In the present book we concern ourselves exclusively with the qualitative analysis and synthesis of recurrent artificial neural networks. There is an abundance of papers, monographs, and books that address qualitative properties of such networks. However, whereas in these works, the analysis of the networks is frequently incidental to the achievement of some other principal goal, in the present book the network itself is the primary object of inquiry. Works in which the primary object of study is the recurrent neural network itself are rare. Consequently, many of the qualitative attributes and limitations of such networks have been ignored and are therefore obscure.