Ogden's lemma, multiple context-free grammars, and the control language hierarchy

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Abstract

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. I also prove a strengthening of my pumping lemma for well-nested MCFLs which places a bound on the combined length of the substrings that can be iterated.

Keywords

Grammars
Ogden's lemma
Multiple context-free grammars
Control languages
Pumping lemma

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This work was supported by JSPS KAKENHI Grant Number 25330020.