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

108 CHAPTER 5. HENNESSY-MILNER LOGIC
Show also that, for each n ≥ 0, the process Clock satisfies the formula
〈tick〉 · · · 〈tick〉 tt .
n-times
Exercise 5.5 (Mandatory) Find a formula in M that is satisfied by a.b.0 + a.c.0,
but not by a.(b.0 + c.0).
Find a formula in M that is satisfied by a.(b.c.0 + b.d.0), but not by a.b.c.0 +
a.b.d.0.
It is sometimes useful to have an alternative characterization of the satisfaction
relation |= presented in Definition 5.2. This can be obtained by defining the binary
relation |= relating processes to formulae by structural induction on formulae thus:
• P |= tt, for each P ,
• P |= ff, for no P ,
• P |= F ∧ G iff P |= F and P |= G,
• P |= F ∨ G iff P |= F or P |= G,
• P |= 〈a〉F iff P a → P ′ for some P ′ such that P ′ |= F , and
• P |= [a]F iff P ′ |= F , for each P ′ such that P a → P ′ .
Exercise 5.6 Show that the above definition of the satisfaction relation is equivalent
to that given in Definition 5.2. Hint: Use induction on the structure of formulae.
Exercise 5.7 Find one labelled transition system with initial state s that satisfies
all of the following properties:
• 〈a〉(〈b〉〈c〉tt ∧ 〈c〉tt),
• 〈a〉〈b〉([a]ff ∧ [b]ff ∧ [c]ff), and
• [a]〈b〉([c]ff ∧ 〈a〉tt).