  We describe how hierarchical concepts can be represented in three types of
layered neural networks. The aim is to support recognition of the concepts when
partial information about the concepts is presented, and also when some of the
neurons in the network might fail. Our failure model involves initial random
failures. The three types of networks are: feed-forward networks with high
connectivity, feed-forward networks with low connectivity, and layered networks
with low connectivity and with both forward edges and "lateral" edges within
layers. In order to achieve fault-tolerance, the representations all use
multiple representative neurons for each concept. We show how recognition can
work in all three of these settings, and quantify how the probability of
correct recognition depends on several parameters, including the number of
representatives and the neuron failure probability. We also discuss how these
representations might be learned, in all three types of networks. For the
feed-forward networks, the learning algorithms are similar to ones used in [4],
whereas for networks with lateral edges, the algorithms are generally inspired
by work on the assembly calculus [3, 6, 7].
