Refuting learning revisited

  • Authors:

  • Wolfgang Merkle;Frank Stephan

  • Affiliations:

  • Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany;Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany

  • Venue:
  • Theoretical Computer Science - Selected papers in honour of Setsuo Arikawa
  • Year:
  • 2003

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Abstract

We consider, within the framework of inductive inference, the concept of refuting learning as introduced by Mukouchi and Arikawa, where the learner is not only required to learn all concepts in a given class but also has to explicitly refute concepts outside the class.In the first part of the paper, we consider learning from text and introduce a concept of limit-refuting learning that is intermediate between refuting learning and reliable learning. We give characterizations for these concepts and show some results about their relative strength and their relation to confident learning.In the second part of the paper we consider learning from texts that for some k contain all positive Πk-formulae that are valid in the standard structure determined by the set to be learned. In this model, the following results are shown. For the language with successor, any countable axiomatizable class can be limit-refuting learned from Π1-texts. For the language with successor and order, any countable axiomatizable class can be reliably learned from Π1-texts and can be limit-refuting learned from Π2-texts, whereas the axiomatizable class of all finite sets cannot be limit-refuting learned from Π1-texts. For the full language of arithmetic, which contains in addition plus and times, for any even k there is an axiomatizable class that can be limit-refuting learned from Πk+1-texts but not from Πk-texts. A similar result with k + 3 in place of k + 1 holds with respect to the language of Presburger's arithmetic.