Minimal and Reduced Reversible Automata

TL;DR

This paper characterizes when regular languages have multiple nonisomorphic minimal reversible automata, introduces the concept of reduced reversible automata, and explores conditions for their existence and uniqueness.

Contribution

It provides a structural condition for multiple minimal reversible automata, introduces reduced reversible automata, and establishes criteria for their uniqueness and infinitude.

Findings

Characterizes regular languages with multiple minimal reversible automata.

Shows existence of reduced reversible automata that are not minimal.

Provides conditions for infinite reduced reversible automata accepting the same language.

Abstract

A condition characterizing the class of regular languages which have several nonisomorphic minimal reversible automata is presented. The condition concerns the structure of the minimum automaton accepting the language under consideration. It is also observed that there exist reduced reversible automata which are not minimal, in the sense that all the automata obtained by merging some of their equivalent states are irreversible. Furthermore, a sufficient condition for the existence of infinitely many reduced reversible automata accepting a same language is given. It is also proved that, when the language is accepted by a unique minimal reversible automaton (that does not necessarily coincide with the minimum deterministic automaton), then no other reduced reversible automata accepting it can exist.