Expressiveness modulo Bisimilarity of Regular Expressions with Parallel Composition (Extended Abstract)

TL;DR

This paper investigates how adding parallel composition to regular expressions enhances their ability to describe automaton behaviors up to bisimilarity, showing that certain extensions make the language class strictly more expressive.

Contribution

It demonstrates that incorporating parallel composition operations into regular expressions increases their expressiveness up to bisimilarity, culminating in a complete characterization of finite automata.

Findings

Pure interleaving increases expressiveness

ACP-style parallel composition further enhances expressiveness

Regular expressions with ACP-style parallelism and encapsulation can express all finite automata

Abstract

The languages accepted by finite automata are precisely the languages denoted by regular expressions. In contrast, finite automata may exhibit behaviours that cannot be described by regular expressions up to bisimilarity. In this paper, we consider extensions of the theory of regular expressions with various forms of parallel composition and study the effect on expressiveness. First we prove that adding pure interleaving to the theory of regular expressions strictly increases its expressiveness up to bisimilarity. Then, we prove that replacing the operation for pure interleaving by ACP-style parallel composition gives a further increase in expressiveness. Finally, we prove that the theory of regular expressions with ACP-style parallel composition and encapsulation is expressive enough to express all finite automata up to bisimilarity. Our results extend the expressiveness results obtained by Bergstra, Bethke and Ponse for process algebras with (the binary variant of) Kleene's star operation.