A Coinductive Version of Milner's Proof System for Regular Expressions Modulo Bisimilarity

TL;DR

This paper introduces a coinductive proof system variant for Milner's regular expression bisimilarity, linking fixed-point rules to process graph proofs and expanding the methods for establishing system completeness.

Contribution

It characterizes the fixed-point rule's derivational power, proposes a coinductive proof system with circular derivations, and establishes a correspondence with Milner's original system.

Findings

Equivalent proof systems cMil and CLC are developed.

Proof interpretations connect derivability with process graph trees.

The approach broadens avenues for graph-based completeness proofs.

Abstract

By adapting Salomaa's complete proof system for equality of regular expressions under the language semantics, Milner (1984) formulated a sound proof system for bisimilarity of regular expressions under the process interpretation he introduced. He asked whether this system is complete. Proof-theoretic arguments attempting to show completeness of this equational system are complicated by the presence of a non-algebraic rule for solving fixed-point equations by using star iteration. We characterize the derivational power that the fixed-point rule adds to the purely equational part $\text{Mil$^{\boldsymbol{-}}$}$ of Milner's system $\text{$\text{Mil}$}$: it corresponds to the power of coinductive proofs over $\text{Mil$^{\boldsymbol{-}}$}$ that have the form of finite process graphs with the loop existence and elimination property $\text{LEE}$. We define a variant system $\text{cMil}$ by replacing the fixed-point rule in $\text{Mil}$ with a rule that permits $\text{LEE}$-shaped circular derivations in $\text{Mil$^{\boldsymbol{-}}$}$ from previously derived equations as a premise. With this rule alone we also define the variant system $\text{CLC}$ for merely combining $\text{LEE}$-shaped coinductive proofs over $\text{Mil$^{\boldsymbol{-}}$}$. We show that both $\text{cMil}$ and $\text{CLC}$ have proof interpretations in $\text{Mil}$, and vice versa. As this correspondence links, in both directions, derivability in $\text{Mil}$ with derivation trees of process graphs, it widens the space for graph-based approaches to finding a completeness proof of Milner's system. This report is the extended version of a paper with the same title presented at CALCO 2021.