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

by   Clemens Grabmayer, et al.

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 Mil^- of Milner's system Mil: it corresponds to the power of coinductive proofs over Mil^- that have the form of finite process graphs with the loop existence and elimination property LEE. We define a variant system cMil by replacing the fixed-point rule in Mil with a rule that permits LEE-shaped circular derivations in Mil^- from previously derived equations as a premise. With this rule alone we also define the variant system CLC for merely combining LEE-shaped coinductive proofs over Mil^-. We show that both cMil and CLC have proof interpretations in Mil, and vice versa. As this correspondence links, in both directions, derivability in 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.



page 1

page 2

page 3

page 4


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

Milner (1984) defined an operational semantics for regular expressions a...

A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity

Robin Milner (1984) gave a sound proof system for bisimilarity of regula...

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

Milner (1984) introduced a process semantics for regular expressions as ...

On Star Expressions and Coalgebraic Completeness Theorems

An open problem posed by Milner asks for a proof that a certain axiomati...

Foundations of regular coinduction

Inference systems are a widespread framework used to define possibly rec...

One-Pass and Tree-Shaped Tableau Systems for TPTL and TPTLb+Past

In this paper, we propose a novel one-pass and tree-shaped tableau metho...

A Complete Axiom System for 1-Free Kleene Star Expressions under Bisimilarity: An Elementary Proof

Grabmayer and Fokkink recently presented a finite and complete axiomatiz...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.