Proceedings of International Joint Conference on Neural Networks, Montreal, Canada, July 31 - August 4, 2005
Modification of the ART-1 Architecture Based on
Category Theoretic Design Principles
Michael J. Healy†, Richard D. Olinger† Robert J. Young‡, Thomas P. Caudell†‡
†Department of Electrical and
Computer Engineering
University of New Mexico
Albuquerque, New Mexico 87131
E-mails: mjhealy@ece.unm.edu,
rolinger@ece.unm.edu
‡Department of Computer Science
University of New Mexico
Albuquerque, New Mexico 87131
E-mails: ryoung@cs.unm.edu,
tpc@ece.unm.edu
Abstract— Many studies have addressed the knowledge representation capability of neural networks. A recently-developed
mathematical semantic theory explains the relationship between
knowledge and its representation in connectionist systems. The
theory yields design principles for neural networks whose behavioral repertoire expresses any desired capability that can
be expressed logically. In this paper, we show how the design
principle of limit formation can be applied to modify the ART1 architecture, yielding a discrimination capability that goes
beyond vigilance. Simulations of this new design illustrate the
increased discrimination ability it provides for multi-spectral
image analysis.
I. I NTRODUCTION
Many studies (see for example [2], [5], [10], [9], [16], and
the review [1]) have addressed the knowledge representation
capability of neural networks. We present an example to
illustrate the improved performance achievable by applying
neural network design principles derived from a recentlydeveloped theory for knowledge representation. The example
is a small but significant modification to an ART-1 network[3],
applied to multi-spectral image analysis. The knowledge representation theory is based upon the mathematical rigor of
category theory applied to neural network semantic modeling.
Because category theory is as yet unfamiliar to many (until
recently being regarded as the ultimate in pure mathematics),
we begin with a brief overview of the topics necessary for
understanding the work described here. Our semantic theory
in its current state of development is described in full in [14],
which contains a more comprehensive overview of category
theory. Our previous work in applying the semantic theory
to neural network analysis and design is described in [11],
[12], [13]. Many applications of category theory exist, both to
physical and computational theory ([7], [8], [19], [21], [22])
and to practice ([15], [23]).
The semantic theory addresses the question of where and
how knowledge is acquired, organized, and stored in connectionist systems. The knowledge has the structure of a
hierarchical system of concepts, directed from the abstract
to the specific. The theory explains knowledge acquisition
and representation in a neural network as an incremental reuse of existing concept representations combined with data to
0-7803-9048-2/05/$20.00 ©2005 IEEE
457
Kurt W. Larson§
§Sandia National Laboratory
Albuquerque, New Mexico
E-mail: kwlarso@sandia.gov
form new representations of both more abstract (or general)
concepts and more specific (or specialized) ones. Applied
to a neural network with many sensors and other functional
sub-networks, the semantic model provides a mathematically
rigorous yet natural explanation of the combining of networkregion-specific hierarchy representations so that the overall
network, if well-designed, acts as if there were a single
knowledge structure guiding its behavior. Here, we focus
upon the incremental knowledge representation in an ART1 network, which has only a single region, associated with a
single input layer. What we show is that the theory can be
applied to improve the performance of even a single-region
network in processing multi-modal information derived from
a single sensor.
The paper is organized as follows. Section II provides a very
brief grounding in the category theory used. In Section III we
show how our categorically-based semantic theory is applied
to neural networks. In Section IV we describe the use of the
theory in obtaining the ART-1 modification for our example.
Section V describes our experimental method, Section VI the
results, and Section VII is the Conclusion.
