today barrier results natural proofs relativization relativize circuit complexity relativizes? oracle gates thm: any oracle A, P^A\subseteq SIZE^A(\poly) but: PARITY\notin AC^0 not clear how to relativize but: lack of relativizing proofs does not rule out the result relativizing Q. barrier results for circuit complexity? communication complexity? Q. "known techniques" = ? communication complexity recall prop: EQ requires \ge n bits of communication Pf(from before): fooling set prop: f(x,y) requires D(f)\ge \log\rank_\F_\R(M_f), any field \F pf partition M_f into rectangles prop: EQ requires \ge n bits of communication pf: \rank M_{EQ}=2^n for any field \F rmk: rank lower bound is natural proof for CC lb constructivity via determinant largeness via counting low rank decompositions over F_2 motivations largeness random functions are hard constructivity most mathematical reasoning is effective fooling set method is not constructive large natural proofs general definition of a natural property draw picture v. few useful polytime efficient accepts 1/poly fraction of elements natural ckt lbs usefulness constructivity largeness non-examples P(f)=0 if f is simple large not constructive P(f)=1 if f is SAT not large is constructive examples AC^0 lb via random restrictions set n-n^\eps input bits is constructive is also large AC^0[2] lb for parity more challenging, but can do it def: OWF PRG PRF thm[GGM]: PRG=>PRF pf sketch thm[RazborovRudich] weakly-exponentially secure PRFs => no natural proofs against SIZE(n^c) for some c pf perspective conjectured PRFs "just beyond" AC^0 relaxing largeness via problem structure ie, downward self-reductions via completeness contradict time-heirarchy theorem used in william's AC^0[m] lb for NEXP relaxing constructivity diagonalization ???