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The complexity of learning SUBSEQ(A)

Published online by Cambridge University Press:  12 March 2014

Stephen Fenner ,
William Gasarch  and
Brian Postow
Stephen Fenner
Affiliation:
Department of Computer Science and Engineering, University of South Carolina, Columbia, Sc 29208, USA, E-mail: fenner@cse.sc.edu
William Gasarch
Affiliation:
Department of Computer Science and Institute for Advanced Computer Studies, University of MarylandAt College Park, College Park, Md 20742, USA, E-mail: gasarch@cs.umd.edu
Brian Postow
Affiliation:
Acordex Imaging Systems, 37 Walker Road, North Andover, Ma 01845, USA, E-mail: postow@acm.org
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Abstract

Higman essentially showed that if A is any language then SUBSEQ(A) is regular, where SUBSEQ(A) is the language of all subsequences of strings in A. Let s1,s2,s3,… be the standard lexico-graphic enumeration of all strings over some finite alphabet. We consider the following inductive inference problem: A(s1),A(s2),A(s3),…, learn, in the limit, a DFA for SUBSEQ(A). We consider this model of learning and the variants of it that are usually studied in Inductive Inference: anomalies, mind-changes, teams, and combinations thereof.

This paper is a significant revision and expansion of an earlier conference version [10].

