Summary: The prefix problem is to compute all the products \(x_1\circ x_2\circ \ldots x_k\) for \(i\leq k\leq n\), where \(\circ\) is an associative operation. A recursive construction is used to obtain a product circuit for solving the prefix problem which has depth exactly \([\log_2n]\) and size bounded by \(4n\). An application yields fast, small Boolean circuits to simulate finite-state transducers. By simulating a sequential adder, a Boolean circuit which has depth \(2[\log_2n]+2\) and size bounded by \(14n\) is obtained for \(n\)-bit binary addition. The size can be decreased significantly by permitting the depth to increase by an additve constant.