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