For a recurrent neural network the answer is No, because the number of memorized patterns is proportional to the size of the system, so, by definition, isn't considered finite.The exception would be, if one were interested in using a recurrent neural network to group these memorized patterns to a finite number of categories. In that case, as in the case of feed-forward networks, the network does, precisely, the work of a finite state automaton, i.e. it classifies an input pattern among the learned categories.