II. C ATEGORY T HEORY: A B RIEF I NTRODUCTION
Category theory (see [20], [6], [17], [18], or the tutorial in
[14]) is based upon the notion of an arrow, or morphism—a
relationship between two objects in a category. A morphism
f : a −→ b has a domain object a and a codomain object b ,
and serves as a sort of directed relationship between a and
b . In a category C , each pair of arrows f : a −→ b and
g : b −→ c (where the codomain b of f is also the domain
of g as indicated) has a composition arrow g ◦ f : a −→ c
whose domain a is the domain of f and whose codomain c
is the codomain of g . Composition is associative, that is, for
three arrows of the form f : a −→ b , g: b −→ c and h: c −→
d , the result of composing them is order-independent, with
h◦(g ◦f ) = (h◦g)◦f . For each object a , there is an identity
morphism ida : a −→ a such that for any arrows f : a −→ b
and g: b −→ a , ida ◦ g = g and f ◦ ida = f . A familiar
example of a category is one called Set, which has sets as its
between categories that preserves compositional structure,
called a functor, formalizes this notion. A functor F : C −→
D associates to each object a of C a unique image object
F (a) of D and to each morphism f : a −→ b of C a
unique morphism F (f ) : F (a) −→ F (b) of D , and is such
that (1) For each composition g ◦C f in C , F (g ◦C f ) =
F (g) ◦D F (f ) , where ◦C and ◦D denote the respective
compositions in C and D ; (2) for each object a of C ,
F (ida ) = idF (a) . It follows that F maps commutative
diagrams of C to commutative diagrams in D . This means
that any structural constraints expressed in C are translated
into D .
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a
r9 O 3 eLLLL
r
r
LL f2
f1 rrr
LL
r
LL
rr
L
r
LL
r
rr
a1Y3
a
E 2
∆ 3
3
3
g3
3
3
g1 3
g2
3
K
3
3 3 3
b
∆ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Fig. 1.
III. A PPLYING C ATEGORY T HEORY TO N EURAL
N ETWORK S EMANTIC A NALYSIS
A limit for a diagram ∆ .
objects, functions as its morphisms, and whose composition is
just the composition of functions, (g ◦ f )(x) = g(f (x)) .
Key notions for the theoretical background of this paper
are commutative diagrams and initial and terminal objects. A
diagram is a collection of objects and morphisms of C . In
a commutative diagram, any two morphisms with the same
domain and codomain, where at least one of the morphisms
is the composition of two or more diagram morphisms, are
equal. An initial object, where one exists in C , is an object i
for which every object a of C is the codomain of a unique
morphism f : i −→ a . A terminal object t has every object
a of C as the domain of a unique morphism f : a −→ t .
An important use of these key notions is in the definition
of limits and colimits. In [11] and [14] we have shown
how colimits model the learning of more complex concepts
through re-use of simpler concepts already represented in the
connection-weight memory of a neural network. In [14] we
show how limits model the learning of simpler, more abstract
concepts through re-use of existing representations. Let ∆ be
a diagram in a category C as shown in Fig. 1, with objects
a1 , a2 , a3 and morphisms f1 : a1 −→ a3 and f2 : a2 −→ a3 .
The diagram ∆ extends ∆ to a commutative diagram with an
apical object b and morphisms gi : b −→ ai (i = 1, . . . , 3) ,
with f1 ◦ g1 = g3 = f2 ◦ g2 , provided additional objects
and morphisms with the requisite properties exist in C . The
conical structure K is called a cone. Cones for ∆ are the
objects of a category cone∆ (whose morphisms are described
in [14]). A limit for the diagram ∆ is a terminal object K
in cone∆ .
The importance of category theory lies in its ability to
formalize the notion that things that differ in substance can
have an underlying similarity of “structural” form. A mapping
458
Knowledge can be seen as a system of symbolic concepts
- descriptions of objects, events, and anything else one can
imagine, at any arbitrary level of generality or specificity. The
system is organized as a hierarchy ordered from the abstract
to the specific. Learning, the acquisition of a knowledge
representation in a neural network, begins at the sensor level of
processing. Concepts associated with sensor elements describe
the sensor primitives. These are far from being the most
complex, yet are not the simplest, concepts possible. Indeed,
the neural network’s learning algorithm effectively re-uses the
sensor concepts in many ways in combination with the input
data to form concepts not yet represented by the connectionweight array of the network. The more complex concept representations are formed via colimit generation in the network
structure. (Obviously, this implies that a category can be used
to represent the diagrams and colimits; neural categories will
be discussed briefly here.) An abstraction process proceeds in
the other direction via limit generation. The abstract concepts
describe items that are shared by the more complex concepts
in the diagrams over which limits are formed. Thus, the
knowledge-representation process proceeds in both directions
— specialization and abstraction — beginning at the sensorpercept level.
