Abstract
An early result in inductive inference shows that the class of Ex-learnable sets is not closed under unions. In this paper we are interested in the following question: For what classes of functions is the union with an arbitrary Ex-learnable class again Ex-learnable, either effectively (in an index for a learner of an Ex-learnable class) or non-effectively? We show that the effective case and the non-effective case separate, and we give a sufficient criterion for the effective case. Furthermore, we extend our notions to considering unions with classes of single functions, as well as to other learning criteria, such as finite learning and behaviorally correct learning.
Furthermore, we consider the possibility of (effectively) extending learners to learn (infinitely) more functions. It is known that all Ex-learners learning a dense set of functions can be effectively extended to learn infinitely more. It was open whether the learners learning a non-dense set of functions can be similarly extended. We show that this is not possible, but we give an alternative split of all possible learners into two sets such that, for each of the sets, all learners from that set can be effectively extended. We analyze similar concepts also for other learning criteria.
This is a preview of subscription content, log in via an institution to check access.
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
Bārzdiņš, J.A.: Two theorems on the limiting synthesis of functions. Theory of Algorithms and Programs, Latvian State University, Riga, USSR 210, 82–88 (1974)
Blum, L., Blum, M.: Toward a mathematical theory of inductive inference. Information and Control 28, 125–155 (1975)
Case, J.: Periodicity in generations of automata. Mathematical Systems Theory 8, 15–32 (1974)
Case, J., Fulk, M.: Maximal machine learnable classes. Journal of Computer and System Sciences 58, 211–214 (1999)
Case, J., Jain, S., Manguelle, S.N.: Refinements of inductive inference by Popperian and reliable machines. Kybernetika 30, 23–52 (1994)
Case, J., Smith, C.: Comparison of identification criteria for machine inductive inference. Theoretical Computer Science 25, 193–220 (1983)
Gold, M.: Language identification in the limit. Information and Control 10, 447–474 (1967)
Kummer, M., Stephan, F.: On the structure of degrees of inferability. Journal of Computer and System Sciences 52, 214–238 (1996)
Minicozzi, E.: Some natural properties of strong-identification in inductive inference. Theoretical Computer Science 2, 345–360 (1976)
Osherson, D., Stob, M., Weinstein, S.: Systems that Learn: An Introduction to Learning Theory for Cognitive and Computer Scientists. MIT Press, Cambridge (1986)
Pitt, L.: Inductive Inference, DFAs, and Computational Complexity. In: Jantke, K.P. (ed.) AII 1989. LNCS (LNAI), vol. 397, pp. 18–44. Springer, Heidelberg (1989)
Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw Hill, New York (1967); (reprinted in 1987)
Stephan, F.: On one-sided versus two-sided classification. Archive for Mathematical Logic 40, 489–513 (2001)
Sharma, A., Stephan, F., Ventsov, Y.: Generalized notions of mind change complexity. Information and Computation 189, 235–262 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jain, S., Kötzing, T., Stephan, F. (2012). Enlarging Learnable Classes. In: Bshouty, N.H., Stoltz, G., Vayatis, N., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2012. Lecture Notes in Computer Science(), vol 7568. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34106-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-34106-9_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34105-2
Online ISBN: 978-3-642-34106-9
eBook Packages: Computer ScienceComputer Science (R0)