Stuttering equivalence

In theoretical computer science, stuttering equivalence,[1] a relation written as

The paths ${\displaystyle \pi }$ and ${\displaystyle \pi '}$ are stuttering equivalent.

${\displaystyle \pi \sim _{st}\pi '}$,

can be seen as a partitioning of paths ${\displaystyle \pi }$ and ${\displaystyle \pi '}$ into blocks, so that states in the ${\displaystyle k^{\mathrm {th} }}$ block of one path are labeled (${\displaystyle L(\cdot )}$) the same as states in the ${\displaystyle k^{\mathrm {th} }}$ block of the other path. Corresponding blocks may have different lengths.

Formally, this can be expressed as two infinite paths ${\displaystyle \pi =s_{0},s_{1},\ldots }$ and ${\displaystyle \pi '=r_{0},r_{1},\ldots }$ being stuttering equivalent (${\displaystyle \pi \sim _{st}\pi '}$) if there are two infinite sequences of integers ${\displaystyle 0=i_{0}<i_{1}<i_{2}<\ldots }$ and ${\displaystyle 0=j_{0}<j_{1}<j_{2}<\ldots }$ such that for every block ${\displaystyle k\geq 0}$ holds ${\displaystyle L(s_{i_{k}})=L(s_{i_{k}+1})=\ldots =L(s_{i_{k+1}-1})=L(r_{j_{k}})=L(r_{j_{k}+1})=\ldots =L(r_{j_{k+1}-1})}$.

Stuttering equivalence is not the same as bisimulation, since bisimulation cannot capture the semantics of the 'eventually' (or 'finally') operator found in linear temporal/computation tree logic (branching time logic)(modal logic). So-called branching bisimulation has to be used.