admin
	project topics due
today
	circuit lower bounds
		intro
		balancing formulas
		random restrictions
goal: NP\not\subseteq P
goal: NP\not\subseteq P/\poly
	equiv: circuit lower bounds
rmk
	avoids TMs
	combinatorics
	thm[Karp-Lipton]
	EXP^NP\not\subseteq P/\poly
Thm[...]: (1+\Omega(1))n ckt lower bounds
rmk: sensitive to gate set
open: \omega(n) lbs
goal: \omega(n) lbs for "interesting" restricted classes
def: non-uniform NC^i
lem: NC^1=formulas
ex
pf
	wlog
		push negations to leaves
		no fan-in 1 gates
		leaves=~size
	find gate w/ # leaves
		\le 2/3 L(F)
		> 2/3 L(F)
hw: parity has n^2-size formula
thm[Subbotovskaya]: parity requires n^{1.5} size formulas
pf
	idea: 
		small formulas are simple
			partial evaluations can simplify formulas
				OR(x,1)=1, AND(x,0)=0
		parity function is not simple
			partial evaluations are just parity
	def: restriction \rho:[n]\to{0,1,\star}
	goal: find \rho st
		do not set too many variables
		f|_\rho is much simpler than f
	attempt 1: formula w/ s leaves => s(1-1/n) size setting 1 variable
		get: (1-1/n)*(1-1/(n-1))*\cdots = 1/Theta(n) shrinkage
	attempt 2:
		def: simplifcation rules
			leaves are variables
			x\AND g => g independent of x
			push negations to bottom
		goal: simplify formula after restricting it
		prop: 
			\rho=select n-k vars
			set to \bits iid
			\E[|f|_\rho|]\le (k/n)^{1.5} |f|
		pf
			k=1:
				\E[|f|_\rho|]\le (1-1.5/n)|f|
				each leaf:
					w/p 1/2 just the leaf dies
					w/p 1/2 also kill other subtree
				no double counting
				then use linearity of expectation
			k>1:
				repeat 
				use bernoulli
			restrict n-1 variables this way, parity is non-constant still
rmk:
	can improve analysis to get size_f(parity)\gesim n^2
	best explicit lb for formula is \gesim n^3
references:
	http://sites.math.rutgers.edu/~sk1233/courses/topics-S13/lec2.pdf
next time:
	lower bounds for constant-depth formulas