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	ps5 due
	ps6 out
today
	barrier results
	relativization
barrier results
	Q. is P vs NP hard to resolve?
	Q. is P vs NP *possible* to resolve?
	proof vs truth
		halting problem
			Thm: is undecidable
			Thm: halting problem statments not all provable in any "reasonable" proof system
			Cor:
				any "reasonable" proof system is "incomplete"
				equiv: these statements are *independent* of the axioms
			Q. could there be natural statements not provable in reasonable systems?
			A. yes
			A. no
			A. mabye
		parallel postulate
			euclid's axioms
			parallel postulate
				at most one parallel line through a point
				two lines joinly perpendicular to another line never meet
			Q. *prove* the parallel postulate?
			A(Bolyai,Gauss,Beltrami,...): no!
			sketch
				create "worlds"/"models" of core axioms where parallel postulate is false
				elliptical geometry
				hyperbolic geometry
			cor: parallel postulate is *independent* of rest of euclidean geometry
		Q. could P vs NP be independent?
		Q. could P vs NP be independent of *known* techniques?
			ie. a barrier result
relativization
	def:
		oracle TM
		TIME^A
		P^A
		NTIME^A
		NP^A
	def: relativization
		"only simulate"
	thm: TIME hierarchy relativizes
	pf
	thm: BPP in PH relativizes
	rmk:
		common set of techniques
			efficiently simulate one TM with another
				"black box"
			enumerate over all TMs
		can attempt to formalize this further
	thm: exist oracle A where P^A=NP^A
	pf:
		idea: make A "big enough" so P vs NP disappears
		A=TQBF
		P^A=PSPACE
		NP^A=NPSPACE
			=PSPACE
	thm: exist oracle B where P^B\ne NP^B
	pf:
		idea: create a "world" from scratch
		idea: the oracle B is an "input" of size 2^n
			each oracle call is a query to this input
			use separation between P^{query} and NP^{query}
				+ diagonalization over all TM's
		define OR^L={1^n: some x\in\bits^n with B(x)=1}
		lem: OR^L\in NP^L, any L
		prop: TM M^L running in time t(n) solving OR^L for any L
			=> t(n)-depth decisision tree computing OR
		pf
		lem: some B, OR^B\notin P^B
		pf
			idea: plug in L where any TM gets OR^L wrong
			enumerate polytime TMs M_i with clocks n^c+c
			find n_i where 
				B|_{\bits^{n_i}} not defined yet
				n^c+c<2^n
			set B so M_i fails
	rmk:
		B is computable
		any resolution of P vs NP must "notice"	lack of oracle
		most known results in complexity theory relativize
			eg BPP\subset PH
		most open questions are known to require non-relativizing techniques
		arithmetization is non-relativizing
			exists A, coNP^A\not\subseteq IP
			coNP\subseteq P^{#SAT}\subseteq IP 
			arithmetization: f:\bits^n\to\bits \mapsto \hat{f}:\F^n\to\F^n	
							\->	would also need to do this to oracle
		algebraization[AaronsonWigderson]
			allow access to low-degree extension to oracle
			most open problems still cannot be resolved
next time
	natural proofs barrier