Lecture Notes in Computer Science 3472

coffee
coin
tea
19 Model Checking 563
coin coin
coffee tea
Fig. 19.3. Two vending machines
where p ranges over a set of atomic propositions AP. Note that sometimes
true is defined to be a special atomic proposition, which is valid for every state.
The semantics [φ] of a formula φ is the set of all runs π for which the property
holds: [φ] = {π | π |= φ}. The semantics is inductively defined on the structure
of the formula.
π |= true
π |= p ⇔ p ∈ I (π0)
π |= ¬φ ⇔ π �|= φ
π |= φ1 ∧ φ2 ⇔ π |= φ1 ∧ π |= φ2
π |= X (φ) ⇔|π| > 1 ∧ π1 |= φ
π |= φ1Uφ2 ⇔∃k ∈ [|π|−1] : � πk |= φ2 ∧∀i ∈ [k − 1] : πi �
|= φ1
Every run π satisfies true and every run π satisfies an atomic proposition, iff
the first state π0 of the run does. The negation and conjunction is interpreted
as usual; further Boolean connectives may be introduced as abbreviations. E.g.
φ1 ∨ φ2, can be introduced as ¬ (¬φ1 ∧¬φ2). The modality X (φ) is called “next
time φ” and requires the property φ to hold for the next situation in the run.
The modality φ1Uφ2 is also denoted as U (φ1,φ2). It is called “φ1 until φ2” and
requires the property φ1 to hold for all situations on the run until finally the
property φ2 holds for some situation.
Besides abbreviations of further Boolean connectivities, the following abbreviations
are common:
F (φ) :=U (true,φ)
G (φ) :=¬F (¬φ)
The modality F (φ) , called “finally φ”, requires φ to hold for some later situation.
The modality G (φ) , called “globally φ”, requires φ to hold for all situations. The
until modality φ1Uφ2 is sometimes called strong until because it requires φ2
to become true finally. In contrast to this modality, there is a different variant,
called weak until, which holds, even if φ2 never holds while φ1 holds forever.
(φ1WUφ2 := φ1Uφ2 ∨ G (φ1)).
Since in system verification one is typically interested whether a specific state
satisfies a certain property, there is the following convention: a state s ∈ S of a
transition system satisfies a formula if every run starting at s satisfies it.