Definition Labelled Transition Systems A labelled transition system (LTS) (sometimes also called a transition graph) is a triple (Proc, Act, -a->), where:
Definition Strong Bisimulation Game Let (Proc, Act, -a->) be a labelled transition system. A strong bisimulation game starting from the pair of states (s1, t1) in Proc x Proc is a two-player game of an 'attacker' and a 'defender'.
The game is played in rounds and configurations of the game are pairs of states from Proc x Proc. In every round exactly one configuration is called current; initially the configuration (s1, t1) is the current one.
In each round the players change the current configuration (s, t) according to the following rules.
Definition Weak Bisimulation Game A weak bisimulation game is defined in the same way as strong bisimulation game, with the only exception that the defender can answer using weak transition relation =a=> instead of only -a-> as in the strong case. The attacker is still allowed to use only the -a-> moves.
A play of the game is a maximal sequence of configurations formed by the players according to the rules described above, and starting from the initial configuration (s1, t1). Note that a bisimulation game can have many different plays according to the choices made by the attacker and the defender. The attacker can choose a side, an action and a transition. The defender's only choice is in selecting one of the available transitions that are labelled with the same action picked by the attacker.
We shall now define when a play is winning for the attacker and when for the defender.
A finite play is lost by the player who is stuck and cannot make amove from the current configuration (s, t) according to the rules of the game. Note that attacker loses only if both s -/-> and t -/->, i.e., there is no transition from both the left and the right side of the configuration. The defender loses if he has (on his side of the configuration) no available transition under the action selected by the attacker.
It can also be the case that none of the players
are stuck in any configuration and the play is
infinite. In this situation the defender is the
winner of the play.
In this game, an infinite play is defined as returning to an already seen
configuration.
A given play is always winning either for the attacker or the defender and it cannot be winning for both at the same time.
The following proposition relates bisimilarity with the corresponding game characterization.
Proposition States s1 and t1
of a labelled transition system are strongly
bisimilar if and only if the defender has a
universal winning strategy in the strong
bisimulation game starting from the configuration
(s1, t1). The
states s1 and
t1 are not
strongly bisimilar if and only if the attacker has a
universal winning strategy.
Similarly for weak bisimilarity.
By universal winning strategy we mean that the player can always win the game, irrelevant of how the other player is selecting his moves. In case that the opponent has more than one choice how to continue from the current configuration, all these possibilities have to be considered.
[ from "Reactive Systems: Modelling, Specification and Verification" by Luca Aceto, Anna Ingólfsdóttir, Kim G. Larsen and Jiri Srba. ]