Bisimulation
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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 the sense that one system simulates the other and vice versa.
Intuitively two systems are bisimilar if they match each other's moves. In this sense, each of the systems cannot be distinguished from the other by an observer.
Contents
Formal definition
Given a labelled state transition system (, Λ, →), a bisimulation relation is a binary relation over (i.e., ⊆ × ) such that both and its converse are simulations.
Equivalently is a bisimulation if for every pair of elements in with in , for all α in Λ:
for all in ,
 implies that there is a in such that
 and ;
and, symmetrically, for all in
 implies that there is a in such that
 and .
Given two states and in , is bisimilar to , written , if there is a bisimulation such that is in .
The bisimilarity relation is an equivalence relation. Furthermore, it is the largest bisimulation relation over a given transition system.
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 . Counterexample (in CCS, describing a coffee machine) : and simulate each other but are not bisimilar.
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 .
Game theoretical 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. For example, 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 firstorder logic invariant under bisimulation (van Benthem's theorem).
See also
References
 Park, David (1981). "Concurrency and Automata on Infinite Sequences". In Deussen, Peter. Theoretical Computer Science. Proceedings of the 5th GIConference, Karlsruhe. Lecture Notes in Computer Science. 104. SpringerVerlag. pp. 167–183. doi:10.1007/BFb0017309. ISBN 9783540105763.
 Milner, Robin (1989). Communication and Concurrency. Prentice Hall. ISBN 0131149849.
Further reading
 Davide Sangiorgi. (2011). Introduction to Bisimulation and Coinduction. Cambridge University Press. ISBN 9781107003637