Күрделі жүйелердің қатынастары және бисимуляциясы

Relation between transition systems in computer science

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.

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"). 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 the notion of 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 (respectively bisimilarity) relationship depending on the context.

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).

Checking that two finite transition systems are bisimilar can be done in polynomial time. The fastest algorithms are quasilinear time using partition refinement through a reduction to the coarsest partition problem.