admin ps1 back today alternation Karp-Lipton last time defn: ATM \Sigma^k P \Pi^k P PH Conj: PH is infinite Q. explanatory power? perspectives proof system ex: min ckt debates completeness soundness completeness prop: \k-\exist-TBQF is \Sigma^k P complete Pf: extended cook-levin oracle machines motivation reduction style defn OTM oracle tape oracle state P^L P^C ex P^NP=P^{\coNP} P^SAT=P^NP Prop \Sigma^k P=NP^{\Sigma^{k-1} P} Prop \Sigma^2 P=NP^{NP} Pf <=: do it note that only one query is used >=: issues: existentialism of machine asking NP questions asking coNP questions interleaving these key idea: guess queries, then verify L\in NP^{SAT} x\in L iff uses non-deterministic choices uses oracle queries asks questions q_1,\ldots,q_k\in\bits^\star gets answers a_1,\ldots,a_k\in\bits^\star reaches accepting state iff exist set \{(q_i,a_i)\} non-deterministic chocies st for those a_i=1 exist satisfying assignment to \varphi_i for those a_i=0 for all assignments to \varphi_i, it isn't satisfying running the NTM on these choices these answers you get these questions accepting path Cor: P=NP => PH=P Pf. Cor: \Sigma^kP=\Pi^kP => PH=\Sigma^k Pf. KarpLipton recall ckts Q. NP\subseteq P/poly Lem: NP\subseteq P/poly => circuit for finding satisfying assignment Thm: NP\subseteq P/poly => PH=\Sigma^2 P Pf guess circuit, use to find satisfying assignment L\in \Pi^2 P x\in L iff \forall y, \exist z P(x,y,z) iff \forall y P(x,y,C(x,y)), for the correct circuit C iff \exists C \forall y P(x,y,C(x,y)), for the any circuit C next time randomness