Alquézar and Sanfeliu (1995),

These authors (http://www.dlsi.ua.es/~mlf/nnafmc/papers/alquezar95algebraic.pdf) show how arbitrary FSM may be represented in Elman nets under the condition that the inputs, the outputs, and the state values are all rational numbers and the sigmoid operates with rational arithmetic, and give a simple recipe to select the weights of the network so that this occurs, which is derived from a representation of the next-state function of the FSM in terms of a system of linear equations; the construction byMinsky (1967) happens to be a special case of the proposed method. The construction needs a split-state representation of the states in the FSM for the reasons given by Goudreau et al. (1994) . Corresponding results for second-order DTRNN are also presented. The authors also indicate how the derived algebraic relations may be used to constrain gradient-descent algorithms to preserve prior knowledge inserted in the DTRNN in form of FSM transitions.