Stutter bisimulation

In Principles of Model Checking, Baier and Katoen define a stutter bisimulation for a single transition system and later adapt it to a relation on two transition systems. A stutter bisimulation for ${\displaystyle {\text{TS}}=(S,{\text{Act}},\to ,I,{\text{AP}},L)}$ is a binary relation R on S such that for all (s₁,s₂) in R:

1. ${\displaystyle s_{1},s_{2}}$ have the same labels
2. If ${\displaystyle s_{1}\to s_{1}'}$ is a valid transition and ${\displaystyle (s_{1}',s_{2})\not \in R}$ then there exists a finite path fragment ${\displaystyle s_{2}u_{1}\cdots u_{n}s_{2}'}$ (${\displaystyle n\geq 0}$) such that each pair ${\displaystyle (s_{1},u_{i})}$ is in R and ${\displaystyle (s_{1}',s_{2}')}$ is in R
3. If ${\displaystyle s_{2}\to s_{2}'}$ is a valid transition and ${\displaystyle (s_{1},s_{2}')\not \in R}$ is not then there exists a finite path fragment ${\displaystyle s_{1}v_{1}\cdots v_{n}s_{1}'}$ (${\displaystyle n\geq 0}$) such that each pair ${\displaystyle (v_{i},s_{2})}$ is in R and ${\displaystyle (s_{1}',s_{2}')}$ is in R

A generalization, the divergent stutter bisimulation, can be used to simplify the state space of a system with the tradeoff that statements using the linear temporal logic operator "next" may change truth value.^[2] A robust stutter bisimulation allows uncertainty over transitions in the system.^[3]