- Bisimulation
In
theoretical computer science a bisimulation is abinary relation betweenstate transition system s, associating systems which 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.
Formal definition
Given a labelled state transition system (, Λ, →), a "bisimulation" relation is a
binary relation over (i.e., ⊆ × ) such that both -1 and 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 inverse of the simulation between and . Counter-example (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 (, Λ, →), a "bisimulation" relation is a
binary relation over (i.e., ⊆ × ) such that:::and::
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 which 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.
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
τ , i.e. actions which 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 .Typically, if the
state transition system gives theoperational semantics of aprogramming 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 offirst-order logic invariant under bisimulation (Van Benthem's theorem).See also
*
Operational semantics
*State transition system s
*Simulation preorder
*Congruence relation Software tools
* [http://www.inrialpes.fr/vasy/cadp CADP - tools to minimize and compare finite-state systems according to various bisimulations]
References
# cite conference
first = David
last = Park
year = 1981
title = Concurrency and Automata on Infinite Sequences.
conference = Proceedings of the 5th GI-Conference Karlsruhe.
booktitle = Theoretical Computer Science.
series =Lecture Notes in Computer Science .
editor = Deussen, P. (ed.)
pages = 167–183
volume = 104
publisher =Springer-Verlag
id = ISBN 978-3-540-10576-3
# cite book
last = Milner
first = Robin
title = Communication and Concurrency.
year = 1989
publisher =Prentice Hall
isbn = 0-13-114984-9
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