today constant-depth formulas context random restrictions polynomial approximations last time: ckt lbs intro Subbotovskayaâ€™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