An Entropy Proof of the Switching Lemma and Tight Bounds on the Decision-Tree Size of AC

The first result of this paper is (a novel proof of) the following switching lemma for m-clause DNF formulas F : (1) P[ DTdepth(F Rp) ≥ t ] = O(p log(m+ 1)) for all p ∈ [0, 1] and t ∈ N where Rp is the p-random restriction and DTdepth denotes decision-tree depth. Our proof replaces the counting arguments in previous proofs of H̊astad’s O(pw) switching lemma for width-w DNFs [5, 8, 2] with a novel entropy argument, which naturally applies to unbounded-width DNFs with a bounded number of clauses. With respect to AC circuits, our m-clause switching lemma has similar applications as H̊astad’s width-w switching lemma, including a 2 1/(d−1)) lower bound for PARITY. Moreover, the entropy proof of (1) is potentially more amenable to certain generalizations of Rp, as we illustrate for the class of p-pseudorandom restrictions. The second result of this paper extends (1) to higher-depth circuits via a combination of H̊astad’s switching and multi-switching lemmas [5, 6]. For boolean functions f : {0, 1} → {0, 1} computable by AC circuits of depth d and size s, we show that (2) P[ DTdepth(f Rp) ≥ t ] = (p ·O(log s)d−1)t for all p ∈ [0, 1] and t ∈ N. As a corollary, we obtain a tight bound on decision-tree size (3) DTsize(f Rp) = O(2 (1−ε)n) where ε = 1/O(log s)d−1. Qualitatively, (2) strengthens a similar inequality of Tal [11] with degree in place of DTdepth, and (3) strengthens a similar inequality of Impagliazzo, Matthews and Paturi [7] with subcube partition number in place of DTsize. ∗Supported by NSERC and a Sloan Research Fellowship