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	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
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