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Learning How to Separate

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Book cover Algorithmic Learning Theory (ALT 2001)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2225))

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Abstract

The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated in which cases it is possible to satisfy the following additional constraints: confidence where the learner converges on all data-sequences; conservativeness where the learner abandons only definitely wrong hypotheses; consistency where also every intermediate hypothesis is consistent with the data seen so far; set-driven learners whose hypotheses are independent of the order and the number of repetitions of the data-items supplied; learners where either the last or even all hypotheses are programs of total recursive functions.

The present work gives an overview of the relations between these notions and succeeds to answer many questions by finding ways to carry over the corresponding results from other scenarios within inductive inference. Nevertheless, the relations between conservativeness andsetdriven inference needed a novel approach which enabled to show the following two major results: (1) There is a class for which recursive separators can be foundin a confident and set-driven way, but no conservative learner finds a (not necessarily total) separator for this class. (2) There is a class for which recursive separators can be foundin a confident and conservative way, but no set-driven learner finds a (not necessarily total) separator for this class.

Supported in part by NUS grant number RP3992710.

Supported by the Deutsche Forschungsgemeinschaft (DFG) under the Heisenberg grant Ste 967/1-1

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References

  1. A. Ambainis, S. Jain, and A. Sharma. Ordinal mind change complexity of language identification. Theoretical Computer Science, 220(2):323–343, 1999.

    Article  MathSciNet  Google Scholar 

  2. D. Angluin and C. Smith. Inductive inference: Theory and methods. Computing Surveys, 15:237–289, 1983.

    Article  MathSciNet  Google Scholar 

  3. J. Bārzdiņš. Two theorems on the limiting synthesis of functions. In Theory of Algorithms and Programs, vol. 1, pages 82–88. Latvian State University, 1974. In Russian.

    Google Scholar 

  4. J. Bārzdiņš and R. Freivalds. On the prediction of general recursive functions. Soviet Mathematics Doklady, 13:1224–1228, 1972.

    Google Scholar 

  5. L. Blum and M. Blum. Towarda mathematical theory of inductive inference. Information and Control, 28:125–155, 1975.

    Article  MathSciNet  Google Scholar 

  6. M. Blum. A machine-independent theory of the complexity of recursive functions. Journal of the ACM, 14:322–336, 1967.

    Article  MathSciNet  Google Scholar 

  7. J. Case, S. Jain, and S. Ngo Manguelle. Refinements of inductive inference by Popperian andreliable machines. Kybernetika, 30:23–52, 1994.

    MathSciNet  MATH  Google Scholar 

  8. J. Case and C. Lynes. Machine inductive inference andlanguage identification. In M. Nielsen and E. M. Schmidt, editors, Proceedings of the 9th International Colloquium on Automata, Languages and Programming, volume 140 of Lecture Notes in Computer Science, pages 107–115. Springer-Verlag, 1982.

    MATH  Google Scholar 

  9. J. Case and C. Smith. Comparison of identification criteria for machine inductive inference. Theoretical Computer Science, 25:193–220, 1983.

    Article  MathSciNet  Google Scholar 

  10. M. Fulk. Prudence and other conditions on formal language learning. Information and Computation, 85:1–11, 1990.

    Article  MathSciNet  Google Scholar 

  11. E. M. Gold. Language identification in the limit. Information and Control, 10:447–474, 1967.

    Article  MathSciNet  Google Scholar 

  12. J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979.

    Google Scholar 

  13. S. Jain, D. Osherson, J. Royer, and A. Sharma. Systems that Learn: An Introduction to Learning Theory. MIT Press, Cambridge, Mass., second edition, 1999.

    Google Scholar 

  14. S. Jain and F. Stephan. Learning how to separate. Technical Report Forschungsberichte Mathematische Logik 51/2001, Mathematical Institute, University of Heidelberg, 2001.

