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A DISTRIBUTED BASIS FOR ANALOGICAL MAPPING

Ross W. Gayler
r.gayler@gmail.com
School of Communication, Arts and Critical Enquiry
La Trobe University
Victoria 3086 Australia

Simon D. Levy
levys@wlu.edu
Department of Computer Science
Washington and Lee University
Lexington, Virginia USA

ABSTRACT

We are concerned with the practical fea-
sibility of the neural basis of analogical map-
ping. All existing connectionist models of ana-
logical mapping rely to some degree on local-
ist representation (each concept or relation is
represented by a dedicated unit/neuron). These
localist solutions are implausible because they
need too many units for human-level compe-
tence or require the dynamic re-wiring of net-
works on a sub-second time-scale.
Analogical mapping can be formalised as
finding an approximate isomorphism between
graphs representing the source and target con-
ceptual structures. Connectionist models of
analogical mapping implement continuous
heuristic processes for finding graph isomor-
phisms. We present a novel connectionist
mechanism for finding graph isomorphisms
that relies on distributed, high-dimensional
representations of structure and mappings.
Consequently, it does not suffer from the prob-
lems of the number of units scaling combinato-
rially with the number of concepts or requiring
dynamic network re-wiring.

GRAPH ISOMORPHISM

Researchers tend to divide the process of
analogy into three stages: retrieval (finding an
appropriate source situation), mapping (identi-
fying the corresponding elements of the source
and target situations), and application. Our
concern is with the mapping stage, which is
essentially about structural correspondence. If
the source and target situations are formally
represented as graphs, the structural corre-
spondence between them can be described as
approximate graph isomorphism. Any mecha-
nism for finding graph isomorphisms is, by
definition, a mechanism for finding structural
correspondence and a possible mechanism for
implementing analogical mapping. We are
concerned with the formal underpinning of
analogical mapping (independently of whether
any particular researcher chooses to describe
their specific model in these terms).
It might be supposed that representing
situations as graphs is unnecessarily restrictive.
However, anything that can be formalised can
be represented by a graph. Category theory,
which is effectively a theory of structure and
graphs, is an alternative to set theory as a
foundation for mathematics (Marquis, 2009),
so anything that can be mathematically repre-
sented can be represented as a graph.
It might also be supposed that by working
solely with graph isomorphism we favour
structural correspondence to the exclusion of
other factors that are known to influence ana-
logical mapping, such as semantic similarity
and pragmatics. However, as any formal struc-
ture can be represented by graphs it follows
that semantics and pragmatics can also be en-
coded as graphs. For example, some models of
analogical mapping are based on labelled
graphs with the process being sensitive to label
similarity. However, any label value can be
encoded as a graph and label similarity cap-

Analogical Mapping with Vector Symbolic Architectures
166
tured by the degree of approximate isomor-
phism. Further, the mathematics of graph iso-
morphism has been extended to include attrib-
ute similarity and is commonly used this way
in computer vision and pattern recognition
(Bomze, Budinich, Pardalos & Pelillo, 1999).
The extent to which analogical mapping
based on graph isomorphism, is sensitive to
different types of information depends on what
information is encoded into the graphs. Our
current research is concerned only with the
practical feasibility of connectionist implemen-
tations of graph isomorphism. The question of
what information is encoded in the graphs is
separable. Consequently, we are not concerned
with modelling the psychological properties of
analogical mapping as such questions belong
to a completely different level of inquiry.

CONNECTIONIST IMPLEMENTATIONS

It is possible to model analogical map-
ping as a purely algorithmic process. However,
we are concerned with physiological plausibil-
ity and consequently limit our attention to
connectionist models of analogical mapping
such as ACME (Holyoak & Thagard, 1989),
AMBR (Kokinov, 1988), DRAMA (Eliasmith
& Thagard, 2001), and LISA (Hummel &
Holyoak, 1997). These models vary in their
theoretical emphases and the details of their
connectionist implementations, but they all
share a problem in the scalability of the repre-
sentation or construction of the connectionist
mapping network. We contend that this is a
consequence of using localist connectionist
representations or processes. In essence, they
either have to allow in advance for all combi-
natorial possibilities, which requires too many
units (Stewart & Eliasmith, in press), or they
have to construct the required network for each
new mapping task in a fraction of a second.

Problems with localist implementation

Rather than review all the major connec-
tionist models of analogical mapping, we will
use ACME and DRAMA to illustrate the prob-
lem with localist representation. Localist and
distributed connectionist models have often
been compared in terms of properties such as
neural plausibility and robustness. Here, we
are concerned only with a single issue: dy-
namic re-wiring (i.e., the need for connections
to be made between neurons as a function of
the source and target situations to be mapped).
ACME constructs a localist network to
represent possible mappings between the
source and target structures. The network is a
function of the source and target representa-
tions, and a new network has to be constructed
for every source and target pair. A localist unit
is constructed to represent each possible map-
ping between a source vertex and target vertex.
The activation of each unit indicates the degree
of support for the corresponding vertex map-
ping being part of the overall mapping be-
tween the source and target. The connections
between the network units encode compatibil-
ity between the corresponding vertex map-
pings. These connections are a function of the
source and target representations and con-
structed anew for each problem. Compatible
vertex mappings are linked by excitatory con-
nections so that support for plausibility of one
vertex mapping transmits support to compati-
ble mappings. Similarly, inhibitory connec-
tions are used to connect the units representing
incompatible mappings. The network imple-
ments a relaxation labelling that finds a com-
patible set of mappings. The operation of the
mapping network is neurally plausible, but the
process of its construction is not.
The inputs to ACME are symbolic repre-
sentations of the source and target structures.
The mapping network is constructed by a
symbolic process that traverses the source and
target structures. The time complexity of the
traversal will be a function of the size of the
structures to be mapped. Given that we believe
analogical mapping is a continually used core
part of cognition and that all cognitive infor-
mation is encoded as (large) graph structures,
we strongly prefer mapping network setup to
require approximately constant time independ-
ent of the structures to be mapped.
DRAMA is a variant of ACME with dis-
tributed source and target representations.