The Uniqueness Condition

The main point is to show that two irreducible, I-O equivalent nets must necessarily be the same net, up to some simple <#382#> internal symmetries<#382#>. Clearly, if a net N is reducible (using one or more of the above three reducibility conditions), then it is I-O equivalent to another net with fewer hidden nodes. Some simple <#383#> internal symmetries<#383#> create various transformations that can be applied to a network N without changing its I-O map. These symmetries are: The maps #tex2html_wrap_inline2783#, #tex2html_wrap_inline2785# generate a finite group #tex2html_wrap_inline2787# of transformations of the set #tex2html_wrap_inline2789# of all nets with m input nodes, n hidden nodes, and u output units. We can call two nets in #tex2html_wrap_inline2797# <#422#> equivalent<#422#> if they are related by a transformation in the group #tex2html_wrap_inline2799#. It is then clear that two equivalent nets are I-O equivalent. Now, Sussmann's major result is:

#theorem1696#

This theorem implies the following corollary: An irreducible #tex2html_wrap_inline2819#-net is minimal. #tex2html_wrap_inline2821#