    Google Scholar 

  15. C. Jockusch. Degrees in which recursive sets are uniformly recursive. Candadian Journal of Mathematics, 24:1092–1099, 1972.

    Article  MathSciNet  Google Scholar 

  16. E. Kinber and F. Stephan. Language learning from texts: Mindc hanges, limited memory and monotonicity. Information and Computation, 123:224–241, 1995.

    Article  MathSciNet  Google Scholar 

  17. R. Klette and R. Wiehagen. Research in the theory of inductive inference by GDR mathematicians-A survey. Information Sciences, 22:149–169, 1980.

    Article  MathSciNet  Google Scholar 

  18. M. Machtey and P. Young. An Introduction to the General Theory of Algorithms. North Holland, New York, 1978.

    Google Scholar 

  19. E. Minicozzi. Some natural properties of strong identification in inductive inference. Theoretical Computer Science, 2:345–360, 1976.

    Article  MathSciNet  Google Scholar 

  20. P. Odifreddi. Classical Recursion Theory. North-Holland, Amsterdam, 1989.

    Google Scholar 

  21. D. Osherson, M. Stob, and S. Weinstein. Systems that Learn: An Introduction to Learning Theory for Cognitive and Computer Scientists. MIT Press, 1986.

    Google Scholar 

  22. H. Rogers. Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967. Reprintedb y MIT Press in 1987.

    Google Scholar 

  23. G. Schäfer-Richter. Über Eingabeabhangigkeit und Komplexität von Inferenzstrategien. PhD thesis, RWTH Aachen, 1984.

    Google Scholar 

  24. A. Sharma, F. Stephan, and Y. Ventsov. Generalizednotions of mindc hange complexity. In Proceedings of the Tenth Annual Conference on Computational Learning Theory, pages 96–108. ACM Press, 1997.

    Google Scholar 

  25. F. Stephan and T. Zeugmann. On the uniform learnability of approximations to non-recursive functions. In O. Watanabe and T. Yokomori, editors, Algorithmic Learning Theory: Tenth International Conference (ALT’ 99), volume 1720 of Lecture Notes in Artificial Intelligence, pages 276–290. Springer-Verlag, 1999.

    Google Scholar 

  26. K. Wexler and P. Culicover. Formal Principles of Language Acquisition. MIT Press, 1980.

    Google Scholar 

  27. R. Wiehagen and W. Liepe. Charakteristische Eigenschaften von erkennbaren Klassen rekursiver Funktionen. Journal of Information Processing and Cybernetics (EIK), 12:421–438, 1976.

    MathSciNet  MATH  Google Scholar 

  28. T. Zeugmann. On Bārzdiņš’ conjecture. In K. P. Jantke, editor, Analogical and Inductive Inference, Proceedings of the International Workshop, volume 265 of Lecture Notes in Computer Science, pages 220–227. Springer-Verlag, 1986.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Computing, National University of Singapore, 119260, Singapore

    Sanjay Jain

  2. Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Frank Stephan

Authors
  1. Sanjay Jain
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  2. Frank Stephan
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Editor information

Editors and Affiliations

  1. IBM, Thomas J.Watson Research Center, Room 33-237, P.O.Box 218, 10598, Yorktown, NY, USA

    Naoki Abe

  2. Dept.of Electrical Engineering and Computer Science, Tufts University, 161 College Ave., 02155, Medford, MA, USA

    Roni Khardon

  3. Medizinische Universität zu Lübeck,Inst.für Theoretische Informatik, Wallstr.40, 23560, Lübeck, Germany

    Thomas Zeugmann

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© 2001 Springer-Verlag Berlin Heidelberg

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Jain, S., Stephan, F. (2001). Learning How to Separate. In: Abe, N., Khardon, R., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2001. Lecture Notes in Computer Science(), vol 2225. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45583-3_18

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  • DOI: https://doi.org/10.1007/3-540-45583-3_18

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42875-6

  • Online ISBN: 978-3-540-45583-7

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