What is a structural
representation? Fourth variation
Initial posting:
Last updated: October 11, 2005 (simplified def. of a struct, i.e. Def. 3).
Complete paper (in PDF
format)
Abstract
We outline a formalism for structural, or symbolic,
representation, the necessity of which has been acutely felt in all sciences,
particularly biology, for quite some time now. At the same time, biology has
been gradually edging to the forefront of sciences, although the reasons
obviously have nothing to do with its state of formalization or maturity—which
is quite primitive as compared, for example, to that of physics. Rather, the reasons have to do with the
growing realization that the objects of biology are not only more important and
interesting, but that they also more explicitly exhibit the evolving nature of
all objects in the Universe. It is this view of objects as evolving structural
processes that we aim to address here, in contrast to the ubiquitous
mathematical view of objects as points in some abstract space.
One can gain an initial intuitive understanding of the proposed
representation by generalizing the (Peano) process of
construction of natural numbers: replace the single structureless
unit out of which a number is built by multiple structural ones. An immediate but critical consequence of the distinguishability/multiplicity of units in the
construction process is that we can now see which unit was attached and
when. Hence, the resulting
representation for the first time embodies temporal structural information in
the form of a formative, or generative, history.
To gain some intuition about the nature of the above
“structural unit”, one needs to open it up, i.e. to observe its
formation at the previous level of representation. To this end, redraw the
popular image of particle collision as follows: substitute for each particle
track a ‘regular’ structural process (paved with its
‘generators’, where each generator, in turn, is composed out of
previous level units), and for the entire collision event, a ‘transformation’
that restructures the incoming regular processes into the resulting ones. Thus,
in particular, the formalism allows for a very natural introduction of
representational levels: a next-level unit corresponds to a previous level
transform.
The concept of class can then be introduced as that of a class
of similar regular processes, where each such process is both a temporal and
structural object representation and their similarity is understood via a
common generative origin. Furthermore, the proposed concept of class representation—which
inspired and directed the development of this formalism—differs radically
from the few inadequate surrogates that have emerged from numeric formalisms.
Indeed, the evolving transformation system (ETS) formalism proposed here is the
first one developed to support that concept: a class representation is a
generating system that outputs “similar” regular processes, i.e.
structured entities serving as object representations (for objects from that
class). The process responsible for the
construction and modification of both object and class representations is the
inductive learning process.
The classical discrete “representations” (strings,
graphs) now appear as incomplete special cases at best, the proper adaptation
of which should incorporate corresponding generative histories, as is done
here.
The gradual emergence of ETS, including the concepts of
structural object and class representations, as well as the associated
inductive learning processes and the representational levels, points to the
beginning of a new field—inductive informatics—which is intended as
a class oriented rival to conventional information processing paradigms.
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