Keywords

machine learninginductive inferenceautomatacomputabilitysubsequence
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 939 - 975
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Ambainis, A., Jain, S., and Sharma, A., Ordinal mind change complexity of language identification, Theoretical Computer Science, vol. 220 (1999), no. 2, pp. 323–343.CrossRefGoogle Scholar
[2]Angluin, D., Inductive inference of formal languages from positive data, Information and Control, vol. 45 (1980), no. 2, pp. 117–135.CrossRefGoogle Scholar
[3]Baliga, G. and Case, J., Learning with higher order additional information, Proceedings of the 5th international workshop on Algorithmic Learning Theory, 1994, pp. 64–75.CrossRefGoogle Scholar
[4]Barzdin, J. M., TWO theorems on the limiting synthesis of functions, Theory of algorithms and programs (Barzdin, J., editor), Latvian State University, Riga, U.S.S.R., 1974, pp. 82–88.Google Scholar
[5]Blum, L. and a, Towards a mathematical theory of inductive inference, Information and Control, vol. 28 (1975), pp. 125–155.CrossRefGoogle Scholar
[6]Case, J., Jain, S., and Manguelle, S. N., Refinements of inductive inference by Popperian and reliable machines, Kybernetika, vol. 30–1 (1994), pp. 23–52.Google Scholar
[7]Case, J. and Smith, C. H., Comparison of identification criteria for machine inductive inference, Theoretical Computer Science, vol. 25 (1983), pp. 193–220.CrossRefGoogle Scholar
[8]Daley, R., Kalyanasundaram, B., and Velauthapillai, M., Breaking the probability 1/2 barrier in FIN-type learning, Journal of Computer and System Sciences, vol. 50 (1995), pp. 574–599.CrossRefGoogle Scholar
[9]Ershov, Y. L., Goncharov, S. S., Nerode, A., and Remmel, J. B. (editors), Handbook of recursive mathematics, Elsevier North-Holland, New York, 1998, Volume 1: Recursive Model Theory; Volume 2: Recursive Algebra, Analysis, and Combinatorics.Google Scholar
[10]Fenner, S. and Gasarch, W., The complexity of learning SUBSEQ(A), Proceedings of the 17th international conference on algorithmic learning theory, Lecture Notes in Artificial Intelligence, vol. 4264, Springer-Verlag, 2006, pp. 109–123.CrossRefGoogle Scholar
[11]Fenner, S., Gasarch, W., and Postow, B., The complexity of finding SUBSEQ(A), Theory of Computing Systems, (2008), to appear.Google Scholar
[12]Fortnow, L., Jain, S., Gasarch, W., Kinber, E., Kummer, M., Kurtz, S., Pleszkoch, M., Slaman, T., Stephan, F., and Solovay, R., Extremes in the degrees of inferability, Annals of Pure and Applied Logic, vol. 66 (1994), no. 3, pp. 231–276.CrossRefGoogle Scholar
[13]Freivalds, R. and Smith, C. H., On the role of procrastination for machine learning, Information and Computation, vol. 107 (1993), no. 2, pp. 237–271.CrossRefGoogle Scholar
[14]Freivalds, R., Smith, C. H., and Velauthapillai, M., Trade-off among parameters affecting inductive inference, Information and Computation, vol. 82 (1989), no. 3, pp. 323–349.CrossRefGoogle Scholar
[15]Gasarch, W., Kinber, E., Pleszkoch, M., Smith, C. H., and Zeugmann, T., Learning via queries, teams, and anomalies, Fundamenta Informaticae, vol. 23 (1995), pp. 67–89, prior version in Computational Learning Theory (COLT), 1990.CrossRefGoogle Scholar
[16]Gasarch, W. and Lee, A., Inferring answers to queries, Proceedings of the 10th annual ACM conference on Computational Learning Theory, 1997, journal version to appear in Journal of Computer and System Sciences in 2008, pp. 275–284.CrossRefGoogle Scholar
[17]Gasarch, W., Pleszkoch, M., and Solovay, R., Learning via queries to [+, <], this Journal, vol. 57 (1992), no. 1, pp. 53–81.Google Scholar
[18]Gasarch, W., Pleszkoch, M., Stephan, F., and Velauthapillai, M., Classification using information, Annals of Mathematics and Artificial Intelligence, vol. 23 (1998), pp. 147–168, earlier version in Proceedings of the 5th international workshop on algorithmic learning theory, 1994, pp. 290-300.CrossRefGoogle Scholar
[19]Gasarch, W. and Smith, C. H., Learning via queries, Journal of the ACM, vol. 39 (1992), no. 3, pp. 649–675, prior version in Proceedings of the 29th IEEE Symposium on Foundations of Computer Science (FOCS), 1988.CrossRefGoogle Scholar
[20]Gold, E. M., Language identification in the limit, Information and Control, vol. 10 (1967), no. 10, pp. 447–474.CrossRefGoogle Scholar
[21]Hartmanis, J., Context-free languages and Turing machine computations. Mathematical aspects of computer science, Proceedings of symposia in Applied Mathematics (Schwartz, J. T., editor), vol. 19, American Mathematical Society, Providence, RI, 1967, pp. 42–51.Google Scholar
[22]Higman, A. G., Ordering by divisibility in abstract algebras, Proceedings of the London Mathematical Society, no. 1, 1952, s3–2, pp. 326–336.Google Scholar
[23]Jain, S., Personal Communication, 2008.Google Scholar
[24]Jain, S. and Sharma, A., Computational limits on team identification of languages, Information and Computation, vol. 130 (1996), no. 1, pp. 19–60.CrossRefGoogle Scholar
[25]Jain, S., Sharma, A., and Velauthapillai, M., Finite identification of functions by teams with success ratio 1/2 and above, Information and Computation, vol. 121 (1995), no. 2, pp. 201–213.CrossRefGoogle Scholar
[26]Kummer, M. and Stephan, F., On the structure of the degrees of inferability. Journal of Computer and System Sciences, vol. 52 (1996), no. 2, pp. 214–238, prior version in Sixth annual conference on Computational Learning Theory (COLT), 1993.CrossRefGoogle Scholar
[27]Metakides, G. and Nerode, A., Effective content offield theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.CrossRefGoogle Scholar
[28]Pitt, L., Probabilistic inductive inference, Journal of the ACM, vol. 36 (1989), no. 2, pp. 383–433.CrossRefGoogle Scholar
[29]Pitt, L. and Smith, C. H., Probability and plurality for aggregations of learning machines, Information and Computation, vol. 77 (1988), pp. 77–92.CrossRefGoogle Scholar
[30]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, 1967, reprinted by MIT Press, 1987.Google Scholar
[31]Sacks, G. E., Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.CrossRefGoogle Scholar
[32]Smith, C. H., The power ofpluralism for automatic program synthesis, Journal of the ACM, vol. 29 (1982), pp. 1144–1165, prior version in Proceedings of the 22nd IEEE symposium on Foundations of Computer Science (FOCS), 1981.CrossRefGoogle Scholar
[33]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[34]Van Leeuwen, J., Effective constructions in well-partially-ordered free monoids, Discrete Mathematics, vol. 21 (1978), pp. 237–252.CrossRefGoogle Scholar