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

1. Decide whether the following statements hold:
• s ?
|= 〈a〉tt,
• s ?
|= 〈b〉tt,
• s ?
|= [a]ff,
• s ?
|= [b]ff,
• s ?
|= [a]〈b〉tt,
• s ?
|= 〈a〉〈b〉tt,
• s ?
|= [a]〈a〉[a][b]ff,
• s ?
|= 〈a〉 〈a〉tt ∧ 〈b〉tt ,
• s ?
|= [a] 〈a〉tt ∨ 〈b〉tt ,
• s ?
|= 〈a〉 [b][a]ff ∧ 〈b〉tt , and
• s ?
|= 〈a〉 [a](〈a〉tt ∧ [b]ff) ∧ 〈b〉ff .
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2. Compute the following sets using the denotational semantics for Hennessy-
Milner logic.
• [[a][b]ff ] = ?
• [〈a〉 〈a〉tt ∧ 〈b〉tt ] = ?
• [[a][a][b]ff ] = ?
• [[a] 〈a〉tt ∨ 〈b〉tt ] = ?
Exercise 5.4 Consider an everlasting clock whose behaviour is defined thus:
Clock def
= tick.Clock .
Prove that the process Clock satisfies the formula
[tick](〈tick〉tt ∧ [tock]ff) .