The Coin Problem and Pseudorandomness for Branching Programs

Abstract

The \emph{Coin Problem} is the following problem: a coin is given, which lands on head with probability either $1/2 + \beta$ or $1/2 - \beta$. 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 $\ge 2/3$. When our computational model is unrestricted, the majority function is optimal, and succeeds when $\beta \ge c /\sqrt{n}$ 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, $\beta$ must be at least $1/ (\log n)^{\Theta(w)}$. 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 \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 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 \emph{regular} branching programs, using seed length $O(w^4 \log n \log \log n + \log n \log (1/\eps))$. For $w=\eps=\Theta(1)$, this seed length 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 Braver man, Rao, Raz and Yehudayoff~\cite{BRRY}.