raw
	rand algo
		PIT
		DNF approx count
		perfect matching
		USTCONN
		maxcut
	BPP vs NEXP
admin
	ps1 due now
	ps2 out tonight
today
	derandomization
	derandomization via
		enumeration
		nonuniformity
		nondeterminism
derandomization
	derandomization
		of algo
			derandomization "complete"
				+\eps-approx counting of circuits
				PIT
			no derandomization known
				x(1+\eps)-approx # bipartite perfect matching
			derandomized
				primality
				x(1+\eps)-approx count of DNF
				USTCONN\in RL
				bipartite matching in RNC
				x(1/2)-approx maxcut
		of complexity classes
			P vs BPP
	def: TIME(f) = languages deterministically decidable in O(f) time
		P=\cup_c TIME(n^c)=TIME(poly(n))
		~P=TIME(2^{poly(log n)})
		SUBEXP=\cap_\eps \TIME(2^{n^\eps})
		EXP=TIME(2^{poly})
	Thm[DTime Hierarchy Theorem]
		f,g:\N\to\N, f\log f=o(g) then TIME(f)\subsetneq TIME(g)
		[[says that having more resources gives you more power]]
	cor: P\subsetneq ~P\subsetneq SUBEXP\subsetneq EXP
	Q. BPP vs P,~P,SUBEXP,EXP?
	open: BPTIME heirarchy? 
enumeration
	thm: L has rand algo in time t, using r(n) random bits
		=> L has det algo in time poly(2^{r(n)},t)
	Pf.
		A algo for L
		A(x,r)
			x = input
			r = randomness
		x\in L => Pr_r [A(x,r)=1]\ge 2/3
		x\notin L => Pr_r [A(x,r)=1]\le 1/3
		algo
			compute b=1/2^{r(n)}\cdot \sum_{r\in\bits^{r(n)}} A(x,r)
			b>1/2		=> yes
			b\le 1/2	=> no
		analysis
			time: clear
			correctness: clear
	Cor: BPP\subseteq EXP
	derandomization goal: reduce from poly(n) random bits to O(log n)
		then derandomization
		we'll see why directly getting to constant is not usually what happens later
nonuniformity
	prop: A(x,r) BPP algo with error <2^{-n}. Then *exists* r_0 such that 
		x\in L		=> 	A(x,r_0)=1
		x\notin L	=> 	A(x,r_0)=0
	[[can always amplify to this low error]]
	pf
		consider random r
		Pr_r[L(x)\ne A(x,r)]<2^{-n}
		Pr_r[\exists L(x)\ne A(x,r)]
			\le \sum_x Pr_r[L(x)\ne A(x,r)]
			< 2^n * 2^{-n}
			=1
		Pr_r[\forall L(x)= A(x,r)]>0
			=> exists desired r_0
	rmk:
		r_0 is 
			non-constructive
			only depends on *input length*, called "advice"
	defn: let a:N->N, C a complexity class.
		C/a = those languages L with advice strings \alpha_1,\alpha_2,\ldots
			such that that x\in L iff (x,\alpha_|x|)\in L
			and |\alpha_i|\le a(i)
		[[one advice string per input, of size a(|x|)]]
	eg, P/poly
	fact: L\in P/poly iff family of boolean circuits {C_n:\bits^n\to \bits} such that
		[[and, or not circuits]]
		C_n:\bits^n\to \bits = L|_{\bits^n}
		|C_n|\le poly(n)
		[[TMs are uniform = one computational object for all inputs]]
		[[ckts are non-uniform = computational device changes based on the input]]
	thm: P\ne P/\poly
	pf.
		L={1^n: n in binary = x, x\in HALT}
		[[advice is whether x in HALT, only one advice per input size needed]]
		[[HALT is undecidable]]
	Cor: BPP\subseteq P/poly
		[[advice = r_0 from previous theorem]]
	Q. How powerful is P/poly?
	Thm[KarpLipton] NP\subseteq P/poly => "almost" P=NP
non-determinism
	NP algo A(x,r)
		x\in L		=> \Pr_r[A(x,r)=1]>0
		x\notin L	=> \Pr_r[A(x,r)=1]=0
	Lem: RP\subseteq NP
	open: BPP\subseteq NP?
	thm: P=NP => BPP=P
	Pf.
		A(x,r) BPP machine with error \le 2^{-n} running in time t(n)
		will design polytime predicate P(x,y,z)
			x\in L	iff \exist y \forall z P(x,y,z)
		=> result
			Q(x,y)=\forall z P(x,y,z)
				in coNP
				in P
			L(x)=\exist y Q(x,y)
				in NP
				in P
		random covering argument
			x\in L
				draw \bits^t
				draw accepting runs
			x\notin L
				draw \bits^t
				draw accepting runs
			idea:
				take many "random shifts" of accepting runs
					x\in L		cover whole space
					x\notin L	won't cover whole space
		formally
			define A_x\subseteq \bits^t by \{r: A(x,r)=L(x)\}
				x\in L => A_x large
				x\notin L => A_x small
			A_x\parity s = \{r\parity s: r\in A_x}
				shift A_x around 
				has same size as A_x
			clm:
				x\in L
					exist s_1,\ldots,s_t
						for all r\in bits^t
							r\in \cup_{i} (A_x\xor s_i)
				x\notin L
					not of ^
					forall s_1,\ldots,s_t
						exist r\in bits^t
							r\notin \cup_{i} (A_x\xor s_i)
				[[this suffices, as ^ is polytime predicate]]
			pf.
				x\in L
					Pr_s[\exists  r\in bits^t r\notin \cup_{i} (A_x\xor s_i)]
						\le 2^n \Pr_s[r\notin \cup_{i} (A_x\xor s_i)]
						= 2^t \Pr_s[r\notin \cup_{i} (A_x\xor s)]^t
						\le 2^n (1/2^n)^t
						<1
					=> exists s
				x\notin L
					volume argument
					Pr_s[\exists  r\in bits^t r\notin \cup_{i} (A_x\xor s_i)]
					Pr_r[r\in \cup_{i} (A_x\xor s_i)
						\le t * \Pr[r\in \cup_{i} (A_x\xor s]
						\le t * 1/2^n
						<1 for large n
					exists r outside union
		review
			if P=NP can solve quantified expressions with \exists\forall statements
			convert BPP to this type of statement
			randomized covering argument
				lots of accepting strings => many random copies cover the space
				few accepting strings => many random copies do not cover the space
today
	derandomization
		enumeration: BPP\in SUBEXP
		nonuniformity: BPP\in P/poly
		nondeterminism: P=NP => BPP=P
next time
	ps2 out tonight
	method of conditional expectations
	pairwise independence