Approximating AC^0 by Small Height Decision Trees and a Deterministic Algorithm for #AC^0SAT

We show how to approximate any function in AC⁰ by decision trees of much smaller height than its number of variables. More precisely, we show that any function in n variables computable by an unbounded fan-in circuit of AND, OR, and NOT gates that has size S and depth d can be approximated by a decision tree of height n - βn to within error exp(-βn), where β = β(S, d) = 2^-O(d log^4/5 S). Our proof is constructive and we use its constructivity to derive a deterministic algorithm for #AC⁰ SAT with multiplicative factor savings over the naive 2ⁿ S algorithm of 2^-Ω(βn), when applied to any n-input AC⁰ circuit of size S and depth d. Indeed, in the same running time we can deterministically construct a decision tree of size at most 2^n-βn that exactly computes the function given by such a circuit. Recently, Impagliazzo, Matthews, and Paturi derived an algorithm for #AC⁰ SAT with greater savings over the naive algorithm but their algorithm is only randomized rather than deterministic. The main technical result we prove to show the above is that for every family F of k-DNF formulas in n variables and every 1 <; C = C(n) ≤ log^poly(k) |F|, one can construct a distribution on restrictions that each set at most n/C variables such that, except with probability at most ^{2-n/(2O(k)Clog} |T|), after application of the restriction, all formulas in F simultaneously reduce to log^poly(k) |F|-juntas where an s-junta is a function whose value depends on only s of its inputs. Previously, Ajtai showed simultaneous approximations for k-DNF formulas by juntas related to the one we show but with a dependence on exp(k) rather than poly(k), resulting in a weaker height-approximation tradeoff than ours.