On different modes of communication
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
One-way functions and the nonisomorphism of NP-complete sets
Theoretical Computer Science
Communication complexity
Reimer's inequality and tardos' conjecture
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
A Dual Version of Reimer's Inequality and a Proof of Rudich's Conjecture
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
On notions of information transfer in VLSI circuits
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Proof of the Van den Berg–Kesten Conjecture
Combinatorics, Probability and Computing
Generic oracles and oracle classes
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
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Let f : {0, 1}n → {0, 1}. Let μ be a product probability measure on {0, 1}n. For ε ≥ 0, we define Dε(f), the ε-approximate decision tree complexity of f, to be the minimum depth of a decision tree T with μ(T(x) ≠ f(x)) ≤ ε. For j = 0 or 1 and for δ ≥ 0, we define Cj,δ(f), the δ-approximate j-certificate complexity of f, to be the minimum certificate complexity of a set A ⊆ Ω with μ(AΔf−1(j)) ≤ ε. Note that if μ(x) 0 for all x then D0(f) = D(f) and Cj,0(f) = Cj(f) are the ordinary decision tree and j-certificate complexities of f, respectively. We extend the well-known result, D(f) ≤ C1(f)C0(f) [Blum and Impagliazzo 1987; Hartmanis and Hemachandra 1991; Tardos 1989], proving that for all ε 0 there exists a δ 0 and a constant K = K(ε, δ) 0 such that for all n, μ, f, Dε(f) ≤ K C1,δ(f)C0,δ (f). We also give a partial answer to a related question on query complexity raised by Tardos [1989]. We prove generalizations of these results to general product probability spaces.