Quotients and atoms of reversible languages

Abstract

We consider several reversible finite automaton models which have been introduced over decades, and study some properties of their languages. In particular, we look at the question whether the quotients and atoms of a specific class of reversible languages also belong to that class or not. We consider bideterministic automata, reversible deterministic finite automata (REV-DFAs), reversible multiple-entry DFAs (REV-MeDFAs), and several variants of reversible nondeterministic finite automata (REV-NFAs). It is known that the class of REV-DFA languages is strictly included in the class of REV-MeDFA languages. We show that the classes of complete REV-DFA languages and complete REV-MeDFA languages are the same. We also prove that differently from the general case of a REV-DFA language, the minimal DFA for a complete REV-DFA language is a complete REV-DFA. We observe that atoms of any regular language are accepted by REV-NFAs with a single initial and a single final state. We also study atoms of atoms of regular languages and show that any atom of an atom of a regular language is a union of atoms of that language.