A category Concept provides the required mathematical
model for the hierarchical structure of knowledge. In actuality,
this is a category whose objects are formal logic theories T
and whose morphisms are theory morphisms s: T −→ T 0 .
Briefly, s is a mapping of the quantities and axioms expressed
in T into the theory T 0 such that the images of the axioms
of T are either axioms or theorems of T 0 (see [14] or any
of [6], [8], [19]). Categories NA,w , where A is a neural
network architecture (such as a specific ART-1 network) and
w is an array of connection weight values for it, provide the
required mathematical model for neural networks in specified
states of learning. The objects of NA,w are the sets of inputs
that “activate” pairs (pi , η) given the weights in w , where
pi is a node of A and η is a set of output values for pi . The
set η is often modeled as an interval of real values where pi
has a real-valued signal function. A member of the activating
set for (pi , η) is an input pattern that causes pi to generate
an output signal in the set η . A morphism m: (pi , η) −→
(pj , η 0 ) is the set of inputs that cause all the nodes lying
along the paths of connections forming a bundle Γ to generate
outputs within specified intervals. The paths in Γ share the
common source and target objects (pi , η) and (pj , η 0 ) . If
A is properly designed and w is an array of weight values
acquired at some stage of learning from input patterns, it will
be possible to define a functor M : Concept −→ NA,w . This
is a mathematical description of the representation of concepts
and their morphisms in A at the stage of learning represented
by w .
Each concept morphism s: T −→ T 0 has an associated model-space morphism, a functor Mod(s): Mod(T 0 ) −→
Mod(T ) . Here, Mod(T ) and Mod(T 0 ) are categories of models, possible worlds or instances within which T and T 0
hold, respectively. Since Mod(s) reverses the direction of s ,
each instance of T 0 has a corresponding instance of T .
This fact has great significance for neural networks. To see
this, suppose that (pi , η) and (pj , η 0 ) are the images of
objects T and T 0 under the functor M , (pi , η) = M (T )
and (pj , η 0 ) = M (T 0 ) , and that m: (pi , η) −→ (pj , η 0 )
is the image of s: T −→ T 0 , m = M (s) . We associate
the activating inputs for the objects (pi , η) = M (T ) and
(pj , η 0 ) = M (T 0 ) with objects in the model categories
Mod(T ) and Mod(T 0 ) , respectively. Given this association,
every input that activates (pj , η 0 ) must also activate (pi , η) ,
a consequence of the existence of the model-space morphism
Mod(s): Mod(T 0 ) −→ Mod(T ) . Let T be the apical concept
of a limit cone for a diagram ∆ in Concept and let
s: T −→ T 0 be one of the leg morphisms for the limit cone.
Then, (pi , η) must be activated whenever (pj , η 0 ) is, where
(pj , η 0 ) can be any object in the diagram image M (∆) .
