Strong, Weak and Branching Bisimulation for Transition Systems and Markov Reward Chains: A Unifying Matrix Approach

TL;DR

This paper introduces a unifying algebraic matrix approach to define and analyze strong, weak, and branching bisimulation for transition systems and Markov reward chains, linking boolean and real matrix theories.

Contribution

It develops a novel algebraic framework that unifies bisimulation concepts across different models using matrix theory, including a new definition for branching bisimulation.

Findings

Strong and weak bisimulations match existing definitions.

Branching bisimulation is newly defined, but its practical usefulness is uncertain.

The matrix approach provides a unified algebraic perspective.

Abstract

We first study labeled transition systems with explicit successful termination. We establish the notions of strong, weak, and branching bisimulation in terms of boolean matrix theory, introducing thus a novel and powerful algebraic apparatus. Next we consider Markov reward chains which are standardly presented in real matrix theory. By interpreting the obtained matrix conditions for bisimulations in this setting, we automatically obtain the definitions of strong, weak, and branching bisimulation for Markov reward chains. The obtained strong and weak bisimulations are shown to coincide with some existing notions, while the obtained branching bisimulation is new, but its usefulness is questionable.