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(2ⁿ) 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 AC0 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 cite [52],[21], strengthened to the 'high-probability' version by [48],[26],[33]. 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 n². We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz [33] 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 ⊆ P/poly would imply the circuit lower bound NEXPnotsubseteq mathcal ⊈ C. This complements Williams's result [55] that any non-trivial Circuit-SAT algorithm for a circuit class mathcal {C} would imply a super polynomial lower bound against mathcal {C} for a language in NEXP also proves such an implication in NEXP.