## Tuesday, February 24, 2009

### Lecture 12

Today we covered: Definitions of $AM[k]$ and $MA[k]$; $AM[k]$ and $MA[k]$ unchanged if perfect completeness required (statement); $MA = MA[2] \subseteq AM[2] = AM$; $AM[k] = AM[2]$ for all constant $k$; indeed, $AM[4r] \subseteq AM[2r+1]$ for any $r = r(n)$; $AM \subseteq \Pi_2$; Boppana-Hastad-Zachos Theorem, $coNP \subseteq AM \Rightarrow PH = AM$; Graph-Non-Isomorphism is not $NP$-complete unless the hierarchy collapses to $\Simga_2$; Goldwasser-Sipser Theorem, $k$-round private-coin interactive proofs are doable with $(k+10)$-round public-coin interactive proofs, hence $AM$ is unchanged if public coins are required; set size lower bound protocol in public coins $AM$.

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