Cs221: Computational Complexity Lecture 20: Circuit Complexity

2 Existential Lower Bounds Theorem 1 (Lupanov, Shannon) For each ε > 0 and for each sufficiently large n, ∃f : {0, 1}n → {0, 1} with circuit complexity (over the full basis B2) at least (1− ε)2n n . Proof: We will prove the required result using a counting argument. In particular, we will show that the number of functions mapping {0, 1}n to {0, 1} is much larger than the number of ’distinct’ circuits of size m = (1 − ε)2n n . Note here that we do not need to explicitly ask for the circuit size to be at most m, since we can always increase the size of circuits by padding with extra gates.