Mining Circuit Lower Bound Proofs for Meta-Algorithms

We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for “easy” Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2 n ) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size $${2^n/n}$$ . We get non-trivial compression for functions computable by AC 0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of “easy” functions, which are useful both for proving circuit lower bounds and for designing “meta-algorithms” (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the “shrinkage under random restrictions” results by Subbotovskaya (Doklady Akademii Nauk SSSR 136(3):553–555, 1961) and Håstad (SIAM J Comput 27:48–64, 1998), strengthened to the “high-probability” version by Santhanam (Proceedings of the Fifty-First Annual IEEE Symposium on Foundations of Computer Science, pp 183–192, 2010), Impagliazzo, Meka & Zuckerman (Proceedings of the Fifty-Third Annual IEEE Symposium on Foundations of Computer Science, pp 111–119, 2012b), and Komargodski & Raz (Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp 171–180, 2013). We give a new, simple proof of the “high-probability” version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n 2. We also use this shrinkage result to get an alternative proof of the result by Komargodski & Raz (Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp 171–180, 2013) of the average-case lower bound against small (de Morgan) formulas. Finally, we show that the existence of any non-trivial compression algorithm for a circuit class $${\mathcal{C} \subseteq \mathsf{P}/ \mathsf{poly}}$$ would imply the circuit lower bound $${\mathsf{NEXP} \not \subseteq \mathcal{C}}$$ ; a similar implication is independently proved also by Williams (Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp 21–30, 2013). This complements the result by Williams (Proceedings of the Forty-Second Annual ACM Symposium on Theory of Computing, pp 231–240, 2010) that any non-trivial Circuit-SAT algorithm for a circuit class $${\mathcal{C}}$$ would imply a superpolynomial lower bound against $${\mathcal{C}}$$ for a language in NEXP.