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Mathematical systems theory

, Volume 29, Issue 6, pp 599–634 | Cite as

Set-driven and rearrangement-independent learning of recursive languages

  • S. Lange
  • T. Zeugmann
Article

Abstract

This paper studies the impact of order independence to the learnability of indexed families\(\mathcal{L}\) of uniformly recursive languages from positive data. In particular, we considerset-driven andrearrangement-independent learners, i.e., learning devices whose output exclusively depends on the range and on the range and length of their input, respectively. The impact of set-drivenness and rearrangement-independence on the behavior of learners to their learning power is studied in dependence on thehypothesis space the learners may use. We distinguish betweenexact learnability (\(\mathcal{L}\) has to be inferred with respect to\(\mathcal{L}\)),class-preserving learning (\(\mathcal{L}\) has to be inferred with respect to some suitably chosen enumeration of all the languages from\(\mathcal{L}\)), andclass-comprising inference (\(\mathcal{L}\) has to be learned with respect to some suitably chosen enumeration of uniformly recursive languages containing at least all the languages from\(\mathcal{L}\)).

Furthermore, we consider the influence of set-drivenness and rearrangement-independence for learning devices that realize thesubset principle to different extents. Thereby we distinguish betweenstrong-monotonic, monotonic, andWeakmonotonic orconservative learning.

The results obtained are threefold. First, rearrangement-independent learning does not constitute a restriction except in the case of monotonic learning. Next, we prove that for all but two of the learning models considered set-drivenness is a severe restriction. However, class-comprising set-drivenconservative learning is exactly as powerful as unrestricted class-comprisingconservative learning. Finally, the power of class-comprising set-driven learning in the limit is characterized by equating the collection of learnable indexed families with the collection of class-comprisingly conservatively inferable indexed families. These results considerably extend previous work done in the field (see, e.g., [20] and [5]).

Keywords

Target Language Hypothesis Space Monotonicity Constraint Correct Hypothesis Canonical Text 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • S. Lange
    • 1
  • T. Zeugmann
    • 2
  1. 1.FB Mathematik und InformatikHTWK LeipzigLeipzigGermany
  2. 2.Department of Informatics, Graduate School of Information Science and EEKyushu UniversityFukuokaJapan

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