The Failure of the Strong Pumping Lemma for Multiple Context-Free Languages
Résumé
Seki et al. (Theoretical Computer Science 88(2):191–229, 1991) showed that every m-multiple context-free language L is weakly 2m-iterative in the sense that either L is finite or L contains a subset of the form $\{u_0 w_1^i u_1 \dots w_{2m}^i u_{2m} \mid i \in \mathbb{N}\}$, where $w_1 \dots w_{2m} \neq \epsilon$. Whether for every m-multiple context-free language L is 2m-iterative, that is to say, whether all but finitely many elements $z$ of L can be written as $z=u_0 w_1 u_1 \dots w_{2m} u_{2m}$ with $w_1 \dots w_{2m} \neq \epsilon$ and $\{u_0 w_1^i u_1 \dots w_{2m}^i u_{2m} \mid i \in \mathbb{N}\} \subseteq L$ has been open. We show that there is a 3-multiple context-free language that is not k-iterative for any k.
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