Rational index of context-free languages in exp [formula omitted]and [formula omitted

Abstract

There are many ways to measure the complexity of languages. Rational index, introduced by Boasson and Nivat in [3], is one of them and behaves well when combined with rational transductions: if L⩾ L' (i.e., there exists a rational transduction τ such that τ( L)= L'), then the rational index ϱ L of L provides an upper bound on ϱ L since ∃cϵN∗∀nϵN∗:cn(ρL(cn+1)⩾ρL(n) . Thus we define the family Pol of languages whose rational indexes grow less than polynomial, and the family Exp of languages whose rational indexes grow more than exponential functions. It was conjectured in [4] that all context-free languages belong to Pol or Exp . The aim of this paper is to show context-free languages whose rational indexes are exp Θ(P n ) andn Θ,((ln n) l p ) proving thereby that this conjecture is false. The results and most of the proofs appearing in this paper are those of [5].