today
	constant-depth formulas
		context
		random restrictions
		polynomial approximations
last time:
	ckt lbs intro
	Subbotovskaya’s random restriction
		restriction makes small formula constant
		parity is non-constant under restrictions
Q. better restrictions for simpler circuits?
def: AC^i
	AC^0
lem: AC^i\subseteq NC^i\subseteq AC^{i+1}
thm[Hastad]: AC^0 lb for parity
pf sketch:
	find restriction to constant, w (\lg s)^d variables
thm[Hastad switching lemma]: switching lemma, it switches DNF and CNF
	[[technical proof]]
	=> lbs
rmk:
	hw: bound is optimal
---
Q. is this interesting?
A. yes
A. no
	can't improve lb if matching ub
	not much interesting stuff going on in AC^0
def: 	mod_m gate
	AC^[m]
fact: interesting stuff in AC^0[2]
Thm[Razborov,Smolensky]: primes p, q constant. mod_q requires depth-d AC^0[p] of size exp(n^{\Omega_p(1/d)}
	we'll do q=2, p=3
idea:
	F small AC^0[3] formula => F is "well-approximated" by a low-degree polynomial over \F_3
	\mod_2 is not """
prop: F AC^0[3] size s, depth d, distribution of F_3-polynomials computing F except w/p error \eps, degree polylog(s/\eps))^d
rmk: exact approx is hard, eg AND
lem: OR has probabilistic poly, constant error
pf
	start w/ 0,1
	move to \pm 1
lem: OR has probabilistic poly, small error
pf of prop:
	F=\neg G
	F=MOD_3(G_1,\ldots,G_k): use frobenius
	F=OR(G_1,\ldots,G_k): compose, do union bound
	F=AND(...):
Cor: average-case easiness for AC^0
Cor: average-case hardness for parity
pf
rmks:
	drastically fails for mod_m gates where m not prime
	basically Valiant-Vazirani/Toda's theorem
	Thm[Williams]: NEXP\not\subseteq \AC^0[m] any constant m