The Inclusion Problem for Some Subclasses of Context-Free Languages

By a reduction to Post's Correspondence Problem we provide a direct proof of the known fact that the inclusion problem for unambiguous context-free grammars is undecidable. The argument or some straightforward modification also applies to some other subclasses of context-free languages such as linear languages, sequential languages, and DSC-languages (i.e., languages generated by context-free grammars with disjunct syntactic categories). We also consider instances of the problem ``Is $L(D_1 )\subseteq L(D_2 )^$ ?'' where $D_1$ and $D_2$ are taken from possibly different descriptor families of subclasses of context-free languages.