Rational indexes of generators of the cone of context-free languages

Abstract

The rational index ϱ L of a non-empty language L is a non-decreasing function from N ∗ into 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 τ, such that τ( L)= L′, then ϱ L the rational index of L, provides an upper bound on ϱ L ′, since ∃c∈ N ∗,∀n∈ N ∗,cn(ϱ L(cn)+1)⩾ϱ 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 Ω( n)∩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 Θ(n 2/ lnn) .