On decision problems concerning contextual insertions and deletions

The notions of stability, anti-stability, and error-correctability of a language that is modified by making contextual insertions in the words of the language were introduced in a previous paper by Bottoni et al. in 2011, where it was shown that these properties are decidable for regular languages. The authors proposed investigating the decidability of these properties for other classes of languages. Here, we derive necessary and sufficient conditions for a class of languages to have decidable stable, anti-stable, and error-correctable properties, and use these conditions to exhibit general classes of languages (strictly greater than the regular languages) for which the properties are decidable, and also simple classes (the first such classes) for which the properties are undecidable. We obtain identical results for the case when contextual deletions (instead of insertions) are made in the words of the language, and also with mixes of insertions and deletions. Our constructions also demonstrate that certain general problems involving nondeterministic generalized sequential machines (GSMs) applied to languages accepted by deterministic machine models are decidable, which is surprising as the deterministic language families do not need to be closed under GSM mappings.