On the State Complexity of the Reverse of ${\mathcal R}$- and ${\mathcal J}$-Trivial Regular Languages

Abstract

The tight bound on the state complexity of the reverse of ${\mathcal R}$-trivial and ${\mathcal J}$-trivial regular languages of the state complexity n is 2^n − 1. The witness is ternary for ${\mathcal R}$-trivial regular languages and (n − 1)-ary for ${\mathcal J}$-trivial regular languages. In this paper, we prove that the bound can be met neither by a binary ${\mathcal R}$-trivial regular language nor by a ${\mathcal J}$-trivial regular language over an (n − 2)-element alphabet. We provide a characterization of tight bounds for ${\mathcal R}$-trivial regular languages depending on the state complexity of the language and the size of its alphabet. We show the tight bound for ${\mathcal J}$-trivial regular languages over an (n − 2)-element alphabet and a few tight bounds for binary ${\mathcal J}$-trivial regular languages. The case of ${\mathcal J}$-trivial regular languages over an (n − k)-element alphabet, for 2 ≤ k ≤ n − 3, is open.