## Rational indexes of generators of the cone of context-free languages.(English)Zbl 0745.68068

Summary: The rational index $$\rho_ L$$ of a non-empty language $$L$$ is a non- decreasing function from $$\mathbb{N} ^*$$ into $$\mathbb{N}$$, whose asymptotic behavior can be used to classify languages. The rational index behaves well when combined with rational transductions: if a language $$L$$ rationally dominates another language $$L'$$ (i.e. there exists a rational transduction $$\tau$$, such that $$\tau(L)=L'$$), then $$\rho_ L$$, the rational index of $$L$$, provides an upper bound on $$\rho_{L'}$$, since $\exists c\in\mathbb{N}^*, \qquad \forall n\in\mathbb{N}^*, \qquad cn(\rho_ L(cn)+1)\geq\rho_{L'}(n).$ Hence all the generators of the rational cone of context-free languages, i.e. the context-free languages which dominate any context-free language, have roughly the same rational indexes, which were known to belong to $$\exp \Omega(n)\cap\exp O(n^ 2)$$. This paper improves these bounds. Indeed the rational index of any generator of the rational cone of context-free languages belongs to $$\exp \Theta(n^ 2/\ln n)$$.

### MSC:

 68Q45 Formal languages and automata
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### References:

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