Let #tex2html_wrap_inline3015# and #tex2html_wrap_inline3017# stand for 2 feedforward nets, for example, see
Figure-#fig1#607>. Also, let the symbol (#tex2html_wrap_inline3019#) stands for
#tex2html_wrap_inline3021#-equivalence in behavior, and (#tex2html_wrap_inline3023#) for
#tex2html_wrap_inline3025#-equivalence in structure.
Albertini, Sontag and Maillot [#5##1#] have stated
that given two networks #tex2html_wrap_inline3027# and #tex2html_wrap_inline3029#, one says that they
are I-O equivalent if the behavior of #tex2html_wrap_inline3031# equal to the behavior of
#tex2html_wrap_inline3033#. In other words, applying the same set of inputs to both
networks will yield the same output. Clearly, networks with identical
structures and identical internal representations (<#613#> weight sets<#613#>)
have the same I-O mappings. Also, nets with different internal
representations can have the same I-O mappings.
Now, what we would like to answer is the following question: When does
#tex2html_wrap_inline3035# imply #tex2html_wrap_inline3037# ?
Sontag [#20##1#] pointed out that Sussmann's
argument takes into account the ``independence property'' (IP)
(defined below), but he did not explicitly state this property in his
work [#8##1#]. Also, they state that to show uniqueness of
the connection weights of a given network, one needs to demonstrate that
the given activation functions satisfy IP.
Sussmann showed that #tex2html_wrap_inline3039# and the related sigmoid #tex2html_wrap_inline3041#
satisfy IP.