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Learning Algebraic Structures from Text Using Semantical Knowledge

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Algorithmic Learning Theory (ALT 1998)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1501))

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Abstract

The present work investigates to which extent semantical knowledge can support the learning of basic mathematical concepts. The considered learning criteria are learning characteristic or enumerable indices for languages from positive data where the learner has to converge either syntactically (Ex) or semantically (BC). The considered classes are the classes of all monoids of a given group, all ideals of a given ring or all subspaces of a given vector space. The following is shown:

  1. (a)

    Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 in fields) and certain parameters (like the dimension of finite dimensional vector spaces). For several natural examples good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all.

  2. (b)

    A recursive commutative ring is Noetherian iff the class of its ideals is BC-learnable. Such a BC-learner can be synthesized from programs for addition and multiplication. In many Noetherian rings, one can Ex-learn characteristic indices for the ideals with an ordinal bound on the number of mind changes. But there are also some Noetherian rings where it is impossible to Ex-learn the ideals or to learn characteristic indices for them.

A more comprehensive version of this paper is available as Forschungsbericht Mathematische Logik 34 of the Mathematisches Insitut at the UniversitÄt Heidelberg (same authors and title). This paper was written when Frank Stephan was visiting Arun Sharma at the School of Computer Science and Engineering, UNSW. The first author is supported by the Deutsche Forschungsgemeinschaft (DFG) grant Am 60/9-2. The second author is supported by the Australian Research Council grant A49600456 to Arun Sharma. The latter grant also supported the first author’s visit to UNSW.

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Author information

Authors and Affiliations

  1. Mathematisches Institut, UniversitÄt Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Frank Stephan

  2. School of Computer Science and Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia

    Yuri Ventsov

Authors
  1. Frank Stephan
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  2. Yuri Ventsov
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Editor information

Editors and Affiliations

  1. AG Künstliche Intelligenz - Expertensysteme, UniversitÄt Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Michael M. Richter

  2. Department of Computer Science, University of Maryland, College Park, MD, 20742, USA

    Carl H. Smith

  3. AG Algorithmischesn Lernen, UniversitÄt Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Rolf Wiehagen

  4. Graduate School of Information Science and Electrical Engineering Department of Informatics, Kyushu University, Kassuga, 816-8580, Japan

    Thomas Zeugmann

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© 1998 Springer-Verlag Berlin Heidelberg

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Stephan, F., Ventsov, Y. (1998). Learning Algebraic Structures from Text Using Semantical Knowledge. In: Richter, M.M., Smith, C.H., Wiehagen, R., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 1998. Lecture Notes in Computer Science(), vol 1501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49730-7_24

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  • DOI: https://doi.org/10.1007/3-540-49730-7_24

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65013-3

  • Online ISBN: 978-3-540-49730-1

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