Topological analysis of infinite learning automata I: classification of neural network behavior

The authors have been developing a theory of learning automata that is applicable to studying goal-seeking systems of many types. The stepwise convergence of such systems to a goal state can be mapped into the training sequences of an appropriate learning automaton. This theory has been expanded to include infinite sets of states. The learning automaton may have an infinite characteristic semigroup of state transitions. The algebraic properties of the automaton are determined by this semigroup. A topology is introduced into the characteristic semigroup, S´. The latter can be considered as a function space of transformations of S. Behavior characteristics of the learning automaton be defined rigorously. In special cases, they can be related to the topological properties of S´ and S, such as continuity of the transition function, separation properties and compactness