Abstract
In this paper we consider uncountable classes recognizable by ω-automata and investigate suitable learning paradigms for them. In particular, the counterparts of explanatory, vacillatory and behaviourally correct learning are introduced for this setting. Here the learner reads in parallel the data of a text for a language L from the class plus an ω-index α and outputs a sequence of ω-automata such that all but finitely many of these ω-automata accept the index α iff α is an index for L.
It is shown that any class is behaviourally correct learnable if and only if it satisfies Angluin’s tell-tale condition. For explanatory learning, such a result needs that a suitable indexing of the class is chosen. On the one hand, every class satisfying Angluin’s tell-tale condition is vacillatory learnable in every indexing; on the other hand, there is a fixed class such that the level of the class in the hierarchy of vacillatory learning depends on the indexing of the class chosen.
We also consider a notion of blind learning. On the one hand, a class is blind explanatory (vacillatory) learnable if and only if it satisfies Angluin’s tell-tale condition and is countable; on the other hand, for behaviourally correct learning there is no difference between the blind and non-blind version.
This work establishes a bridge between automata theory and inductive inference (learning theory).
The first and fourth author are supported in part by NUS grant R252-000-308-112; the third and fourth author are supported by NUS grant R146-000-114-112.
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
Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45(2), 117–135 (1980)
Bárány, V., Kaiser, Ł., Rubin, S.: Cardinality and counting quantifiers on omega-automatic structures. In: Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science, STACS 2008, pp. 385–396 (2008)
Bārzdiņš, J.: Two theorems on the limiting synthesis of functions. Theory of Algorithms and Programs 1, 82–88 (1974)
Blumensath, A., Grädel, E.: Automatic structures. In: 15th Annual IEEE Symposium on Logic in Computer Science, Santa Barbara, CA, pp. 51–62. IEEE Computer Society Press, Los Alamitos (2000)
Blumensath, A., Grädel, E.: Finite presentations of infinite structures: automata and interpretations. Theory of Computing Systems 37(6), 641–674 (2004)
Richard Büchi, J.: Weak second-order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 6, 66–92 (1960)
Richard Büchi, J.: On a decision method in restricted second order arithmetic. In: Logic, Methodology and Philosophy of Science (Proceedings 1960 International Congress), pp. 1–11. Stanford University Press, Stanford (1962)
Case, J.: The power of vacillation in language learning. SIAM Journal on Computing 28(6), 1941–1969 (1999) (electronic)
Mark Gold, E.: Language identification in the limit. Information and Control 10, 447–474 (1967)
Jain, S., Luo, Q., Stephan, F.: Learnability of automatic classes. Technical Report TRA1/09, School of Computing, National University of Singapore (2009)
Khoussainov, B., Nerode, A.: Automata theory and its applications. Birkhäuser Boston, Inc., Boston (2001)
Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)
Osherson, D.N., Stob, M., Weinstein, S.: Systems that learn. An introduction to learning theory for cognitive and computer scientists. Bradford Book—MIT Press, Cambridge (1986)
Vardi, M.Y.: The Büchi complementation saga. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 12–22. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jain, S., Luo, Q., Semukhin, P., Stephan, F. (2009). Uncountable Automatic Classes and Learning. In: Gavaldà, R., Lugosi, G., Zeugmann, T., Zilles, S. (eds) Algorithmic Learning Theory. ALT 2009. Lecture Notes in Computer Science(), vol 5809. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04414-4_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-04414-4_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04413-7
Online ISBN: 978-3-642-04414-4
eBook Packages: Computer ScienceComputer Science (R0)