Weak bisimulation approximants

• Authors: Will Harwood;Faron Moller;Anton Setzer
• Venue: CSL'06 Proceedings of the 20th international conference on Computer Science Logic
• Year: 2006

Bisimilarity and weak bisimilarity ≈ are canonical notions of equivalence between processes, which are defined co-inductively, but may be approached – and even reached – by their (transfinite) inductively-defined approximants ~α and ≈α. For arbitrary processes this approximation may need to climb arbitrarily high through the infinite ordinals before stabilising. In this paper we consider a simple yet well-studied process algebra, the Basic Parallel Processes (BPP), and investigate for this class of processes the minimal ordinal α such that ≈ = ≈α. The main tool in our investigation is a novel proof of Dickson’s Lemma. Unlike classical proofs, the proof we provide gives rise to a tight ordinal bound, of ωn, on the order type of non-increasing sequences of n-tuples of natural numbers. With this we are able to reduce a long-standing bound on the approximation hierarchy for weak bisimilarity ≈ over BPP, and show that ${\approx} = {\approx_{\omega^\omega}}$.