Abstract
This paper investigates closedness properties in relation with team learning of total recursive functions. One of the first problems solved for any new identification types is the following: “Does the identifiability of classes U 1 and U 2 imply the identifiability of U 1∪U 2?” In this paper we are interested in a more general question: “Does the identifiability of every union of n−1 classes out of U 1,...,U n imply the identifiability of U 1∪...∪U n?” If the answer is positive, we call such identification type n-closed. We show that n-closedness can be equivalently formulated in terms of team learning. After that we find for which n team identification in the limit and team finite identification types are n-closed. In the case of team finite identification only teams in which at least half of the strategies must be successful are considered. It turns out that all these identification types, though not closed in the usual sense, are n-closed for some n>2.
This research was supported by Latvian Science Council Grant No. 93.599.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Ambainis. Probabilistic and team PFIN-type learning: general properties. Proceedings of the Ninth Conference on Computational Learning Theory, pp. 157–168, ACM, 1996.
K. Apsītis. Topological considerations in composing teams of learning machines. In K. Jantke, S. Lange, editors, Algorithmic Learning for Knowledge Based Systems. LNAI, vol. 961, pp. 146–154. Springer-Verlag, 1992.
K. Apsītis, R. Freivalds, M. Kriķis, R. Simanovskis, and J. Smotrovs. Unions of identifiable classes of total recursive functions. In K. Jantke, editor, Analogical and Inductive Inference. LNAI, vol. 642, pp. 99–107. Springer-Verlag, 1992.
K. Apsītis, R. Freivalds, R. Simanovskis, and J. Smotrovs. Unions of identifiable families of languages. In L. Miclet, C. de la Higuera, editors, Grammatical Inference: Learning Syntax from Sentences. LNAI, vol. 1147, pp. 48–58. Springer-Verlag, 1996.
J. Bārzdiņs. Two theorems on the limiting synthesis of functions. In J. Bārzdiņ>s, editor, Theory of Algorithms and Programs, vol. 1, pp. 82–88. Latvian State University, Rīga, 1974. (In Russian.)
L. Blum and M. Blum. Toward a mathematical theory of inductive inference. Information and Control, vol. 28, pp. 125–155, 1975.
R. Daley, L. Pitt, M. Velauthapillai, and T. Will. Relations between probabilistic and team one-shot learners. In M. Warmuth and L. Valiant, editors, Proceedings of the 1991 Workshop on Computational Learning Theory, pp. 228–239, Morgan Kaufmann Publishers, 1991.
R. Freivalds. Functions computable in the limit by probabilistic machines. Lecture Notes in Computer Science, vol. 28, pp. 77–87, 1975.
R. Freivalds and R. Wiehagen. Inductive inference with additional information. Elektronische Informationsverabeitung und Kybernetik, vol. 15 (4), pp. 179–184, 1979.
E. M. Gold. Language identification in the limit. Information and Control, vol. 10, pp. 447–474, 1967.
S. Jain and A. Sharma. Finite learning by a team. In M. Fulk and J. Case, editors, Proceedings of the Third Annual Workshop on Computational Learning Theory, pp. 163–177, Morgan Kaufmann Publishers, 1990.
D. Osherson, M. Stob, and S. Weinstein. Aggregating inductive expertise. Information and Control, vol. 70, pp. 69–95, 1986.
L. Pitt and C. Smith. Probability and plurality for aggregations of learning machines. Information and Computation, vol. 77, pp. 77–92, 1988.
H. Rogers, Jr. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.
C. Smith. The power of pluralism for automatic program synthesis. Journal of the ACM, vol. 29, pp. 1144–1165, 1982.
J. Smotrovs. Closedness properties in EX-identification of recursive functions. Submitted to the ECML-97 conference, Prague.
R. Smullyan. Theory of Formal Systems. Annals of Mathematical Studies, vol. 47, Princeton, 1961.
M. Velauthapillai. Inductive inference with a bounded number of mind changes. In R. Rivest, D. Haussler, and M. Warmuth, editors, Proceedings of the 1989 Workshop on Computational Learning Theory, pp. 200–213, Morgan Kaufmann, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Smotrovs, J. (1997). Closedness properties in team learning of recursive functions. In: Ben-David, S. (eds) Computational Learning Theory. EuroCOLT 1997. Lecture Notes in Computer Science, vol 1208. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62685-9_8
Download citation
DOI: https://doi.org/10.1007/3-540-62685-9_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62685-5
Online ISBN: 978-3-540-68431-2
eBook Packages: Springer Book Archive