Operational complexity and right linear grammars

Abstract

For a regular language L, let \({{\,\mathrm{Var}\,}}(L)\) be the minimal number of nonterminals necessary to generate L by right linear grammars. Moreover, for natural numbers \(k_1,k_2,\ldots ,k_n\) and an n-ary regularity preserving operation f, let \(g_f^{{{\,\mathrm{Var}\,}}}(k_1,k_2,\ldots ,k_n)\) be the set of all numbers k such that there are regular languages \(L_1,L_2,\ldots , L_n\) such that \({{\,\mathrm{Var}\,}}(L_i)=k_i\) for \(1\le i\le n\) and \({{\,\mathrm{Var}\,}}(f(L_1,L_2,\ldots , L_n))=k\). We completely determine the sets \(g_f^{{{\,\mathrm{Var}\,}}}\) for the operations reversal, Kleene-closures \(+\) and \(*\), and union; and we give partial results for product and intersection.