Abstract
It is shown that every stochastic automaton generates a semigroup of continuous linear operators in order to give conditions for the convergence of certain operators connected with infinite input sequences. The main results of this note are conditions for the infinitesimal stability of stochastic automata. Since under suitable conditions the behaviour of a learning system can be represented by a stochastic automaton, these results apply to the asymptotic stability of learning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E.-E. Doberkat, Optimal Strategies in Measurable Learning Systems on Metric Spaces.J. Appl. Prob. 14, 795–805 (1977).
E.-E. Doberkat, Gute Strategien für bewertete Zustandsautomaten über metrischen Räumen. Submitted for publication.
N. Dunford, and J. T. Schwartz, Linear Operators I.Interscience, New York, 1957.
T. Hayashi, Y. Inagaki, and T. Fukumura, Some Notes on Stability Problems of Probabilistic Automata.J. Cybernetics 1, 94–114 (1971).
Y. Inagaki, T. Fukumura, and H. Matuura, Some Aspects of Linear Space Automata.Information and Control 20, 439–479 (1972).
C. Kuratowski, Topology II. Academic Press, New York, 1968.
W. Menzel, Theorie der Lernsysteme. Springer Verlag, Berlin, 1968.
W. Menzel, An Extension of the Theory of Learning Systems.Acta Informatica 2, 357–381 (1973).
A. Paz, and M. Rabinovitz, Linear Automata Approximation Problems.IEEE Transactions on Computers, Vol. C-23, 249–255 (1974).
K. Sugino, Y. Inagaki, and T. Fukumura, A Note on the Linear Space Automata Stability Problem.IEEE Transactions on Computers, Vol. C-25, 678–683 (1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Doberkat, EE. Convergence theorems for stochastic automata and learning systems. Math. Systems Theory 12, 347–359 (1978). https://doi.org/10.1007/BF01776582
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01776582