Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

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74 CHAPTER 3.5. GAME CHARACTERIZATION OF BISIMILARITY Definition 3.5 [Strong bisimulation game] Let (Proc, Act, { α →| α ∈ Act}) be a labelled transition system. A strong bisimulation game starting from the pair of states (s1, t1) ∈ Proc × 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 × 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. 1. The attacker chooses either the left- or the right-hand side of the current configuration (s, t) and an action α from Act. • If the attacker chose left then he has to perform a transition s α → s ′ for some state s ′ ∈ Proc. • If the attacker chose right then he has to perform a transition t α → t ′ for some state t ′ ∈ Proc. 2. In this step the defender must provide an answer to the attack made in the previous step. • If the attacker chose left then the defender plays on the right-hand side, and has to respond by making a transitions t α → t ′ for some t ′ ∈ Proc. • If the attacker chose right then the defender plays on the left-hand side and has to respond by making a transitions s α → s ′ for some s ′ ∈ Proc. 3. The configuration (s ′ , t ′ ) becomes the current configuration and the game continues for another round according to the rules described above. 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). (A sequence of configurations is maximal if it cannot be extended while following the rules of the game.) 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.
CHAPTER 3.5. GAME CHARACTERIZATION OF BISIMILARITY 75 A finite play is lost by the player who is stuck and cannot make a move from the current configuration (s, t) according to the rules of the game. Note that the attacker loses a finite play only if both s and t , i.e., there is no transition from both the left- and the right-hand side of the configuration. The defender loses a finite play 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 is stuck in any configuration and the play is infinite. In this situation the defender is the winner of the play. Intuitively, this is a natural choice of outcome because if the play is infinite then the attacker has been unable to find a ‘difference’ in the behaviour of the two systems— which will turn out to be bisimilar. 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 strong bisimilarity with the corresponding game characterization (see, e.g., (Stirling, 1995; Thomas, 1993)). Proposition 3.3 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. By universal winning strategy we mean that the player can always win the game, regardless of how the other player is selecting his moves. In case the opponent has more than one choice for how to continue from the current configuration, all these possibilities have to be considered. The notion of a universal winning strategy is best explained by means of an example. Example 3.5 Let us recall the transition system from Example 3.1. s a t a a b s1 s2 t1 We will show that the defender has a universal winning strategy from the configuration (s, t) and hence, in light of Proposition 3.3, that s ∼ t. In order to do so, we have to consider all possible attacker’s moves from this configuration and define b b
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Reactive Systems: Modelling, Specif
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iv CONTENTS 5 Hennessy-Milner logic
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List of Figures 2.1 The interface f
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List of Tables 2.1 An alternative f
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xii PREFACE It has been very satisf
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xiv PREFACE a textbook, and offers
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xvi PREFACE studies in Edinburgh, a
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Part I A classic theory of reactive
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