Rational subsets of polycyclic monoids and valence automata

TL;DR

This paper characterizes the language classes accepted by valence automata with rational target sets over polycyclic monoids, showing they accept exactly the context-free languages for rank ≥ 2 and a broader class for rank 1.

Contribution

It provides a complete description of rational subsets of polycyclic monoids and their automata, including decidability and closure properties, and clarifies the language classes accepted.

Findings

Automata over polycyclic monoids of rank ≥ 2 accept exactly context-free languages.

For rank 1, they accept languages beyond partially blind one-counter languages.

Rational subset membership is decidable, and rational subsets are closed under intersection and complement.

Abstract

We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind one-counter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem for these monoids, and the closure of the class of rational subsets under intersection and complement.