Abstract:
We introduce the notion of {\em natural} proof. We argue that the known proofs of lower bounds on the complexity of explicit Boolean functions in non-monotone models fall within our definition of natural. We show based on a hardness assumption that natural proofs can't prove superpolynomial lower bounds for general circuits. Without the hardness assumption, we are able to show that they can't prove exponential lower bounds (for general circuits) applicable to the discrete logarithm problem. We show that the weaker class of $AC^0$-natural proofs which is sufficient to prove the parity lower bounds of Furst, Saxe, and Sipser; Yao; and Hastad is inherently incapable of proving the bounds of Razborov and Smolensky. We give some formal evidence that natural proofs are indeed natural by showing that every formal complexity measure which can prove super-polynomial lower bounds for a single function, can do so for almost all functions, which is one of the key requirements to a natural proof in our sense.