Learning Algorithms from Natural Proofs.

Based on Hastad's (1986) circuit lower bounds, Linial, Mansour, and Nisan (1993) gave a quasipolytime learning algorithm for AC(0) (constant-depth circuits with AND, OR, and NOT gates), in the PAC model over the uniform distribution. It was an open question to get a learning algorithm (of any kind) for the class of AC(0)[p] circuits (constant-depth, with AND, OR, NOT, and MOD, gates for a prime p). Our main result is a quasipolytime learning algorithm for AC(0)[p] in the PAC model over the uniform distribution with membership queries. This algorithm is an application of a general connection we show to hold between natural proofs (in the sense of Razborov and Rudich (1997)) and learning algorithms. We argue that a natural proof of a circuit lower bound against any (sufficiently powerful) circuit class yields a learning algorithm for the same circuit class. As the lower bounds against AC(0)[p] by Razborov (1987) and Smolensky (1987) are natural, we obtain our learning algorithm for AC(0)[p].