What is a structural representation?
A proposal for a representational
formalism
Fifth variation
Initial posting:
Last updated: May 8, 2006 (further substantial improvements).
Complete paper (in PDF
format)
Abstract
We outline formalism for structural, or
symbolic, representation, the necessity of which has been acutely felt not just
in artificial intelligence and pattern recognition, but also in the natural
sciences, particularly biology. 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. Rather, the
reasons have to do with the growing realization that the objects of biology are
not only more important (to society) and interesting (to science), 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 formally address here, in
contrast to the ubiquitous mathematical view of objects as points in some
abstract space. In light of the above,
the paper is addressed to a very broad group of scientists.
One can gain an initial intuitive understanding of the proposed
representation by generalizing the temporal process of the (Peano) construction
of natural numbers: replace the single structureless unit
out of which a number is built by multiple structural ones. An
immediate and important consequence of the distinguishability (or multiplicity)
of units in the construction process is that we can now see which unit
was attached and when. Hence, the resulting (object) 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” —to observe its formation at the previous stage of
representation, at which it is called a ‘transformation’. To this end, imagine this opened unit as
representing a generalization of the event captured by a vertex in a
Feynman diagram. Now substitute for each
particle a process depicted as an elongated network composed out of analogous
structural units of the previous stage, and for the vertex itself, a
‘transformation’, which restructures the incoming processes into
the outgoing ones. Hence, in particular,
the formalism allows for a very natural introduction of representational stages:
a next-stage unit corresponds to a previous stage transformation.
We introduce the new concept
of class representation via the concept of class generating system,
which outputs structural entities belonging to that class. Hence the concept of class is
introduced as that of a class of similar structural processes, where the
“similarity” of such entities is ensured by them being outputs of
the same class generating system. Moreover,
the concept of class is introduced in such a way that—in contrast to all
existing formalisms—no two classes have elements in common. The evolving transformation system (ETS)
formalism proposed here is the first one developed to support such a new vision
of classes. The process responsible for
the construction and modification of both object and class representations is
the inductive learning process.
In light of ETS, the classical discrete
“representations” (strings, graphs) appear as incomplete special
cases at best, the proper adaptation of which should incorporate corresponding
formative histories, as is done here. We
believe that, without the latter, the process of induction becomes
intrinsically and computationally intractable.
Moreover, such discrete representations cannot be used as a basis for
constructing structural hierarchical representations, as becomes naturally
possible within ETS.
The gradual emergence of
ETS—including the concepts of structural object and class
representations, the resulting radically different view of “data”,
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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