Bisimulation

In theoretical computer science a bisimulation is a relation between the states of a transition system that puts those states relate to each other, which behave the same. This means that can simulate the transition state of the other one and vice versa.

Figuratively speaking, two states are bisimilar if their possible moves match. In this sense, they can not be distinguished from each other by an outside observer.

Formal definition

Given a kantenbeschriftetes transition system (S, Λ, → ). A bisimulation is a binary relation R on S ( ie, R ⊆ S × S) with:

For each pair of states p, q ∈ S: If (p, q ) ∈ R, then for all α ∈ Λ: Is there any p ' ∈ S with

So there is a q ' ∈ S with

And (p ', q' ) ∈ R. Similarly, there are for each Q ' ∈ S

A p ' ∈ S with

And (p ', q' ) ∈ R.

Equivalent to this is: Both R and R-1 are simulation quasi orders.

Given two states p and q in S, then p is bisimilar to q, written p ~ q if (q p ) ∈ R is a bisimulation R with.

The Bisimilaritätsrelation ~ is an equivalence relation. Furthermore, it is the largest bisimulation on a given transition system.

Variants of bisimulations

In special situations, the notion of bisimulation is sometimes refined by further conditions are added. Contains example, the transition system a silent transition, often characterized by τ, ie, a transition that is not visible to an outside observer, then the bisimulation can be weakened to a weak bisimulation, the silent transitions ignored.

• Automata Theory