We continue our theoretical examination of the problem of gene network identification, which we introduced in a previous paper. Here we consider a purely binary model of gene networks, without the assumption of sensitivity side information made in our previous paper. We present the following somewhat intuitive result: A general acyclic binary gene network can be identified by a brute force approach (in which every assignment for all subsets of k genes is made, where k is the maximum number of genes by which a gene is controlled, followed by the measurement of steady-state expression response). Our proof shows that the result is not straightforward because of certain side-effects. We also describe a natural characterization of the set of non-acyclic networks that can be identified. Moreover, we show that without new assumptions, this brute force approach has optimal complexity in the worst case.