An entropy-based switching lemma for m-clause DNFs
نویسنده
چکیده
We show that every m-clause DNF F satisfies P[ DTdepth(F Rp) ≥ t ] = O(p log(m+ 1)) where Rp is the p-random restriction and DTdepth(·) is decision-tree depth. Similar to H̊astad’s O(pw) switching lemma for width-w DNFs, we analyze the canonical decision tree of F Rp. However, in contrast to previous switching lemma proofs (H̊astad [4], Razborov [5], Beame [2]), our bound is based on an entropy argument. Since our switching lemma implies an O(pw) bound for conjunctions and disjunctions of 2 many depth-w decision trees, it provides an alternative proof of the classic 2 1/(d−1)) lower bound on the depth-d circuit size of PARITY. ∗Supported by NSERC and a Sloan Research Fellowship
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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 nove...
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