On the State Complexity of the Reverse of R- and J-trivial Regular Languages

TL;DR

This paper investigates the maximum state complexity of reversing R-trivial and J-trivial regular languages, establishing tight bounds depending on alphabet size and providing characterizations for various cases.

Contribution

It proves new bounds on the reverse state complexity for R- and J-trivial languages, especially over small alphabets, and characterizes when these bounds are tight.

Findings

Tight upper bound of 2^{n-1} for reverse of R- and J-trivial languages.

Binary R-trivial languages cannot meet the bound.

Tight bounds are established for J-trivial languages over (n-2)-element alphabet.

Abstract

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