# Trading monotonicity demands versus mind changes

## Abstract

The present paper deals with with the learnability of indexed families \({\cal L}\) of uniformly recursive languages *from positive data*. We consider the influence of three monotonicity demands to the efficiency of the learning process. The efficiency of learning is measured in dependence on the number of mind changes a learning algorithm is allowed to perform. The three notions of monotonicity reflect different formalizations of the requirement that the learner has to produce better and better generalizations when fed more and more data on the target concept.

We distinguish between *exact* learnability (\({\cal L}\) has to be inferred with respect to \({\cal L}\)), *class preserving* learning (\({\cal L}\) has to be inferred with respect to some suitable chosen enumeration of all the languages from \({\cal L}\)), and *class comprising* inference (\({\cal L}\) has to be learned with respect to some suitable chosen enumeration of uniformly recursive languages containing at least all the languages from \({\cal L}\)).

In particular, we prove that a relaxation of the relevant monotonicity requirement may result in an arbitrarily large speed-up.

## Keywords

Initial Segment Inductive Inference Hypothesis Space Monotonicity Constraint Computational Learn Theory## Preview

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## References

- Angluin, D. (1980), Inductive inference of formal languages from positive data,
*Information and Control*,**45**(1980), 117–135.Google Scholar - Barzdin, Ya.M., and Freivalds, R.V. (1972), On the prediction of general recursive functions,
*Sov. Math. Dokl*.**13**, 1224–1228.Google Scholar - Barzdin, Ya.M., Kinber, E.B., and Podnieks, K.M. (1974), Об ускорении синтеза и прогнозирования φункций,
*in*“Теория Алгоритмов и Программ,” Vol. 1 (Ya.M. Barzdin, Ed.) Latvian State University, Riga, pp. 117–128.Google Scholar - Case, J., and Smith, C.H. (1983), Comparison of identification criteria for machine inductive inference,
*Theoretical Computer Science***25**, 193–220.Google Scholar - Gasarch, W.I., and Velauthapillai, M. (1992), Asking questions versus verifiability,
*in*“Proceedings 3rd International Workshop on Analogical and Inductive Inference,” (K.P. Jantke, ed.) Lecture Notes in Artificial Intelligence Vol. 642, pp. 197–213, Springer-Verlag, Berlin.Google Scholar - Gold, M.E. (1967), Language identification in the limit,
*Information and Control***10**, 447–474.Google Scholar - Hopcroft, J.E., and Ullman, J.D. (1969), “Formal Languages and their Relation to Automata,” Addison-Wesley, Reading, Massachusetts.Google Scholar
- Jantke, K.P. (1991), Monotonic and non-monotonic inductive inference,
*New Generation Computing***8**, 349–360.Google Scholar - Kapur, S., and Bilardi, G. (1992), Language learning without overgeneralization,
*in*“Proceedings 9th Annual Symposium on Theoretical Aspects of Computer Science,” (A. Finkel and M. Jantzen, Eds.), Lecture Notes in Computer Science Vol. 577, pp. 245–256, Springer-Verlag, Berlin.Google Scholar - Kinber, E.B. (1994), Monotonicity versus efficiency for learning languages from texts,
*in*“Proceedings 5th Workshop on Algorithmic Learning Theory,” (S. Arikawa and K.P. Jantke, Eds.), Lecture Notes in Artificial Intelligence Vol. 872, pp. 395–406.Google Scholar - Lange, S. (1994), The representation of recursive languages and its impact on the efficiency of learning,
*in*“Proceedings 7th Annual ACM Conference on Computational Learning Theory, New Brunswick, July 1994,” pp. 256–267, ACM Press, New York.Google Scholar - Lange, S., and Zeugmann, T. (1993a), Monotonic versus non-monotonic language learning,
*in*“Proceedings 2nd International Workshop on Non-monotonic and Inductive Logic, December 1991, Reinhardsbrunn,” (G. Brewka, K.P. Jantke and P.H. Schmitt, Eds.), Lecture Notes in Artificial Intelligence Vol. 659, pp. 254–269, Springer-Verlag, Berlin.Google Scholar - Lange, S., and Zeugmann, T. (1993b), Learning recursive languages with bounded mind changes,
*International Journal of Foundations of Computer Science***4**, 157–178.Google Scholar - Lange, S., and Zeugmann, T. (1993c), Language learning in dependence on the space of hypotheses,
*in*“Proceedings 6th Annual ACM Conference on Computational Learning Theory,” pp. 127–136, ACM Press, New York.Google Scholar - Lange, S., and Zeugmann, T. (1994), Trading monotonicity demands versus efficiency,
*Bulletin of Informatics and Cybernetics*, to appear.Google Scholar - Lange, S., Zeugmann, T., and Kapur, S. (1994), Monotonic and dual-monotonic language learning,
*Theoretical Computer Science*, to appear.Google Scholar - Machtey, M., and Young, P. (1978), “An Introduction to the General Theory of Algorithms,” North-Holland, New York.Google Scholar
- Mukouchi, Y. (1992), Inductive inference with bounded mind changes,
*in*“Proceedings 3rd Workshop on Algorithmic Learning Theory,” Tokyo, Japan, JSAI, pp. 125–134.Google Scholar - Mukouchi, Y. (1994), Inductive inference of recursive concepts, Ph.D. Thesis, R1FIS, Kyushu University 33, RIFIS-TR-CS-82, March 25th.Google Scholar
- Osherson, D., Stob, M., and Weinstein, S. (1986), “Systems that Learn, An Introduction to Learning Theory for Cognitive and Computer Scientists,” MIT-Press, Cambridge, Massachusetts.Google Scholar
- Shinohara, T. (1990), Inductive Inference from Positive Data is Powerful,
*in*“Proc. 3rd Annual Workshop on Computational Learning Theory,” pp. 97–110, Morgan Kaufmann Publishers Inc.Google Scholar - Wiehagen, R. (1991), A thesis in inductive inference,
*in*“Proceedings First International Workshop on Nonmonotonic and Inductive Logic,” (J. Dix, K.P. Jantke and P.H. Schmitt, Eds.), Lecture Notes in Artificial Intelligence 543, pp. 184–207, Springer-Verlag, Berlin.Google Scholar - Wiehagen, R., and Zeugmann, T. (1994), Ignoring data may be the only way to learn efficiently,
*Journal Experimental and Theoretical Artificial Intelligence***6**, 131–144.Google Scholar