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

6.2. SYNTAX AND SEMANTICS OF HML WITH RECURSION 123
p1
a
a
b
p2
p3
a
Figure 6.2: A process
Example 6.2 Consider the formula F = 〈a〉X and let Proc be the set of states
in the transition graph in Figure 6.2. If X is satisfied by p1, then 〈a〉X will be
satisfied by p3, i.e., we expect that
O 〈a〉X({p1}) = {p3} .
If the set of states satisfying X is {p1, p2} then 〈a〉X will be satisfied by {p1, p3}.
Therefore we expect to have that
O 〈a〉X ({p1, p2}) = {p1, p3} .
What is the set O [b]X({p2})?
The above intuition is captured formally in the following definition.
Definition 6.1 Let (Proc, Act, { a → | a ∈ Act}) be a labelled transition system.
For each S ⊆ Proc and formula F , we define OF (S) inductively as follows:
OX(S) = S
Ott(S) = Proc
Off(S) = ∅
OF1∧F2 (S) = OF1 (S) ∩ OF2 (S)
OF1∨F2 (S) = OF1 (S) ∪ OF2 (S)
O 〈a〉F (S) = 〈·a·〉OF (S)
O [a]F (S) = [·a·]OF (S) .
A few words of explanation for the above definition are in order here. Intuitively,
the first equality in Definition 6.1 expresses the trivial observation that if we assume
that S is the set of states that satisfy X, then the set of states satisfying X is S!