Lecture 12

Today we covered: Definitions of $AM\left[k\right]$ and $MA\left[k\right]$; $AM\left[k\right]$ and $MA\left[k\right]$ unchanged if perfect completeness required (statement); $MA=MA\left[2\right]\subseteq AM\left[2\right]=AM$; $AM\left[k\right]=AM\left[2\right]$ for all constant $k$; indeed, $AM\left[4r\right]\subseteq AM[2r+1]$ for any $r=r\left(n\right)$; $AM\subseteq {\Pi }_2$; Boppana-Hastad-Zachos Theorem, $coNP\subseteq AM\Rightarrow PH=AM$; Graph-Non-Isomorphism is not $NP$-complete unless the hierarchy collapses to $\backslash Simg{a}_2$; 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 $AM$.