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The rational index of the Dyck language \(D_ 1^{'*}\). (English) Zbl 0632.68072
In order to measure the complexity of languages, the rational index was introduced by L. Boasson and M. Nivat [C. R. Acad. Sci., Paris, Sér. A. 284, 559-562, 625-628, 703-705 (1977; Zbl 0359.68095, Zbl 0359.68096, Zbl 0354.68107)]. The rational index of a nonempty language L is a function \(\rho_ L\) of \({\mathbb{N}}-\{0\}\) into \({\mathbb{N}}\). The asymptotical behaviour of this function allows to classify languages. More precisely, let n be a positive integer. For each rational language K recognized by a finite nondeterministic automaton with n states and not disjoint with L, let us consider \(\delta_{L\cap K}:\) the length of a shortest word in \(L\cap K\). Then \(\rho_ L(n)\) is the maximum of \(\delta_{L\cap K}\) for all such K.
On the other hand, the restricted Dyck language \(D_ 1^{'*}\) is the set of all well-parenthesized words in \(\{a,b\}^*\), considering a and b respectively as left and right parentheses. Then L. Boasson, B. Courcelle, and M. Nivat have shown [Proc. Conf. Theoretical Comput. Sci., Waterloo/Ontario 1977, 130-138 (1977; Zbl 0431.68077)] that \(O(n^ 2)\leq \rho_{D_ 1^{'*}}(n)\leq O(n^ 3)\) and conjectured that \(\rho_{D_ 1^{'*}}(n)=O(n^ 2)\), which we shall prove here.

MSC:
68Q45 Formal languages and automata
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[1] Boasson, L.; Nivat, M.; Boasson, L.; Nivat, M.; Boasson, L.; Nivat, M., Ordres et types de langage, I, II, III, Notes CRAS, Série A tome 284, Notes CRAS, Série A tome 284, Notes CRAS, Série A tome 284, 703-705, (1977) · Zbl 0354.68107
[2] Boasson, L.; Courcelle, B.; Nivat, M., A new complexity measure, (), 130-138
[3] Boasson, L.; Courcelle, B.; Nivat, M., The rational index, a complexity measure for languages, SIAM J. comput., 10, 2, 284-296, (1981) · Zbl 0469.68083
[4] Gabarro, J., Index rationnel, centre et langages algébriques, (), Rapport L.I.T.P. No. 81-54 · Zbl 0505.68033
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