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Learning in Friedberg Numberings

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Algorithmic Learning Theory (ALT 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4754))

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Abstract

In this paper we consider learnability in some special numberings, such as Friedberg numberings, which contain all the recursively enumerable languages, but have simpler grammar equivalence problem compared to acceptable numberings. We show that every explanatorily learnable class can be learnt in some Friedberg numbering. However, such a result does not hold for behaviourally correct learning or finite learning. One can also show that some Friedberg numberings are so restrictive that all classes which can be explanatorily learnt in such Friedberg numberings have only finitely many infinite languages. We also study similar questions for several properties of learners such as consistency, conservativeness, prudence, iterativeness and non U-shaped learning. Besides Friedberg numberings, we also consider the above problems for programming systems with K-recursive grammar equivalence problem.

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References

  1. Angluin, D.: Finding patterns common to a set of strings. Journal of Computer and System Sciences 21, 46–62 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45, 117–135 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baliga, G., Case, J., Merkle, W., Stephan, F., Wiehagen, R.: When unlearning helps. Technical Report Technical Report TRA5/06, School of Computing, National University of Singapore (2005)

    Google Scholar 

  4. Bārzdin, J.: Inductive inference of automata, functions and programs. In: International Congress of Mathematicians, Vancouver, pp. 771–776 (1974)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Carlucci, L., Case, J., Jain, S., Stephan, F.: U-shaped learning may be necessary. Journal of Computer and System Sciences (to appear)

    Google Scholar 

  8. Carlucci, L., Jain, S., Kinber, E., Stephan, F.: Variations on U-shaped learning. Information and Computation 204(8), 1264–1294 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Case, J.: The power of vacillation in language learning. SIAM Journal on Computing 28(6), 1941–1969 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Case, J., Lynes, C.: Machine inductive inference and language identification. In: Nielsen, M., Schmidt, E.M. (eds.) Proceedings of the 9th International Colloquium on Automata, Languages and Programming. LNCS, vol. 140, pp. 107–115. Springer, Heidelberg (1982)

    Chapter  Google Scholar 

  11. Friedberg, R.: Three theorems on recursive enumeration. Journal of Symbolic Logic 23(3), 309–316 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wiehagen, R., Kinber, E., Freivalds, R.: Inductive Inference and Computable One-one Numberings. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28, 463–479 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Goncharov, S.: Nonequivalent constructivizations. In: Proceedings of the Mathematical Institute, Siberian Branch of Russian Academy of Sciences, Nauka, Novosibirsk (1982)

    Google Scholar 

  16. Jain, S., Osherson, D., Royer, J., Sharma, A.: Systems that Learn: An Introduction to Learning Theory, 2nd edn. MIT Press, Cambridge (1999)

    Google Scholar 

  17. Jain, S., Sharma, A.: Characterizing language learning in terms of computable numberings. Annals of Pure and Applied Logic 84(1), 51–72 (1997), Special issue on Asian Logic Conference (1993)

    Google Scholar 

  18. Jantke, K.-P.: Monotonic and non-monotonic inductive inference. New Generation Computing 8, 349–360 (1991)

    Article  MATH  Google Scholar 

  19. Jantke, K.-P., Beick, H.-R.: Combining postulates of naturalness in inductive inference. Journal of Information Processing and Cybernetics (EIK) 17, 465–484 (1981)

    MathSciNet  MATH  Google Scholar 

  20. Kummer, M.: Beiträge zur Theorie der Numerierungen: Eindeutige Numerierungen. PhD Thesis, Karlsruhe (1989)

    Google Scholar 

  21. Odifreddi, P.: Classical Recursion Theory, North-Holland, Amsterdam (1989)

    Google Scholar 

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

    Google Scholar 

  23. Osherson, D., Weinstein, S.: Criteria of language learning. Information and Control 52, 123–138 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967), (Reprinted by MIT Press in 1987)

    Google Scholar 

  25. Schäfer-Richter, G.: Some results in the theory of effective program synthesis - learning by defective information. In: Bibel, W., Jantke, K.P. (eds.) Mathematical Methods of Specification and Synthesis of Software Systems 1985. LNCS, vol. 215, pp. 219–225. Springer, Heidelberg (1986)

    Google Scholar 

  26. Wexler, K., Culicover, P.W.: Formal Principles of Language Acquisition. MIT Press, Cambridge (1980)

    Google Scholar 

  27. Wiehagen, R.: Limes-Erkennung rekursiver Funktionen durch spezielle Strategien. Journal of Information Processing and Cybernetics (EIK) 12, 93–99 (1976)

    MathSciNet  MATH  Google Scholar 

  28. Wiehagen, R.: A thesis in inductive inference. In: Dix, J., Schmitt, P.H., Jantke, K.P. (eds.) Nonmonotonic and Inductive Logic, 1st International Workshop. LNCS, vol. 543, pp. 184–207. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Zeugmann, T., Lange, S.: A guided tour across the boundaries of learning recursive languages. In: Lange, S., Jantke, K.P. (eds.) Algorithmic Learning for Knowledge-Based Systems. LNCS, vol. 961, pp. 190–258. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, National University of Singapore, Singapore 117590, Republic of Singapore

    Sanjay Jain

  2. Department of Computer Science and Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore

    Frank Stephan

Authors
  1. Sanjay Jain
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  2. Frank Stephan
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Editor information

Editors and Affiliations

  1. RSISE @ ANU and SML @ NICTA, Canberra,, ACT, 0200, Australia

    Marcus Hutter

  2. Columbia University, NY, P.O. Box, New York, USA

    Rocco A. Servedio

  3. Graduate School of Information Sciences, Tohoku University,, Sendai 980-8579, Japan

    Eiji Takimoto

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© 2007 Springer-Verlag Berlin Heidelberg

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Jain, S., Stephan, F. (2007). Learning in Friedberg Numberings. In: Hutter, M., Servedio, R.A., Takimoto, E. (eds) Algorithmic Learning Theory. ALT 2007. Lecture Notes in Computer Science(), vol 4754. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75225-7_10

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  • DOI: https://doi.org/10.1007/978-3-540-75225-7_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-75224-0

  • Online ISBN: 978-3-540-75225-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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