Insertion systems or insertion grammars are a generative formalism in which words can only be generated by starting with some axioms and by iteratively inserting strings subject to certain contexts of a fixed maximal length. It is known that languages generated by such systems are always context sensitive and that the corresponding language classes are incomparable with the regular languages. On the other hand, it is possible to generate non-semilinear languages with systems having contexts of length two. Here, we study decidability questions for insertion systems. On the one hand, it can be seen that emptiness and universality are decidable. Moreover, the fixed membership problem is solvable in deterministic polynomial time. On the other hand, the usually studied decidability questions such as, for example, finiteness, inclusion, equivalence, regularity, inclusion in a regular language, and inclusion of a regular language turn out to be undecidable. Interestingly, the latter undecidability results can be carried over to other models which are basically able to handle the mechanism of inserting strings depending on contexts. In particular, new undecidability results are obtained for pure grammars, restarting automata, clearing restarting automata, and forgetting automata.

中文翻译：

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插入系统和相关模型的可确定性问题

插入系统或插入语法是一种生成形式主义，其中只能通过以某些公理开头并通过迭代插入固定最大长度的特定上下文下的字符串来生成单词。已知由这样的系统生成的语言总是上下文相关的，并且相应的语言类别与常规语言是无法比拟的。另一方面，有可能使用具有长度为2的上下文的系统来生成非半线性语言。在这里，我们研究插入系统的可判定性问题。一方面，可以看出，空性和普遍性是可以决定的。而且，固定隶属度问题可以在确定的多项式时间内解决。另一方面，通常研究的可判定性问题，例如有限性，包含性，等价性，规则性，包含常规语言以及包含常规语言是不确定的。有趣的是，后者的不确定性结果可以转移到其他模型中，这些模型基本上能够根据上下文处理插入字符串的机制。特别是，对于纯语法，重新启动自动机，清除重新启动的自动机以及忘记自动机，获得了新的不确定性结果。