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

52 CHAPTER 3. BEHAVIOURAL EQUIVALENCES
• define an LTS and a binary relation over states that is not reflexive but is a
strong bisimulation;
• define an LTS and a binary relation over states that is not symmetric but is a
strong bisimulation; and
• define an LTS and a binary relation over states that is not transitive but is a
strong bisimulation.
Are the relations you have constructed the largest strong bisimulations over your
labelled transition systems?
Exercise 3.9 (Recommended) A binary relation R over the set of states of an LTS
is a string bisimulation iff whenever s1 R s2 and σ is a sequence of actions in Act:
- 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 .
Two states s and s ′ are string bisimilar iff there is a string bisimulation that relates
them.
Prove that string bisimilarity and strong bisimilarity coincide. That is, show
that two states s and s ′ are string bisimilar iff they are strongly bisimilar.
Exercise 3.10 Assume that the defining equation for the constant K is K def
= P .
Show that K ∼ P holds.
Exercise 3.11 Prove that two strongly bisimilar processes afford the same traces,
and thus that strong bisimulation equivalence satisfies the requirement for a behavioural
equivalence we set out in equation (3.1). Hint: Use your solution to
Exercise 3.9 to show that, for each trace α1 · · · αk (k ≥ 0),
P ∼ Q and α1 · · · αk ∈ Traces(P ) imply α1 · · · αk ∈ Traces(Q) .
Is it true that strongly bisimilar processes have the same completed traces? (See
Exercise 3.2 for the definition of the notion of completed trace.)
Exercise 3.12 (Recommended) Show that the relations listed below are strong
bisimulations:
{(P | Q, Q | P ) | where P, Q are CCS processes}
{(P | 0, P ) | where P is a CCS process}
{((P | Q) | R, P | (Q | R)) | where P, Q, R are CCS processes} .