Bisimulation

In theoretical Informatique a bisimulation is a Relation of equivalence between systems of transition from states, associating the systems which behave in the same way with the direction that one of the systems simulates the other and vice versa.

Intuitively two systems are bisimilaires if they are able to imitate one the other. Accordingly, each system cannot be distinguished from different by an observer.

Formal definition

Being given a labelled System of transition from states (S, Λ, →), a relation of bisimulation is a binary Relation R on S (c.à.d R ⊆ S × S) such as at the same time R and R^-1 is préordres of simulation.

In an equivalent way R is a bisimulation so for each pair of elements p, Q in S, if (p, Q) is in R then for all α in Λ, and for all p' in S,

\begin{matrix} & \ alpha & \ \ p & \ rightarrow & p' \ \ \end{matrix}

imply that there exists a q' in S such as

\begin{matrix} & \ alpha & \ \ Q & \ rightarrow & q' \ \ \end{matrix}

and (p', q') in R, and for all q' in S,

\begin{matrix} & \ alpha & \ \ Q & \ rightarrow & q' \ \ \end{matrix}

imply that there exists a p' in S such as

\begin{matrix} & \ alpha & \ \ p & \ rightarrow & p' \ \ \end{matrix}

and (p', q') in R.

Being given two states p and Q in S, p is bisimilaire with Q, noted p ∼ Q, if there exists a bisimulation R such as (p, Q) that is to say in R.

The relation of bisimilarity ∼ is a Relation of equivalence. Moreover, it is the greatest relation of bisimulation on a given system of transition.

Let us note that the fact that p simulates Q and Q simulates p is not always enough so that they are bisimilaires. So that p and Q are bisimilaires, simulation between p and Q must be the opposite of simulation between Q and p.

Alternatives of the bisimulation

In particular contexts the concept of bisimulation is sometimes refined by adding additional constraints. For example if the system of transition from states includes a concept of quiet actions , often indicated by τ, c.à.d. of the actions which are not visible by the external observers, then the bisimulation can be weakened to become the weak bisimulation , in which the quiet actions are ignored.

Typically, if the Système of transition from states gives the operational Sémantique of a Computer programming language, then the precise definition of the bisimulation will be specific to the restrictions of the computer programming language. Consequently, in general, there can be more than one kind of relation of bisimulation (resp. bisimilarity) according to the context.

See too

References

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