##### 抄録

We present two approaches to the analysis of the
relationship between a recurrent neural network (RNN) and the finite state
machine \( {\cal M} \) the network is able to exactly mimic. First, the
network is treated as a state machine and the relationship between the RNN
and \( {\cal M} \) is established in the context of algebraic theory of
automata. In the second approach, the RNN is viewed as a set of
discrete-time dynamical systems associated with input symbols of \( {\cal
M} \). In particular, issues concerning network representation of loops
and cycles in the state transition diagram of \( {\cal M} \) are shown to
provide a basis for the interpretation of learning process from the point
of view of bifurcation analysis. The circumstances under which a loop
corresponding to an input symbol \( x \) is represented by an attractive
fixed point of the underlying dynamical system associated with \( x \) are
investigated. For the case of two recurrent neurons, under some
assumptions on weight values, bifurcations can be understood in the
geometrical context of intersection of increasing and decreasing parts of
curves defining fixed points. The most typical bifurcation responsible for
the creation of a new fixed point is the saddle node bifurcation.
(Also cross-referenced as UMIACS-TR-95-1)