IV. A N ART-1 N ETWORK M ODIFIED WITH L IMITS AT F1
In the following, we apply limits to supplement the ART1 vigilance mechanism. This enhances the resolving power
of ART-1, allowing it to control information loss in specific
regions of the templates as they form. Use of the vigilance
mechanism alone allows control only over information loss in
a whole template. Applying limits requires that we discuss
ART-1 as an architecture that can be extended to have a
categorical representation capable of containing the image of
a functor from the Concept category. This is not a trivial
task with any existing artificial neural architecture, but this
need not prevent us from testing a partial categorically-based
extension. Accordingly, we have modified an ART-1 network
by applying limits to discrete diagrams (i.e., having objects
only) comprising disjoint subsets of nodes whose union is
the entire F1 layer. Each node is associated with an object
whose output set η includes all positive outputs. This allows
the apical objects of the limit cones to be shown simply as
nodes (see Fig. 2, where the apical objects are labelled “ SMi
limit”, SM standing for “sub-modality”). These form a new
layer which we call F−1 . The limit cone leg morphisms
are represented by bundles consisting of a single connection
each, projecting from a limit node SMi to each of the F1
nodes representing an object in its diagram. Each feedforward
459
Fig. 2. A resonating template in an ART-1 network modified to extract
abstract concepts corresponding to sub-modalities in each input pattern.
The F1 nodes represent input concepts and the SM nodes represent the
extracted sub-modality concepts.
connection is paired with a feedback connection. If the neural
cones are the functor images of limit cones in the category
Concept , the reciprocal feedback connections represent the
model-space morphisms corresponding to each of the limit
cone leg morphisms. The feedforward connections have small
positive weights and their reciprocal connections have unit
positive weight, so that activity in SMi has a minimal impact
upon its F1 nodes while they, in contrast, provide excitatory
input to it proportional to the number of them that are active.
Letting the set of all nonzero outputs from each F−1 node
represent the limit object of that node ensures the enforcement
of the property that a concept morphism is accompanied
by model-space morphisms: That is, if any one of its F1
nodes is active, a node SMi will be active. Thus, the F1
nodes represent neural category objects which in turn represent
concepts specifying sensor input properties, while the subset
constituting the diagram for an SMi limit node represents a
property comprising a group of input properties. Each SMi
node in F−1 represents an aspect of the group property shared
by the F1 nodes in its diagram.
The basic idea in making use of the F−1 nodes was to have
them supplement the vigilance node’s F2 reset capability.
This is the purpose of the connections from the F−1 nodes
to node V via node S as shown in Fig. 2. If resonance
between the current input and a template pattern is about to
occur, but one of the F−1 nodes is inactive because none
of its F1 correspondents is active, the resulting lack of an
inhibitory signal to S can allow its (tonic) activity to activate
V , thereby effectively vetoing the resonance. In this way, each
sub-modality is required to maintain at least one binary 1 in
each template. A further idea is to require an arbitrary number
of binary 1s for each sub-modality in each template by having
an adjustable but uniform threshold value t for the F−1
nodes. Here, all of the m sub-modality regions F1,i of F1
(i = 1, . . . , m) have the same number of nodes s , so that
n = ms where n is the number of nodes in F1 (hence, also
in F0 ). The bit-wise “AND” I ∧ T k of the current input
pattern I and choice template T k consists of sub-patterns
Ii ∧ Tik . Let k X k be the number of 1-bits in a binary pattern
X . To avoid activating V , then, each sub-modality i must
satisfy the inequality k Ii ∧ Tik k ≥ ts/2 (the factor 1/2 is
based upon our use of complement-coded input patterns [10]).
This requirement allows the user of the network to exercise a
more specific control over template information loss during recoding than is allowed by having a vigilance parameter alone,
which requires only that k I ∧ T k k ≥ ρk I k , where ρ is the
usual vigilance parameter. Just as ρ can be applied to control
the amount of specialization versus generalization allowed in
the templates (a higher value means fewer input exemplars per
template, hence, greater specialization), t can be applied to
control the specialization versus generalization allowed in each
sub-modality region of the templates (a higher value means
greater specialization within the sub-modality).
Given that an F1 activity pattern I ∧ T k is made up of
the activity patterns Ii ∧ Tik , and any F−1 node can activate
V if its activity falls below its threshold t , it is natural to ask
if t can be used to eliminate ρ altogether. It can be shown
that the usual test involving ρ is indeed redundant if t ≥ ρ .
