circuits motivation definition circuit vs formula size family SIZE P/poly lem: size(f)\le n2^n pf via DNF prop: TIME(f)\subseteq SIZE(t^2) pf computation history computation tableu local update rule Cor: P\subseteq P/\poly rmk: no f more than exp-hard, compare to TIME lem: P\subsetneq P/\poly defn: TMs w/ advice untrusted vs trusted per-input vs per-input-size Cor: P/poly=P/poly prop: exist f needing size 2^\Omega{n} pf counting probabilistic method cor: circuit size heirarchy Thm[Cook-Levin] NP\le CKT-SAT Prop: CKT-SAT\le 3-SAT Rmk: completes PSPACE\le TQBF