The monoid of queue actions

TL;DR

This paper studies the algebraic structure of transformations induced by queue operations, providing a formal description, analyzing properties like conjugacy, and exploring decidability and recognizability of subsets.

Contribution

It introduces a formal semi-Thue system for the monoid of queue actions and analyzes its algebraic and computational properties, including decidability and recognizability.

Findings

The monoid is described by a confluent and terminating semi-Thue system.

Certain properties of rational subsets are undecidable, but their uniform membership problem is NL-complete.

The monoid is not Thurston-automatic.

Abstract

We investigate the monoid of transformations that are induced by sequences of writing to and reading from a queue storage. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties, e.g., conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is NL-complete. Furthermore, we present an algebraic characterization of this monoid's recognizable subsets. Finally, we prove that it is not Thurston-automatic.