The Failure of the Strong Pumping Lemma for Multiple Context-Free Languages

Kanazawa M, Kobele GM, Michaelis J, Salvati S, Yoshinaka R. The Failure of the Strong Pumping Lemma for Multiple Context-Free Languages. Theory of Computing Systems. 2014;55(1):250-278.Seki et al. (Theor. Comput. Sci. 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 {u0w1iu1w2miu2m|i≥0}, where w1···w2m≠ε. Whether 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=u0w1u1w2mu2m with w1···w2m≠ε and {u0w1iu1···w2miu2m|i≥0}⊆L, has been open. We show that there is a 3-multiple context-free language that is not k-iterative for any k