Set-driven and rearrangement-independent learning of recursive languages

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]).