So I asked a similar question on MathOverflow, but I now realize I asked the wrong question and asked it in the wrong place.
Anyway, I'm searching for a class of languages containing the boolean closure of CFLs that has a 'pumping-type' condition on it. What I've found so far are:
- MCFGs (multiple context-free languages), which have a nice pumping lemma and contain CFLs (though it's not obvious if they contain the boolean closure of CFLs, since they're not boolean-closed themselves),
- ET0L languages (which contain CFLs, have a 'pumping-type' lemma, and are (full) AFLs themselves--but I don't think they necessarily contain the boolean closure of CFLs),
- Mildly CSLs (which apparently have pumping-type lemmas?),
- Indexed languages (which have a 'shrinking' lemma), etc.
Now, I'm certainly no specialist in formal language theory, so I don't know too much about these classes. Anyway, does such a class exist, and, if so, where can I find papers/etc. on it?