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]).
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This is a substantially revised version of the authors' paper inProceedings of the 5th International Workshop on Algorithmic Learning Theory (ALT '94), Lecture Notes in Artificial Intelligence, Vol. 872, pp. 453–468, Springer-Verlag, Berlin. The first author has been partially supported by the German Ministry for Research and Technology (BMFT) under Grant No. 01 IW 101.
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Lange, S., Zeugmann, T. Set-driven and rearrangement-independent learning of recursive languages. Math. Systems Theory 29, 599–634 (1996). https://doi.org/10.1007/BF01301967
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DOI: https://doi.org/10.1007/BF01301967