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

TL;DR

This paper introduces a coinductive reformulation of Milner's proof system for regular expressions under bisimilarity, aiming to address completeness issues by replacing fixed-point rules with cyclic derivations.

Contribution

It develops a coinductive rule for Milner's system that captures fixed-point reasoning via cyclic derivations satisfying LLEE, enabling a new proof-theoretic approach.

Findings

cMil and Mil are theorem equivalent

Effective proof transformations between Mil and cMil

Linking coinductive proofs to coalgebraic solutions

Abstract

Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete for equality of regular expressions under the language semantics. Apart from most equational axioms, Milner's system Mil inherits from Salomaa's system a non-algebraic rule for solving single fixed-point equations. Recognizing distinctive properties of the process semantics that render Salomaa's proof strategy inapplicable, Milner posed completeness of the system Mil as an open question. As a proof-theoretic approach to this problem we characterize the derivational power that the fixed-point rule adds to the purely equational part Mil$^-$ of Mil. We do so by means of a coinductive rule that permits cyclic derivations that consist of a finite process graph with empty steps that satisfies the layered loop existence and elimination property LLEE, and two of its Mil$^{-}$-provable solutions. With this rule as replacement for the fixed-point rule in Mil, we define the coinductive reformulation cMil as an extension of Mil$^{-}$. In order to show that cMil and Mil are theorem equivalent we develop effective proof transformations from Mil to cMil, and vice versa. Since it is located half-way in between bisimulations and proofs in Milner's system Mil, cMil may become a beachhead for a completeness proof of Mil. This article extends our contribution to the CALCO 2022 proceedings. Here we refine the proof transformations by framing them as eliminations of derivable and admissible rules, and we link coinductive proofs to a coalgebraic formulation of solutions of process graphs.