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The Coin Problem, and Pseudorandomness for Branching Programs
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@MISC{Brody_thecoin,
author = {Joshua Brody and Elad Verbin},
title = {The Coin Problem, and Pseudorandomness for Branching Programs},
year = {}
}
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Abstract
is given, which lands on head with probability either 1/2+β or 1/2 − β. We are given the outcome of n independent tosses of this coin, and the goal is to guess which way the coin is biased, and to answer correctly with probability ≥ 2/3. When our computational model is unrestricted, the majority function is optimal, and succeeds when β ≥ c / √ n foralargeenough constant c. The coin problem is open and interesting in models that cannot compute the majority function. In this paper we study the coin problem in the model of read-once width-w branching programs. We prove that in order to succeed in this model, β must be at least 1/(log n) Θ(w). For constant w this is tight by considering the recursive tribes function, and for other values of w this is nearly tight by considering other read-once AND-OR trees. We generalize this to a Dice Problem, where instead of independent tosses of a coin we are given independent tosses of one of two m-sided dice. We prove that if the distributions are too close and the mass of each side of the dice is not too small, then the dice cannot be distinguished by small-width read-once branching programs. We suggest one application for this kind of theorems: we prove that Nisan’s Generator fools width-w read-once regular branching programs, using seed length O(w 4 log n log log n + log n log(1/ε)). For w = ε = Θ(1), this seedlength is O(log n log log n). The coin theorem and its relatives might have other connections to PRGs. This application is related to the independent, but chronologically-earlier, work of Braverman, Rao, Raz and Yehudayoff [1]. Keywords-pseudorandomness; branching programs; lower bounds I.
Keyphrases
coin problem branching program independent toss majority function dice problem width-w read-once regular branching program read-once and-or tree log log log log log recursive tribe function small-width read-once branching program read-once width-w branching program computational model m-sided dice foralargeenough constant seed length coin theorem