However, this is not the case when t < ρ , and therefore the
parameter ρ cannot be eliminated.
V. T HE E XPERIMENTAL M ETHOD
A multi-spectral image was given as a set of 10 optical
band amplitudes for each pixel ( m = 10 ). This was to be
used to produce a false color image. Our method for this was
as follows. First, the 10-dimensional vector of analog values
for each pixel was converted to a binary input pattern for an
ART-1 network by converting the values to complement-coded
stack numerals; each stack code consists of a 0-1 binary array
which is activated in contiguous fashion, with the number of
binary 1s representing an amplitude (this is known widely as
“thermometer code”), together with an array with the same
number of binary values representing the complement of the
first array (see [10], where ART-1 with this representation was
proven equivalent to fuzzy ART.). If there are N bits for the
“positive” stack representing the amplitude, then there are s =
2N bits in the complement-coded stack numeral and, hence,
ms = 10 x 2N = 20N bits in the resulting input pattern for
ART-1 (hence, 20N input F0 nodes and 20N F1 nodes). An
ART-1 network sorts the input patterns so formed into clusters
so that the templates can be decoded into hyperbox regions in
10-dimensional space. The hyperboxes all lie within the 10cube defined by lower and upper bounds on the variation in
band values. The color code for a pixel was then selected by
first assigning a color code to the template with which it was
associated following training on all pixels, and then using that
color for the pixel. The color codes for the templates were
460
Fig. 3. Two-dimensional hyperboxes generated by ART-1 with submodality limits: vigilance = 0.47, F−1 threshold = 0.32 .
assigned by first sorting the templates in decreasing order of
the number of pixels with which they were associated, and then
assigning colors starting with blue and proceeding through
the visible spectrum to red and, for templates associated with
fewer than 10 pixels, white (therefore, the color white can be
associated with more than one template).
Two versions of ART-1 were used in the experiment, the
modified ART-1 network with an F−1 layer as described and
an unmodified ART-1 network. For the modified network, the
spectral bands were the sub-modalities, with each F−1 node
serving as a limit node for the set of 2N F1 nodes representing
its complement-coded band value. An activation threshold was
used for the F−1 nodes, allowing control over information
loss in all bands at an arbitrary, uniform level by the user. All
ART-1 simulations for this experiment were performed with
our recently-developed network specification and simulation
package eLoom[4].
VI. R ESULTS
To illustrate the formation of hyperbox templates with the
modified ART-1 network, a simpler, two-dimensional example
was processed first at several combinations of vigilance and
F−1 threshold values (see Figs. 3 and 4). A data file of
500 random 2D points was created in Matlab using the
rand function with lower and upper bounds on x and y
component variation of 0.0 and 128.0. These points were then
preprocessed in Matlab to generate complement-coded binary
input vectors based upon an N = 32-bit “positive” stack for
each dimension, resulting in 2 x 2N = 4N = 128 bits per input
pattern to ART-1. The resulting hyperboxes are shown along
with the points in each in Figs. 3 and 4 for a vigilance level
of 0.47 and thresholds of 0.32 and 0.40, respectively.
A single multi-spectral image was used in the 10-band
image experiment. Each “positive” binary stack had N = 8
Fig. 4. Two-dimensional hyperboxes generated by ART-1 with submodality limits: vigilance = 0.47, F−1 threshold = 0.40 .
Fig. 6. False color image generated by ART-1 modified with band limits
at F−1 , vigilance = 0.55 , F−1 threshold = 0.55 .
Fig. 5. False color image generated by unmodified ART-1, vigilance
= 0.55 .
