The N-N-N conjecture in ART1

In this paper we consider the ART1 neural network architecture introduced by Carpenter and Grossberg. In their original paper, Carpenter and Grossberg made the following conjecture: In the fast learning case, if the F"2 layer in ART1 has at least N nodes, then each member of a list of N input patterns presented cyclically at the F"1 layer of ART1 will have direct access to an F"2 layer node after at most N list presentations. In this paper, we demonstrate that the conjecture is not valid for certain large L values, where L is a network parameter associated with the adaptation of the bottom-up traces in ART1. It is worth noting that previous work has shown the conjecture to be true for small L values.