A Spectrum of Approximate Probabilistic Bisimulations

TL;DR

This paper introduces and compares various approximate probabilistic bisimulation notions for labeled Markov chains, including weak, branching, and epsilon-perturbed bisimulations, revealing their interrelations and connections to existing concepts.

Contribution

It presents new approximate bisimulation definitions for LMCs and analyzes their relationships with established notions, enhancing understanding of probabilistic system equivalences.

Findings

Defined approximate weak and branching bisimulation variants.

Introduced epsilon-perturbed bisimulation for small probability perturbations.

Established connections between new notions and existing epsilon-bisimulation.

Abstract

This paper studies various notions of approximate probabilistic bisimulation on labeled Markov chains (LMCs). We introduce approximate versions of weak and branching bisimulation, as well as a notion of $\varepsilon$-perturbed bisimulation that relates LMCs that can be made (exactly) probabilistically bisimilar by small perturbations of their transition probabilities. We explore how the notions interrelate and establish their connections to other well-known notions like $\varepsilon$-bisimulation.