Mike Healy



Michael J. Healy
Research Scholar
University of New Mexico
Department of Electrical and Computer Engineering
MSC01 1100
Albuquerque, NM 87131
E-mail: mjhealy@ece.unm.edu

Areas of Interest



Category theory, topology, logic
Mathematical semantics, knowledge representation, ontology
Neural networks, neuroscience

Professional


American Mathematical Society
Institute for Electrical and Electronic Engineers
Cognitive Neuroscience Society


Our Tutorial at IJCNN 2009

We gave a tutorial at the International Joint Conference on Neural Networks 2009. The title is Tutorial A1_T11: Fundamentals of Categorical Neural Semantic Theory, and it was given jointly with my colleague Tom Caudell during the tutorial sessions. It consisted of three slide presentations: An Introduction and Background; Fundamentals of the Theory; and Category Theory Applied to Improving the Performance of the Neocognitron and ART networks. Handouts were provided except for Fundamentals of the Theory. The fundamentals are described in publications listed below including UNM Technical Report EECE-TR-04-020, the 2006 Axiomathes paper, and the Neurocomputing preprint, which recounts also the ART network performance improvement. Each of these papers provides a good introduction to the theory; many of the other papers relating to the theory (usually with ``category theory'' or ``categorical'' in the title) contain mini-tutorials. The paper on category theory as a mathematics for ontology provides a background discussion and tutorial on category theory itself in case this is of interest; however, the previously-mentioned documents introduce all the category theory presently required for understanding our theory. Finally, UNM Technical Report EECE-TR-08-0010, written for cognitive scientists, provides an interesting application and introduction to the theory.

Research

My research is in the mathematical semantics of biological and computational systems. The mathematical discipline involved is principally category theory, together with formal logic and topology in the categorical context. With the exception of formal logic, which is applied in the form of theorem-proving software and other formal methods by computer scientists, the use of these disciplines outside mathematics itself is still regarded as a novelty. Yet, they are an excellent fit for studying the structure of the knowledge underlying the structure and processing flow in biological and computational systems. This knowledge can be regarded as an expression of the semantics of systems.

Two major areas of application for mathematical semantics are knowledge representation and the technological evolution of the philosophical notion of ontology. These have many computer science applications such as the semantic web. In this context, the slides for my talk at the recently-held Levels of Reality conference at the Mitteleuropa Foundation in Bolzano are located here . Past research with colleagues in industrial applications for knowledge-based systems and knowledge-based software synthesis is detailed in the publications listed below.

Also listed below are articles on research in the semantics of neural systems performed jointly with my colleague Tom Caudell. We are developing a new theory, the categorical neural semantic theory (CNST), to express the underlying meaning of the structure and processing in artificial and biological neural systems, from simple, engineering-oriented computational network models to complex, ultimately cognitive, neural networks. Through investigations in analyzing and improving upon current artificial network architectures, we are exploring the use of the CNST in neural network analysis and design. A preprint of a paper to appear in the journal Neurocomputing provides an example (see link below) in which we applied the categorical constructs known as limits and colimits to improve upon a well-known neural network architecture. Working jointly with cognitive psychologists and neuroscientists, we are also exploring the use of the CNST as a mathematical base for cognitive neuroscience. The UNM Technical Report EECE-TR-08-0010 (see link below) describes a preliminary experiment in cognitive science testing the theory.

Technical questions and comments are welcome.

Publications in this line of research

M. J. Healy, R. D. Olinger, R. J. Young, S. E. Taylor, T. P. Caudell, and K. W. Larson (2009)
Applying Category Theory to Improve the Performance of a Neural Architecture,
Neurocomputing, vol. 72, pp. 3158-3173.
Authors' Preprint (PDF),
NOTICE: This is the authors' version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication.

M. J. Healy, T. P. Caudell, and T. E. Goldsmith (2008)
A Model of Human Categorization and Similarity Based Upon Category Theory,
UNM Technical Report EECE-TR-08-0010, DSpaceUNM, University of New Mexico.

