Abstract
In this paper we investigate inductive inference of recursive real-valued functions from data. A recursive real-valued function is regarded as a computable interval mapping, which has been introduced by Hirowatari and Arikawa (1997), and modified by ApsĪtis et al (1998). The learning model we consider in this paper is an extension of the Gold’s inductive inference. We first introduce some criteria for successful inductive inference of recursive real-valued functions. Then we show a recursively enumerable class of recursive real-valued functions which is not inferable in the limit. This should be an interesting contrast to the result by Wiehagen (1976) that every recursively enumerable subset of recursive functions from N to N is consistently inferable in the limit. We also show that every recursively enumerable class of recursive real-valued functions on a fixed rational interval is consistently inferable in the limit. Furthermore we show that our consistent inductive inference coincides with the ordinary inductive inference, when we deal with recursive real-valued functions on a fixed closed rational interval.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Alefeld and J. Herzberger. Introduction to Interval Computations. Academic Press, London, England, 1983.
K. ApsĪtis, S. Arikawa, R. Freivalds, E. Hirowatari, and C. H. Smith. Inductive inference of real functions. (to appear in TCS).
K. ApsĪtis, R. Freivalds, and C. H. Smith. On the inductive inference of real valued functions. In Proceedings of the Eighth Annual Conference on Computational Learning Theory, pp. 170–177, 1995.
L. Blum and M. Blum. Toward a mathematical theory of inductive inference. Information and Control, Vol. 28, pp. 125–155, 1975.
E.M. Gold. Language identification in the limit. Information and Control, Vol. 10, pp. 447–474, 1967.
A. Grzegorczyk. Computable functionals. Fundamenta Mathematicae, Vol. 42, pp. 168–202, 1955.
A. Grzegorczyk. On the definitions of computable real continuous functions. Fundamenta Mathematicae, Vol. 44, pp. 61–71, 1957.
D. Haussler. Decision theoretic generalizations of the PAC model for neural net and other learning applications. Information and Computation, Vol. 100, pp. 78–150, 1992.
E. Hirowatari and S. Arikawa. Inferability of recursive real-valued functions. In Proceedings of International Workshop on Algorithmic Learning Theory, (Lecture Notes in Artificial Intelligence 1316 Springer-Verlag), pp. 18–31. Springer-Verlag, 1997.
K. Ko. Complexity Theory of Real Functions. BirkhÄuser, 1991.
R.E. Moore. Interval Analysis. Prentice-Hall, 1966.
A. Mostowski. On computable sequences. Fundamenta Mathematicae, Vol. 44, pp. 37–51, 1957.
M. B. Pour-El and J.I. Richards. Computability in Analysis and Physics. Springer-Verlag, 1988.
R. Wiehagen. Limes-erkennung rekursiver Funktionen durch spezielle Strategien. Elektronische Informationsverarbeitung und Kybernetik, Vol. 12, pp. 93–99, 1976.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hirowatari, E., Arikawa, S. (1998). A Comparison of Identification Criteria for Inductive Inference of Recursive Real-Valued Functions. In: Richter, M.M., Smith, C.H., Wiehagen, R., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 1998. Lecture Notes in Computer Science(), vol 1501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49730-7_20
Download citation
DOI: https://doi.org/10.1007/3-540-49730-7_20
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65013-3
Online ISBN: 978-3-540-49730-1
eBook Packages: Springer Book Archive