Decidability of operation problems for T0L languages and subclasses

We investigate the decidability of the operation problem for T0L languages and subclasses. Fix an operation on formal languages. Given languages from the family considered (0L languages, T0L languages, or their propagating variants), is the application of this operation to the given languages still a language that belongs to the same language family? Observe, that all the Lindenmayer language families in question are anti-AFLs, that is, they are not closed under homomorphisms, inverse homomorphisms, intersection with regular languages, union, concatenation, and Kleene closure. Besides these classical operations we also consider intersection and substitution, since the language families under consideration are not closed under these operations, too. We show that for all of the above mentioned language operations, except for the Kleene closure, the corresponding operation problems of 0L and T0L languages and their propagating variants are not even semidecidable. The situation changes for unary 0L languages. In this case we prove that the operation problems with respect to Kleene star, complementation, and intersection with regular sets are decidable.