M. J. Healy (preprint of draft under review)
Category Theory as a Mathematics for Formalizing Ontologies,
Position paper.

M. J. Healy and T. P. Caudell (2007)
"Generalized Lattices Express Parallel Distributed Concept Learning",
Computational Intelligence Based on Lattice Theory, Kaburlasos, V. G. and Ritter, G. X. (Eds.),
Studies in Computational Intelligence, Vol. 67, Springer-Verlag:Heidelberg and New York, pp. 59-77.

Shawn E. Taylor, Michael J. Healy, and Thomas P. Caudell (2007)
"Categorical Mapping from Ontology to Neural Network: Initial Studies of Simple Neural Networks' Concept Capacity",
Proceedings of International Joint Conference on Neural Networks, Orlando, Florida.

Dulany B. Weaver, Michael J. Healy, and Thomas P. Caudell (2007)
"An Application of Category-Theoretic Design Methods to the Control of a Simulated Robot",
Proceedings of International Joint Conference on Neural Networks, Orlando, Florida.

Robert J. Young, Mike Ritthaler, Peter Zimmer, John McGraw, Michael J. Healy, and Thomas P. Caudell (2007)
"Comparison of Adaptive Resonance Theory Neural Networks for Astronomical Region of Interest Detection and Noise Characterization",
Proceedings of International Joint Conference on Neural Networks, Orlando, Florida.

M. J. Healy and T. P. Caudell (2006a)
Ontologies and Worlds in Category Theory: Implications for Neural Systems,
Axiomathes, vol. 16, nos. 1-2, pp. 165-214.


M. J. Healy and T. P. Caudell (2006b)
"Knowledge Representation and Possible Worlds for Neural Networks",
Proceedings of the 2006 International Joint Conference on Neural Networks, Vancouver, BC, Canada. pp. 5354-5361.

M. J. Healy and T. P. Caudell (2006c)
"Generalized Lattices Express Parallel Distributed Concept Learning",
Proceedings of the 2006 IEEE International Conference on Fuzzy Systems, Vancouver, BC, Canada. pp. 797-804.

M. J. Healy, R. D. Olinger, R. J. Young, T. P. Caudell, and K. W. Larson (2005)
"Modification of the ART-1 Architecture Based on Category Theoretic Design Principles",
Proceedings of International Joint Conference on Neural Networks, Montreal. pp. 457-462.

M. J. Healy and T. P. Caudell (2004)
Neural Networks, Knowledge and Cognition: A Mathematical Semantic Model Based upon Category Theory,
UNM Technical Report EECE-TR-04-020, DSpaceUNM, University of New Mexico.

T. P. Caudell, Y. Xiao, and M. J. Healy (2003)
" eLoom and Flatland: Specification, Simulation and Visualization Engines for the Study of Arbitrary Hierarchical Neural Architectures",
Neural Networks, vol. 16, pp. 617-624.

Y. Xiao, T. P. Caudell, and M. J. Healy (2003)
"eLoom: A Specification, Simulation and Vizualization Engine for Modeling Arbitrary Hierarchical Neural Architectures",
Proceedings of the IJCNN 2003: International Joint Conference on Neural Networks, Portland, Oregon. IEEE Press. pp. 3048-3053.

M. J. Healy, T. P. Caudell, and Y. Xiao (2003)
"From Categorical Semantics to Neural Network Design",
Proceedings of the IJCNN 2003: International Joint Conference on Neural Networks, Portland, Oregon. IEEE Press. pp. 1981-1986.

M. J. Healy and T. P. Caudell (2002)
"Aphasic Compressed Representations: A Functorial Semantic Design Principle for Coupled ART Networks",
IJCNN02:International Joint Con- ference on Neural Networks, Honolulu, Hawaii. The Printing House, Inc.:Stoughton, WI. Page (Number 2656 on CD-ROM).

