Weakly and Strongly Irreversible Regular Languages

TL;DR

This paper investigates classes of finite automata based on their reversibility properties, characterizes k-reversible languages, and provides decision procedures and transformations for these automata.

Contribution

It introduces conditions for k-reversible languages, offers a decision procedure for weak and strong irreversibility, and presents a method to convert non-k-reversible automata into k-reversible ones.

Findings

Established an infinite hierarchy of weakly irreversible languages.

Provided a decision procedure for classifying automata as weakly or strongly irreversible.

Developed a construction to convert certain non-k-reversible automata into k-reversible automata.

Abstract

Finite automata whose computations can be reversed, at any point, by knowing the last k symbols read from the input, for a fixed k, are considered. These devices and their accepted languages are called k-reversible automata and k-reversible languages, respectively. The existence of k-reversible languages which are not (k-1)-reversible is known, for each k>1. This gives an infinite hierarchy of weakly irreversible languages, i.e., languages which are k-reversible for some k. Conditions characterizing the class of k-reversible languages, for each fixed k, and the class of weakly irreversible languages are obtained. From these conditions, a procedure that given a finite automaton decides if the accepted language is weakly or strongly (i.e., not weakly) irreversible is described. Furthermore, a construction which allows to transform any finite automaton which is not k-reversible, but which accepts a k-reversible language, into an equivalent k-reversible finite automaton, is presented.