Divergence and unique solution of equations

TL;DR

This paper investigates proof techniques for bisimilarity based on unique solutions of equations, extending results from CSP to CCS and other models, and exploring their applications in various equivalences and calculi.

Contribution

It generalizes the unique solution theorem for equations to operational settings and different equivalences, providing refined and abstract proof techniques for bisimilarity.

Findings

Refined the theorem distinguishing divergence forms

Derived an abstract formulation for generic LTSs

Applied techniques to name-passing calculi and lambda calculus encoding

Abstract

We study proof techniques for bisimilarity based on unique solution of equations. We draw inspiration from a result by Roscoe in the denotational setting of CSP and for failure semantics, essentially stating that an equation (or a system of equations) whose infinite unfolding never produces a divergence has the unique-solution property. We transport this result onto the operational setting of CCS and for bisimilarity. We then exploit the operational approach to: refine the theorem, distinguishing between different forms of divergence; derive an abstract formulation of the theorems, on generic LTSs; adapt the theorems to other equivalences such as trace equivalence, and to preorders such as trace inclusion. We compare the resulting techniques to enhancements of the bisimulation proof method (the `up-to techniques'). Finally, we study the theorems in name-passing calculi such as the asynchronous $\pi$-calculus, and use them to revisit the completeness part of the proof of full abstraction of Milner's encoding of the $\lambda$-calculus into the $\pi$-calculus for L\'evy-Longo Trees.