K. Williamson, M. Healy and R. Barker (2001)
"Industrial Applications of Software Synthesis via Category Theory-Case Studies Using Specware",
Automated Software Engineering, vol. 8, no. 1, pp. 7-30.

Z. Gao, M. J. Healy and T. P. Caudell (2001)
"A Study of Autonomous Robot Behavior using the LAPART Neural Architecture",
IJCNN'01:International Joint Conference on Neural Networks, Washington, DC. IEEE Press. Vol. 3, pp. 2171-2175.

M. J. Healy and T. P. Caudell (2001)
"A Categorical Semantic Analysis of ART Architectures",
IJCNN'01:International Joint Conference on Neural Networks, Washington, DC. IEEE Press. Vol. 1, pp. 38-43.

S. J. Verzi, G. L. Heileman, M. Georgiopoulos and M. J. Healy (2001)
"Rademacher Penalization Applied to Fuzzy ARTMAP and Boosted ARTMAP",
IJCNN'01:International Joint Conference on Neural Networks, Washington, DC. IEEE Press. Vol. 2, pp. 1191-1196.

T. P. Caudell and M. J. Healy (2000)
"Lateral Priming Adaptive Resonance Theory (LAPART-2): Innovation in ART",
Chapter 6 in Recent Advances in Artificial Neural Networks: Design and Applications (Jain and Fanelli, eds.), CRC Press LLC: Boca Raton, FL.

M. J. Healy (2000)
"Category Theory Applied to Neural Modeling and Graphical Representations",
Proceedings of the International Joint Conference on Neural Networks (IJCNN 2000), Como, Italy. IEEE CS Press, vol. III, pp. 35-40.

S. J. Verzi, G. L. Heileman, M. Georgiopoulos and M. J. Healy (2000)
"Hierarchical ARTMAP",
Proceedings of the International Joint Conference on Neural Networks (IJCNN 2000), Como, Italy, IEEE CS Press, vol. VI, pp. 41-46.

M. Healy and K. Williamson (2000)
"Applying Category Theory to Derive Engineering Software from Encoded Knowledge",
in G. Goos, J. Hartmanis and J. van Leeuwen, ed., Algebraic Methodology and Software Technology, 8th International Conference (Proceedings of AMAST 2000, Iowa City, Iowa, USA). Lecture Notes in Computer Science 1816, Springer-Verlag:Heidelberg and New York. pp. 484-498.

K. Williamson and M. Healy (2000)
"Deriving engineering software from requirements",
Journal of Intelligent Manufacturing, vol. 11, no. 1, pp. 3-28.

T. P. Caudell and M. J. Healy (1999)
"Studies of Generalization for the LAPART-2 Architecture",
Proceedings of the International Joint Conference on Neural Networks (IJCNN'99), Washington, DC (Number 328 on CD-ROM).

K. Williamson and M. Healy (1999)
"Industrial Applications of Software Synthesis via Category Theory-Case Studies Using Specware",
Proceedings of the Automated Software Engineering Conference. Best Paper Award Nominee.

M. J. Healy (1999)
"Formal Semantic Model for Neural Networks",
in the survey article "Connectionist Symbol Processing: Dead or Alive?" (A. Jagota, T. Plate, L. Shastri and R. Sun, eds.), Neural Computing Surveys, vol. 2, pp. 1-40.

M. J. Healy (1999)
"Colimits in Memory: Category Theory and Neural Systems",
in Proceedings of the International Joint Conference on Neural Networks (IJCNN'99), Washington, DC, pp. 492-496.

M. J. Healy (1999)
"A Topological Semantics for Rule Extraction with Neural Networks",
Connection Science, vol. 11, no. 1, pp. 91-113.

M. J. Healy and T. P. Caudell (1998)
"Guaranteed Two-Pass Convergence for Supervised and Inferential Learning",
IEEE Transactions on Neural Networks, vol. 9, no. 1, pp. 195-204.

