Structure-Constrained Process Graphs for the Process Semantics of Regular Expressions

TL;DR

This paper refines the process semantics of regular expressions to produce process graphs with structural properties, enabling better representation and reasoning about expressible graphs, especially for 1-free regular expressions.

Contribution

It introduces a refined process semantics with 1-transitions to recover the LEE property for a broader class of regular expressions.

Findings

Process graphs with 1-transitions can satisfy the LEE property.

Refined semantics enable structural reasoning about expressible process graphs.

The approach extends the applicability of process semantics to more regular expressions.

Abstract

Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike for the language semantics, where every regular (that is, DFA-accepted) language is the interpretation of some regular expression, there are finite process graphs that are not bisimilar to the process interpretation of any regular expression. For reasoning about graphs that are expressible by regular expressions modulo bisimilarity it is desirable to have structural representations of process graphs in the image of the interpretation. For `1-free' regular expressions, their process interpretations satisfy the structural property LEE (loop existence and elimination). But this is not in general the case for all regular expressions, as we show by examples. Yet as a remedy, we describe the possibility to recover the property LEE for a close variant of the process interpretation. For this purpose we refine the process semantics of regular expressions to yield process graphs with 1-transitions, similar to silent moves for finite-state automata. This report accompanies the paper with the same title in the post-proceedings of the workshop TERMGRAPH 2020. Here we give the proofs of not only one but of both of the two central theorems.