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2012 IEEE 27th Conference on Computational Complexity (2012)
Porto Portugal
June 26, 2012 to June 29, 2012
ISSN: 1093-0159
ISBN: 978-0-7695-4708-4
pp: 117-125
We show how to approximate any function in AC^0 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 - \beta n to within error exp(-\beta n), where \beta = \beta (S, d) = 2^{-O(d log^{4/5}S)}. Our proof is constructive and we use its constructivity to derive a deterministic algorithm for #AC^0SAT with multiplicative factor savings over the naive 2^n.S algorithm of 2^{-\beta n}, when applied to any n-input AC^0 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-\beta n} that exactly computes the function given by such a circuit. Recently, Impagliazzo, Matthews, and Paturi derived an algorithm for #AC^0SAT 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
Decision trees, Constant-depth circuits, Satisfiability algorithms
"Approximating AC^0 by Small Height Decision Trees and a Deterministic Algorithm for #AC^0SAT", 2012 IEEE 27th Conference on Computational Complexity, vol. 00, no. , pp. 117-125, 2012, doi:10.1109/CCC.2012.40
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