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In theoretical computer science a **bisimulation** is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa.

Intuitively two systems are bisimilar if they, assuming we view them as playing a *game* according to some rules, match each other's moves. In this sense, each of the systems cannot be distinguished from the other by an observer.

## Formal definition

Given a labelled state transition system (, , →),
where is a set of states, is a set of labels and → is a set of labelled transitions (i.e., a subset of ),
a **bisimulation** is a binary relation ,
such that both and its converse are simulations. From this follows that the symmetric closure of a bisimulation is a bisimulation, and that each symmetric simulation is a bisimulation. Thus some authors define bisimulation as a symmetric simulation.^{[1]}

Equivalently, is a **bisimulation** iff for every pair of states in and all labels α in :

**if , then there is such that**;

**if , then there is such that**.

Given two states and in , is **bisimilar** to , written , iff there is a bisimulation such that . This means that the bisimilarity relation is the union of all bisimulations: precisely when for some bisimulation .

The set of bisimulations is closed under union;^{[Note 1]} therefore, the bisimilarity relation is itself a bisimulation. Since it is the union of all bisimulations, it is the unique largest bisimulation. Bisimulations are also closed under reflexive, symmetric, and transitive closure; therefore, the largest bisimulation must be reflexive, symmetric, and transitive. From this follows that the largest bisimulation — bisimilarity — is an equivalence relation.^{[2]}

Note that it is not always the case that if simulates and simulates then they are bisimilar. For and to be bisimilar, the simulation between and must be the converse of the simulation between and . Counter-example in CCS: and simulate each other but are not bisimilar.

**^**Meaning the union of two bisimulations is a bisimulation.

## Alternative definitions

### Relational definition

Bisimulation can be defined in terms of composition of relations as follows.

Given a labelled state transition system , a *bisimulation* relation is a binary relation over (i.e., ⊆ × ) such that

- and

From the monotonicity and continuity of relation composition, it follows immediately that the set of the bisimulations is closed under unions (joins in the poset of relations), and a simple algebraic calculation shows that the relation of bisimilarity—the join of all bisimulations—is an equivalence relation. This definition, and the associated treatment of bisimilarity, can be interpreted in any involutive quantale.

### Fixpoint definition

Bisimilarity can also be defined in order theoretical fashion, in terms of fixpoint theory, more precisely as the greatest fixed point of a certain function defined below.

Given a labelled state transition system (, Λ, →), define to be a function from binary relations over to binary relations over , as follows:

Let be any binary relation over . is defined to be the set of all pairs in × such that:

and

Bisimilarity is then defined to be the greatest fixed point of .

### Ehrenfeucht–Fraïssé game definition

Bisimulation can also be thought of in terms of a game between two players: attacker and defender.

"Attacker" goes first and may choose any valid transition, , from . I.e.:

or

The "Defender" must then attempt to match that transition, from either or depending on the attacker's move. I.e., they must find an such that:

or

Attacker and defender continue to take alternating turns until:

- The defender is unable to find any valid transitions to match the attacker's move. In this case the attacker wins.
- The game reaches states that are both 'dead' (i.e., there are no transitions from either state) In this case the defender wins
- The game goes on forever, in which case the defender wins.
- The game reaches states , which have already been visited. This is equivalent to an infinite play and counts as a win for the defender.

By the above definition the system is a bisimulation if and only if there exists a winning strategy for the defender.

### Coalgebraic definition

A bisimulation for state transition systems is a special case of coalgebraic bisimulation for the type of covariant powerset functor. Note that every state transition system is bijectively a function from to the powerset of indexed by written as , defined by

Let be -th projection mapping to and respectively for ; and the forward image of defined by dropping the third component

where is a subset of . Similarly for .

Using the above notations, a relation is a **bisimulation** on a transition system if and only if there exists a transition system on the relation such that the diagram

commutes, i.e. for , the equations

hold where is the functional representation of .

## Variants of bisimulation

In special contexts the notion of bisimulation is sometimes refined by adding additional requirements or constraints. An example is that of stutter bisimulation, in which one transition of one system may be matched with multiple transitions of the other, provided that the intermediate states are equivalent to the starting state ("stutters").^{[3]}

A different variant applies if the state transition system includes a notion of *silent* (or *internal*) action, often denoted with , i.e. actions that are not visible by external observers, then bisimulation can be relaxed to be *weak bisimulation*, in which if two states and are bisimilar and there is some number of internal actions leading from to some state then there must exist state such that there is some number (possibly zero) of internal actions leading from to . A relation on processes is a weak bisimulation if the following holds (with , and being an observable and mute transition respectively):

This is closely related to bisimulation up to a relation.

Typically, if the state transition system gives the operational semantics of a programming language, then the precise definition of bisimulation will be specific to the restrictions of the programming language. Therefore, in general, there may be more than one kind of bisimulation, (bisimilarity resp.) relationship depending on the context.

## Bisimulation and modal logic

Since Kripke models are a special case of (labelled) state transition systems, bisimulation is also a topic in modal logic. In fact, modal logic is the fragment of first-order logic invariant under bisimulation (van Benthem's theorem).

## Algorithm

Checking that two finite transition systems are bisimilar can be done in polynomial time.^{[4]} The fastest algorithms are linear-time using partition refinement through a reduction to the coarsest partition problem.

## See also

## References

**^**Jančar, Petr and Srba, Jiří (2008). "Undecidability of Bisimilarity by Defender's Forcing".*J. ACM*. New York, NY, USA: Association for Computing Machinery.**55**(1): 26. doi:10.1145/1326554.1326559. ISSN 0004-5411. S2CID 14878621.CS1 maint: multiple names: authors list (link)**^**Milner, Robin (1989).*Communication and Concurrency*. USA: Prentice-Hall, Inc. ISBN 0131149849.**^**Baier, Christel; Katoen, Joost-Pieter (2008).*Principles of Model Checking*. MIT Press. p. 527. ISBN 978-0-262-02649-9.**^**Baier & Katoen (2008), Cor. 7.45, p. 486.

## Further reading

- Park, David (1981). "Concurrency and Automata on Infinite Sequences". In Deussen, Peter (ed.).
*Theoretical Computer Science*. Proceedings of the 5th GI-Conference, Karlsruhe. Lecture Notes in Computer Science.**104**. Springer-Verlag. pp. 167–183. doi:10.1007/BFb0017309. ISBN 978-3-540-10576-3. - Milner, Robin (1989).
*Communication and Concurrency*. Prentice Hall. ISBN 0-13-114984-9. - Davide Sangiorgi. (2011).
*Introduction to Bisimulation and Coinduction*. Cambridge University Press. ISBN 9781107003637

## External links

### Software tools

- CADP: tools to minimize and compare finite-state systems according to various bisimulations
- mCRL2: tools to minimize and compare finite-state systems according to various bisimulations
- The Bisimulation Game Game