Abstract
Categorical shape theory adopts a minimalist approach to pattern recognition. It assumes given a collection of archetypes and a collection of ‘objects of interest’, each collection being provided with an internal means of comparison between objects. The two collections are externally linked so as to allow comparison of objects of interest with archetypes. The structure of archetypes and of objects of interest is only observable via comparisons within these two categories. The categorical form of infinitary language is reviewed, including the ideas of equational theories. It is shown that categorical shape theory is the syntax of the abstract recognition process.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.C.M. Baeten and W.D. Weijland,Process Algebra (Kluwer, Deventer, 1986).
J.A. Bergstra and J.W. Klop, Process algebra for synchronous communication, Information and Control 69 (1984) 109–137.
D. Bourn and J.-M. Cordier, Distributeurs et théorie de la forme, Cahiers Top. et Géom. Diff. 21 (1980) 161–189.
J.-M. Cordier and T. Porter,Shape Theory: Categorical Methods of Approximation, Mathematics and its Application (Ellis Horwood, 1989).
A.C. Ehresmann and J.-P. Vanbremeersch, Hierarchical evolutive systems: a mathematical model for complex systems, Bull. Math. Biol. 49 (1987) 13–50.
A.C. Ehresmann and J.-P. Vanbremeersch, Systèmes hiérarchiques évolutifs à mémoire autorégulée,Synergie et cohérence dans les systèmes biologiques, Séminaire Transdisciplinaire, Centre Interuniversitaire Jussieu-St. Bernard, May 1989.
J.-Y. Girard, Linear logic, Theor. Comput. Sci. 50 (1987) 1–102.
J.A. Goguen, Types as theories,Proc. Symp. on General Topology and its Applications, Oxford, June 1989 (Oxford University Press, 1990).
J.A. Goguen, A categorical manifesto, Math. Struct. Comp. Sci. 1 (1991) 49–67.
G.M. Kelly,The Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes, No. 64 (Cambridge University Press, 1983).
F.E.J. Linton, Some aspects of equational categories,Proc. Conf. Categorical on Algebra, La Jolla, 1965 (Springer, 1966) pp. 84–95.
S. MacLane,Categories for the Working Mathematician, Grad. Texts in Maths. Vol. 5 (Springer, 1971).
M. Makkai and R. Paré,Accessible Categories: The Foundations of Categorical Model Theory, Contemporary mathematics, Vol. 104 (1989).
E.G. Manes and M.A. Arbib,Algebraic Approaches to Program Semantics, Texts and Monographs in Computer Science (Springer, 1986).
N. Martí-Oliet and J. Meseguer, From Petri nets to linear logic, Math. Struct. Comp. Sci. 1 (1991) 69–101.
J. Meseguer and U. Montari, Petri nets are monoids, Informat. Computat. 88 (1990) 105–155.
R. Paré, Some applications of categorical model theory, categories in computer science and logic,Proc. of a Summer Research Conf., June 14–20, 1987, Contemporary Mathematics, Vol. 92 (1989).
M. Pavel,Fundamentals of Pattern Recognition (Marcel Dekker, 1989) Pure and Applied Mathematics, Vol. 124.
D.E. Rydeheard and J.G. Stell, Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms,Category Theory and Computer Science, Lecture Notes in Computer Science, Vol. 283 (1987).
S. Vickers,Topology via Logic, Cambridge Tracts in Theoretical Computer Science 5 (Cambridge University Press, 1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Porter, T. Categorical shape theory as a formal language for pattern recognition?. Ann Math Artif Intell 10, 25–54 (1994). https://doi.org/10.1007/BF01530943
Issue Date:
DOI: https://doi.org/10.1007/BF01530943