Fig. 7. False color image generated by unmodified ART-1, vigilance
= 0.795 , same number of template colors as in Fig. 6.
bits, yielding a binary input pattern for ART-1 for each pixel
having 20N = 160 bits. The modified ART-1 network was
trained several times and the resulting template color codes
were used to form false color images. Several combinations of
values for vigilance and F−1 threshold were used, to produce
a variety of false color images from which the best could be
selected by human visualization. The same process but without
threshold values was used with the unmodified ART-1 network
for comparison. A “best” false color image (having greatest
discernible resolution) occurred for the modified network at
vigilance values of zero to 0.55 and an F−1 threshold of
0.55. It was based upon 452 templates, essentially produced
by F−1 -threshold resets. For comparison, the image generated
with the unmodified ART-1 network at a vigilance value of
0.55 is shown in Fig. 5 and the image generated with the
modified ART-1 network is shown in Fig. 6. The latter is
clearly superior in resolution; the former was produced with
far fewer templates. To obtain an even-handed comparison between modified and unmodified networks, successively higher
vigilance values were tried with the unmodified network to
approximate the number of templates yielded by the modified
network as shown in Fig. 6. A “best” image generated with the
unmodified ART-1 network, which generated 399 templates
at a vigilance value of 0.795, is shown in Fig. 7. This
461
and higher vigilance values, generating even more templates,
produced roughly the same false color image quality. The
ART-1 network modified with the thresholded band limit nodes
at F−1 yielded a superior image over the unmodified ART-1
network.
Finally, notice the color bar legend, labelled with positive
integers, at the far right of each image in figs. 5 - 7. With
the exception of white, each color is associated with a single
template and the number of pixel input patterns associated with
it is shown. All templates associated with 10 or fewer pixels
are colored white in Fig. 6. To achieve a fair comparison,
a higher breaking point for templates associated with fewer
pixels was used in Fig. 7; again, all such templates are colored
white. This has the effect of producing a color bar equivalent
to that for Fig. 6, with both having 32 colors and therefore an
equivalent color code. Colors for each figure are assigned to
the templates in the order of decreasing number of associated
pixels, going from blue to red and then white, where the total
number of pixels for all templates with fewer pixels (i.e., white
templates) is shown.
VII. D ISCUSSION AND C ONCLUSION
The objective of this paper was to illustrate the potential in
designing neural networks or improving upon existing designs
by applying a mathematical semantic model for neural networks based upon category theory. The categorical constructs
of the semantic model determine neural representations of
knowledge structures involving concepts and their relationships, or morphisms. This puts constraints on architectural
design and operational properties.
The result of the experiment discussed here illustrates the
potential in applying these constraints. Through a relatively
simple modification, the vigilance capability of an ART-1
network has been supplemented to provide increased discrimination in clustering. Limits were provided for discrete
diagrams in the F1 layer, producing the layer we refer
to as F−1 . In the context of concept representation, this
layer performs an abstraction process by representing subconcepts shared by the concepts represented in the diagrams
at F1 . In the experiment, the concepts are aspects of multispectral image data. We have shown, by visual inspection
of the results, that modifying an ART-1 network to include
limits can yield increased performance. At the cost of an
increase in the number of templates generated, the modified
network, applying a threshold value uniformly over the limitrepresenting nodes at F−1 , yields a false color image with
superior resolution compared with the resolution achievable
with an unmodified ART-1 network. This example shows that
the mathematical semantic model can be a useful guide for
improving the ART-1 design.
ACKNOWLEDGEMENT
This work was supported in part by Sandia National Laboratories, Albuquerque, New Mexico, under contract no. 238984.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States
462
Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.
R EFERENCES
[1] R. Andrews, J. Diederich, and A. B. Tickle. Survey and critique of
techniques for extracting rules from trained artificial neural networks.
Knowledge-Based Systems, 8(6):373–389, 1995.
[2] G. Bartfai. Hierarchical clustering with ART neural networks. In
Proceedings of the IEEE International Conference on Neural Networks,
June 28–July 2, 1994, volume II, pages 940–944. IEEE, 1994.
[3] G. A. Carpenter and S. Grossberg. A massively parallel architecture for
a self-organizing neural pattern recognition machine. Computer Vision,
Graphics, and Image Processing, 37:54–115, 1987.
