On Star Expressions and Coalgebraic Completeness Theorems

TL;DR

This paper explores coalgebraic methods to analyze and compare different proof techniques for the completeness of axiomatisations related to bisimilarity in regular expressions, building on Milner's open problem.

Contribution

It provides an abstract, coalgebraic framework for understanding and comparing two proof methods for completeness in regular expression axiomatizations.

Findings

Introduces local and global coalgebraic proof approaches

Analyzes Grabmayer and Fokkink's proof within a coalgebraic setting

Identifies conditions for substituting one proof method with the other

Abstract

An open problem posed by Milner asks for a proof that a certain axiomatisation, which Milner showed is sound with respect to bisimilarity for regular expressions, is also complete. One of the main difficulties of the problem is the lack of a full Kleene theorem, since there are automata that can not be specified, up to bisimilarity, by an expression. Grabmayer and Fokkink (2020) characterise those automata that can be expressed by regular expressions without the constant 1, and use this characterisation to give a positive answer to Milner's question for this subset of expressions. In this paper, we analyse Grabmayer and Fokkink's proof of completeness from the perspective of universal coalgebra, and thereby give an abstract account of their proof method. We then compare this proof method to another approach to completeness proofs from coalgebraic language theory. This culminates in two abstract proof methods for completeness, what we call the local and global approaches, and a description of when one method can be used in place of the other.