M. Uschold, M. Healy, K. Williamson, P. Clark and S. Woods (1998)
"Ontology Reuse and Application",
Proceedings of the International Conference on Formal Ontology in Information Systems (FOIS'98), Trento, Italy.

M. Uschold, P. Clark, M. Healy, K. Williamson and S. Woods (1998)
"An Experiment in Ontology Reuse",
Proceedings of the 11th Banff Knowledge Acquisition for Knowledge-Based Systems Workshop, vol. 2, Banff. SRDG Publications, Department of Computer Science, University of Calgary: Calgary, Alberta, CA.

M. J. Healy (1997)
"Continuous Functions and Neural Network Semantics",
(Proceedings of the Second World Congress of Nonlinear Analysts--WCNA96, Athens, 1996), Nonlinear Analysis, Vol. 30, No. 3, pp. 1335-1341.

M. J. Healy and T. P. Caudell (1997)
"Acquiring Rule Sets as a Product of Learning in a Logical Neural Architecture",
IEEE Transactions on Neural Networks, vol. 8, no. 3, pp. 461-474.

C. T. Abdallah, G. L. Heileman and M. Healy (1997)
"On the Stability of LAPART",
Proceedings of the Systems, Man, and Cybernetics Conference, Orlando, FL.

G. L. Heileman, M. Georgiopoulos, M. J. Healy and S. J. Verzi (1997)
"The Generalization Capabilities of ARTMAP",
Proceedings of the IEEE International Conference on Neural Networks (ICNN97), Houston, TX.

J. Choi, S. Ly, M. J. Healy and S. Smith (1996)
"Prediction of Cutter Condition Monitoring Using LAPART",
Proceedings of the 1996 IEEE International Conference on Neural Networks, Washington, DC.

T. P. Caudell and M. J. Healy (1996)
"Studies of Inference Rule Creation Using LAPART",
IEEE International Conference on Neural Networks, Washington, DC. (Published in the Proceedings of the Fifth IEEE International Conference on Fuzzy Systems, New Orleans, LA). pp. ICNN 1-ICNN 6.

T. P. Caudell and M. J. Healy (1994)
"Adaptive Resonance Theory Networks in the Encephalon Autonomous Vision System",
Proceedings of the International Conference on Neural Networks (World Congress on Computational Intelligence), Orlando, Florida.

M. J. Healy and T. P. Caudell (1993)
"Discrete Stack Interval Representations and Fuzzy ART1",
Proceedings of the World Congress on Neural Networks, Portland, Oregon.

M. J. Healy, T. P. Caudell and S. D. G. Smith (1993)
"A Neural Architecture for Pattern Sequence Verification Through Inferencing",
IEEE Transactions on Neural Networks, Vol 4, No. 1, pp. 9-20.

M. J. Healy and T. P. Caudell (1992)
"An Efficient Neural Network Architecture for Recognition of Spatial Pattern Invariants",
Proceedings of the International Joint Conference on Neural Networks, Baltimore. Vol. 4, pp. IV-208 to IV-213.

M. J. Healy (1991)
"A Logical Architecture for Supervised Learning",
Proceedings of the International Joint Conference on Neural Networks, Singapore. Vol. I, pp. 190-195.

M. J. Healy and T. P. Caudell (1990)
"On the Semantics of Pattern Recognition Neural Networks",
Northcon/90 Conference Record, Seattle, WA. North Hollywood, CA: Western Periodicals Company, pp. 245-250.

M. J. Healy (1989)
"The Elements of Adaptive Neural Expert Systems",
SPIE Vol. 1095 Applications of Artificial Intel ligence VII, Orlando, FL., pp. 830-837.

M. J. Healy (1988)
"An Investigation of Knowledge Representation in a Neural Network",
Northcon/88 Conference Record, Volume II, Seattle, WA. North Hollywood, CA: Western Periodicals Company, pp. 848-867.