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

3.4. WEAK BISIMILARITY 69
Show that s ≈ t by finding a weak bisimulation containing the pair (s, t).
Exercise 3.26 Show that, for all P, Q, the following equivalences, which are usually
referred to as Milner’s τ-laws, hold:
α.τ.P ≈ α.P (3.9)
P + τ.P ≈ τ.P (3.10)
α.(P + τ.Q) ≈ α.(P + τ.Q) + α.Q . (3.11)
Hint: Build appropriate weak bisimulations.
Exercise 3.27 Show that, for all P, Q, if P τ ⇒ Q and Q τ ⇒ P , then P ≈ Q.
Exercise 3.28 We say that a CCS process is τ-free iff none of the states that it can
reach by performing sequences of transitions affords a τ-labelled transition. For
example, a.0 is τ-free, but a.(b.0 | ¯ b.0) is not.
Prove that no τ-free CCS process is observationally equivalent to a.0 + τ.0.
Exercise 3.29 Prove that, for each CCS process P , the process P \ (Act − {τ})
is observationally equivalent to 0. Does this remain true if we consider processes
modulo strong bisimilarity?
Exercise 3.30 (Mandatory) Show that observational equivalence is the largest
symmetric relation R satisfying that whenever s1 R s2 then for each action α
(including τ), if s1 α ⇒ s ′ 1 , then there is a transition s2 α ⇒ s ′ 2 such that s′ 1 R s′ 2 .
This means that observational equivalence may be defined like strong bisimilarity,
but over a labelled transition system whose transitions are α ⇒, with α ranging
over the set of actions including τ.
Exercise 3.31 For each sequence σ of observable actions in L, and states s, t in
an LTS, define the relation σ ⇒ thus:
• s ε ⇒ t iff s τ ⇒ t, and
• s aσ′
⇒ t iff s a ⇒ s ′ σ′
⇒ t, for some s ′ .
A binary relation R over the set of states of an LTS is a weak string bisimulation
iff whenever s1 R s2 and σ is a (possibly empty) sequence of observable actions in
L:
- if s1 σ ⇒ s ′ 1 , then there is a transition s2 σ ⇒ s ′ 2 such that s′ 1 R s′ 2 ;
- if s2 σ ⇒ s ′ 2 , then there is a transition s1 σ ⇒ s ′ 1 such that s′ 1 R s′ 2 .