  We present results from numerical studies of supervised learning operations
in recurrent networks considered as graphs, leading from a given set of input
conditions to predetermined outputs. Graphs that have optimized their output
for particular inputs with respect to predetermined outputs are asymptotically
stable and can be characterized by attractors which form a representation space
for an associative multiplicative structure of input operations. As the mapping
from a series of inputs onto a series of such attractors generally depends on
the sequence of inputs, this structure is generally non-commutative. Moreover,
the size of the set of attractors, indicating the complexity of learning, is
found to behave non-monotonically as learning proceeds. A tentative relation
between this complexity and the notion of pragmatic information is indicated.
