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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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