In language learning in the limit, the most common type of hypothesis is to give an enumerator for a language. This so-called $W$-index allows for naming arbitrary computably enumerable languages, with the drawback that even the membership problem is undecidable. In this paper we use a different system which allows for naming arbitrary decidable languages, namely programs for characteristic functions (called $C$-indices). These indices have the drawback that it is now not decidable whether a given hypothesis is even a legal $C$-index. In this first analysis of learning with $C$-indices, we give a structured account of the learning power of various restrictions employing $C$-indices, also when compared with $W$-indices. We establish a hierarchy of learning power depending on whether $C$-indices are required (a) on all outputs; (b) only on outputs relevant for the class to be learned and (c) only in the limit as final, correct hypotheses. Furthermore, all these settings are weaker than learning with $W$-indices (even when restricted to classes of computable languages). We analyze all these questions also in relation to the mode of data presentation. Finally, we also ask about the relation of semantic versus syntactic convergence and derive the map of pairwise relations for these two kinds of convergence coupled with various forms of data presentation.

中文翻译：

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用可判定的假设学习语言

在极限语言学习中，最常见的假设类型是给出一种语言的枚举器。这个所谓的 $W$-index 允许命名任意可计算可枚举的语言，其缺点是即使成员问题也是不可判定的。在本文中，我们使用了一个不同的系统，它允许命名任意可判定语言，即特征函数程序（称为 $C$-indices）。这些指数的缺点是现在无法确定给定的假设是否甚至是合法的 $C$ 指数。在对使用 $C$-indices 进行学习的第一次分析中，我们对使用 $C$-indices 的各种限制的学习能力进行了结构化说明，并与 $W$-indices 进行了比较。我们根据是否需要 $C$-indices 来建立学习能力的层次结构 (a) 所有输出；(b) 仅在与要学习的类相关的输出上，以及 (c) 仅在作为最终正确假设的极限中。此外，所有这些设置都比使用 $W$-indices 学习要弱（即使仅限于可计算语言的类别）。我们还分析了与数据呈现方式相关的所有这些问题。最后，我们还询问语义与句法收敛的关系，并推导出这两种收敛以及各种形式的数据呈现的成对关系图。