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

60 CHAPTER 3. BEHAVIOURAL EQUIVALENCES
Exercise 3.17 (Simulation) Let us say that a binary relation R over the set of
states of an LTS is a simulation iff whenever s1 R s2 and α is an action:
- if s1 α → s ′ 1 , then there is a transition s2 α → s ′ 2 such that s′ 1 R s′ 2 .
We say that s ′ simulates s, written s ❁ ∼ s ′ , iff there is a simulation R with s R s ′ .
Two states s and s ′ are simulation equivalent, written s s ′ , iff s ❁ ∼ s ′ and s ′ ❁ ∼ s
both hold.
1. Prove that ❁ ∼ is a preorder and is an equivalence relation.
2. Build simulations showing that
Do the converse relations hold?
a.0 ❁ ∼ a.a.0 and
a.b.0 + a.c.0 ❁ ∼ a.(b.0 + c.0) .
3. Show that strong bisimilarity is included in simulation equivalence—that is,
that for any two strongly bisimilar states s and s ′ it holds that s ′ simulates
s. Does the converse inclusion also hold?
Is there a CCS process that can simulate any other CCS process?
Exercise 3.18 (Ready simulation) Let us say that a binary relation R over the set
of states of an LTS is a ready simulation iff whenever s1 R s2 and α is an action:
- if s1 α → s ′ 1 , then there is a transition s2 α → s ′ 2 such that s′ 1 R s′ 2 ; and
- if s2 α →, then s1 α →.
We say that s ′ ready simulates s, written s ❁ ∼RS s ′ , iff there is a ready simulation R
with s R s ′ . Two states s and s ′ are ready simulation equivalent, written s RS s ′ ,
iff s ❁ ∼RS s ′ and s ′ ❁ ∼RS s both hold.
1. Prove that ❁ ∼RS is a preorder and RS is an equivalence relation.
2. Do the following relations hold?
a.0 ❁ ∼RS a.a.0 and
a.b.0 + a.c.0 ❁ ∼RS a.(b.0 + c.0) .
3. Show that strong bisimilarity is included in ready simulation equivalence—
that is, that for any two strongly bisimilar states s and s ′ it holds that s ′
ready simulates s. Does the converse inclusion also hold?