Labelled Transition Systems

ion In concurrency theory it is often useful to distinguish between internal actions, that do not admit interactions with the outside world, and external ones. As normally there is no need to distinguish the internal actions from each other, they all have the same name, namely τ . If A is the set of external actions a certain class of systems may perform, then Aτ := A . ∪ {τ}. Systems in that class are then represented by labelled transition systems over Aτ and a set of predicates P . The variant of bisimulation equivalence that treats τ just like any action of A is called strong bisimulation equivalence. Often, however, one wants to abstract from internal actions to various degrees. A system doing two τ actions in succession is then considered equivalent to a system doing just one. However, a system that can do either a or b is considered different from a system that can do either a or first τ and then b, because if the former system is placed in an environment where b cannot happen, it can still do a instead, whereas the latter system may reach a state (by executing the τ action) in which a is no longer possible. Several versions of bisimulation equivalence that formalize these desiderata occur in the literature. Branching bisimulation equivalence [2], like strong bisimulation, faithfully preserves the branching structure of related systems. The notions of weak and delay bisimulation equivalence, which were both introduced by Milner under the name observational equivalence, make more identifications, motivated by observable machine-behavior according to certain testing scenarios. Write s =⇒ t for ∃n≥ 0 : ∃s0, ..., sn : s= s0 τ −→ s1 τ −→ · · · τ −→ sn = t, i.e. a (possibly empty) path of τ -steps from s to t. Furthermore, for a ∈ Aτ , write s (a) −→ t for s a −→ t ∨ (a = τ ∧ s = t). Thus (a) −→ is the same as a −→ for a ∈ A, and (τ) −→ denotes zero or one τ -steps. Definition 9 Let (S,→, |=) be an LTS over Aτ and P . Two states s, t ∈ S are branching bisimulation equivalent, denoted s↔b t, if they are related by a binary relation R ⊆ S × S (a branching bisimulation), satisfying: ∧ if sRt and s |= p with p ∈ P , then there is a t1 with t =⇒ t1 |= p and sRt1, ∧ if sRt and t |= p with p ∈ P , then there is a s1 with s =⇒ s1 |= p and s1Rt, ∧ if sRt and s a −→ s with a∈Aτ , then there are t1, t2, t with t =⇒ t1 (a) −→ t2 = t, sRt1 and sRt, ∧ if sRt and t a −→ t with a∈Aτ , then there are s1, s2, s with s=⇒s1 (a) −→s2 = s, s1Rt and sRt. Delay bisimulation equivalence, ↔d , is obtained by dropping the requirements sRt1 and s1Rt. Weak bisimulation equivalence [5], ↔w , is obtained by furthermore relaxing the requirements t2 = t ′ and s2 = s ′ to t2 =⇒ t and s2 =⇒ s. These definition stem from concurrency theory. On Kripke structures, when studying modal or temporal logics, normally a stronger version of the first two conditions is imposed: ∧ if sRt and p ∈ P , then s |= p⇔ t |= p. For systems without τ ’s all these notions coincide with strong bisimulation equivalence. Concurrency When applied to parallel systems, capable of performing different actions at the same time, the versions of bisimulation discussed here employ interleaving semantics: no distinction is made between true parallelism and its nondeterministic sequential simulation. Versions of bisimulation that do make such a distinction have been developed as well, most notably the ST-bisimulation [2], that