Ogden's Lemma, Multiple Context-Free Grammars, and the Control Language Hierarchy

I present a simple example of a multiple context-free language for which a very weak variant of generalized Ogden's lemma fails. This language is generated by a non-branching (and hence well-nested) 3-MCFG as well as by a (non-well-nested) binary-branching 2-MCFG; it follows that neither the class of well-nested 3-MCFLs nor the class of 2-MCFLs is included in Weir's control language hierarchy, for which Palis and Shende proved an Ogden-like iteration theorem. I then give a simple sufficient condition for an MCFG to satisfy a natural analogue of Ogden's lemma, and show that the corresponding class of languages is a substitution-closed full AFL which includes Weir's control language hierarchy. My variant of generalized Ogden's lemma is incomparable in strength to Palis and Shende's variant and is arguably a more natural generalization of Ogden's original lemma.