What is a hybrid model?

[Lev Goldfarb]

On Fri, 29 Mar 1996, Ron Sun wrote:

> lev goldfarb at unb.ca wrote:

> >Please note that the question is not really tricky. The question simply
> >suggests that there is no need to attach the term "hybrid" to the model,
> >because the combination (hybrid model) is both "ugly" and is likely to
> >lead almost all researchers involved in the wrong direction
>
> I really don't care that much what label one uses, and a label simply
> cannot ``lead all researchers in the wrong direction". Sorry. :-)

The labels we use betray our ignorance and prejudices and mislead many
ignorant followers.

> >really no "hybrid" mathematical structures, but rather "symbiotic
> >structures", e.g. topological group
>
> There have been _a lot of_  different
> mathematical structures being proposed that purport
> to capture all the essential properties of both symbolic and neural models.
> I would hesitate to make any such claim right now: we simply do not
> know enough yet about even the basics to make such a sweeping claim.
> What we can do is working toward such an end.

I'm all for "working toward such end". But lets try to do it in a
competent manner. Contrary to the above, there have been NO fundamentally
new and relevant MATHEMATICAL STRUCTURES (except transformation system)
proposed so far that embodies a "natural" symbiosis of symbolic and
numeric mathematical structures. To properly understand the last statement
one has to know first of all what the meaning of the "mathematical
structure" is.

An outstanding group of French mathematicians, who took the pseudonim of
Nicolas Bourbaki, contributed significantly to the popularization of the
emerging (during the first half of this century) understanding of
mathematical structures. Presently a mathematical structure (e.g. totally
ordered set, group, vector space, topological space) is typically
understood as a set - carrier of the structure - together with a set of
operations, or relations, defined on it. The relations/operations are
actually specified by means of axioms. (Frankly, it takes MANY hour to
become comfortable with the term "mathematical structure" through the
study of several typical structures. I don't know any of the newer
introductory books, which I'm sure are many, but from the older ones I
recommend, for example, Algebra, by Roger Godement, 1968.)

In the case of more complex and more interesting classical "symbiotic"
structures, such as the topological group, one defines this new structure
by imposing on the "old" structure (the group) a new structure (the
topology) in such a way that the new structure is CONSISTENT with the old
(in a certain well defined sense, e.g. algebraic operations must be
continuous wrt introduced topology).

Why is it that we are faced with considerable difficulties when trying to
"combine" the symbolic and the numeric mathematical structures into one
"natural" structure that is of relevance to us? It turns out that the two
classes of structures are not "combinable" in the sense which we are used
to in the classical mathematics. (Please, no hacks: I'd like to talk
science now.)
Why? While each element in the classical "symbiotic" math. structure belongs
simultaneously to two earlier structures, in this case this is definitely
not true: symbols are not numbers (and symbolic operations are
FUNDAMENTALLY different from numeric operations).

It appears that THE ONLY NATURAL WAY to accomplish the "symbiosis" in this
case is to associate with each symbolic operation a weight and to
introduce into the corresponding set of symbolic objects, or structs, the
distance measure that takes into consideration the operation weights.  The
distance is defined by means of the sequences of weighted operations, i.e.
NUMBERS enter the mathematical structure through the DISTANCES DEFINED ON
the basic objects, STRUCTS. I'm quite confident that the new structure
thus defined is (in a well defined sense) more general than any of the
classical numeric structures, and hence cannot be "isomorphic" to any of
them. We have discussed the axiomatics of the new structure on the
INDUCTIVE list.

It is also not difficult to see that in applications of the new
mathematical structure, the symbolic representations, or structs, begin to
play much more fundamental role as compared with the classical models,
e.g. the NN. Of course, this implies, in particular, that in applications
of the model we need fundamentally different measurement devices, which
are in nature realized chemically, but can at present be simulated
directly on top of the classical measurement devices. We are completing a
paper "Inductive theory of vision" in which, among other things, we
discuss more formally (and with some illustrations) these issues.

  -- Lev

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