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Description

The emph{Coin Problem} is the following problem: a coin is given, which lands on head with probability either $1/2 + eta$ or $1/2 - eta$. 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 be correct with probability $ge 2/3$. When our computational model is unrestricted, the majority function is optimal, and succeeds when $eta ge c /sqrt{n}$ for a large enough 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 emph{read-once width-$w$ branching programs}. We prove that in order to succeed in this model, $eta$ must be at least $1/ (log n)^{Theta(w)}$. For constant $w$ this is tight by considering the recursive tribes function.

We generalize this to a emph{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, then the dice cannot be distinguished by a small-width read-once branching program.

We suggest one application for this kind of theorems: we prove that Nisan's Generator fools width-$w$ read-once emph{permutation} branching programs, using seed length $O(w^4 log n log log n + log n log (1/eps))$. For $w=eps=Theta(1)$, 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 (which might be submitted to this FOCS).

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