = ⇒ NP ( P/ poly 1 Debates Suppose that we have a statement x (“Universal health care good for the economy.”) which party A (Obama) believes to be true and party B (McCain) believes to be false. A verifier V (the voters) must decide which is correct. A and B therefore decide to hold a debate. However, they must agree on a format for the debate. In particular, how many speeches should each candidate be allowed to make? What order should the candidates give their speeches in? Does it matter? We now formalize this idea of a debate. For a language L , we fix a polynomial-time verifier V and the length of the debate, i . Then, given an input x , the two parties in the debate x ’s membership in L . A tries to convince V that x ∈ L , while B tries to convince V that x 6∈ L . A and B are both assumed to be infinitely powerful. Figure 1 : The basic structure of a debate. Initially, A broadcasts a message c 1 . B then responds with a message c 2 . A then sends c 3 , B sends c 4 , and so on until i messages have been broadcast. V now takes the input x and the messages c 1 ...c i , and decides its final answer....
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- Spring '09
- Computational complexity theory, Πi, Σi, IHA