[4] Thomas Preston Caudell, Yunhai Xiao, and Michael John Healy. eloom
and flatland: Specification, simulation and vizualization engines for the
study of arbitrary hierarchical neural architectures. Neural Networks,
16:617–624, 2003.
[5] M. W. Craven and J. W. Shavlik. Learning symbolic rules using artificial
neural networks. In Proceedings of the 10th International Machine
Learning Conference, Amherst, MA, pages 73–80, San Mateo, CA, 1993.
Morgan Kaufmann.
[6] R L Crole. Categories for Types. Cambridge University Press, 1993.
[7] A. C. Ehresmann and J.-P. Vanbremeersch. Information Processing and
Symmetry-Breaking in Memory Evolutive Systems. BioSystems, 43:25–
40, 1997.
[8] J. A. Goguen and R. M. Burstall. Institutions: Abstract model theory
for specification and programming. Journal of the Association for
Computing Machinery, 39(1):95–146, 1992.
[9] Michael J. Healy. A topological semantics for rule extraction with neural
networks. Connection Science, 11(1):91–113, 1999.
[10] Michael J. Healy and Thomas P. Caudell. Acquiring rule sets as a
product of learning in a logical neural architecture. IEEE Transactions
on Neural Networks, 8(3):461–474, 1997.
[11] Michael J. Healy and Thomas P. Caudell. A categorical semantic analysis of ART architectures. In IJCNN’01:International Joint Conference on
Neural Networks, Washington, DC, volume 1, pages 38–43. IEEE,INNS,
IEEE Press, 2001.
[12] Michael J. Healy and Thomas P. Caudell. Aphasic compressed representations: A functorial semantic design principle for coupled ART
networks. In The 2002 International Joint Conference on Neural
Networks (IJCNN’02). Honolulu. (CD-ROM Proceedings), page 2656.
IEEE,INNS, IEEE Press, 2002.
[13] Michael J. Healy and Thomas P. Caudell. From categorical semantics
to neural network design. In The Proceedings of the IJCNN 2003
International Joint Conference on Neural Networks. Portland, OR, USA.
(CD-ROM Proceedings), pages 1981–1986. IEEE,INNS, IEEE Press,
2003.
[14] Michael John Healy and Thomas Preston Caudell. Neural networks,
knowledge, and cognition: A mathematical semantic model based upon
category theory. Technical Report EECE-TR-04-020, Department of
Electrical and Computer Engineering, University of New Mexico, June
2004.
[15] R. Jullig and Y. V. Srinivas. Diagrams for software synthesis. In
Proceedings of KBSE ‘93: The Eighth Knowledge-Based Software
Engineering Conference, pages 10–19. IEEE Computer Society Press,
1993.
[16] N. K. Kasabov. Adaptable neuro production systems. Neurocomputing,
13:95–117, 1996.
[17] F. W. Lawvere and S. H. Schanuel. Conceptual Mathematics: A First
Introduction to Categories. Cambridge University Press, 1995.
[18] S. Mac Lane. Categories for the Working Mathematician. Springer,
1971. This is the standard reference for mathematicians, written by one
of the two co-discoverors of category theory. The other was S. Eilenberg.
[19] J. Meseguer. General logics. In Logic Colloquium ’87, pages 275–329.
Science Publishers B. V. (North-Holland), 1987.
[20] B. C. Pierce. Basic Category Theory for Computer Scientists. MIT
Press, 1991.
[21] Viggo Stoltenberg-Hansen, Ingrid Lindstroem, and Edward R. Griffor.
Mathematical Theory of Domains. Cambridge University Press, 1994.
[22] Stephen Vickers. Topology via Logic. Cambridge University Press, 1993.
[23] Keith Williamson, Michael Healy, and Richard Barker. Industrial
applications of software synthesis via category theory-case studies using
specware. Automated Software Engineering, 8(1):7–30, 2001.