LEHRSTUHL FÜR INFORMATIK 2
RHEINISCHWESTFÄLISCHE
TECHNISCHE
HOCHSCHULE
AACHEN
RWTH Aachen · D-52056 Aachen
http://www-i2.informatik.rwth-aachen.de/
2
Prof. Dr. Ir. J.-P. Katoen
M. Neuhäußer
Introduction to Model Checking
Winter term 08/09
– Series 7 –
Hand in on December 19 before the exercise class.
Exercise 1
(4 points)
Let AP = {a, b, c}. Consider the transition system T S over AP outlined below
TS :
{a, b}
s0
{b}
s1
s3
{a, b}
s4
{b}
s2 {a}
s5 ∅
and the LTL fairness assumption f air = (23 (a ∧ b) → 23¬c) ∧ (32 (a ∧ b) → 23¬b).
a) Specify the fair paths of T S!
b) Decide for each of the following LTL formulas ϕi whether it holds T S |=f air ϕi :
ϕ1 = ¬a → 32a
ϕ2 = bU2¬b
ϕ3 = bW2¬b.
In case T S 6|=f air ϕi , indicate a path π ∈ F airP aths(T S) for which π 6|= ϕ holds.
Exercise 2
(3 points)
For the LTL-formula
ϕ = a ∧(bUa) ,
give closure(ϕ) and the set of all elementary sets B ⊆ closure(ϕ).
Exercise 3
(2 points)
Let ϕ be an LTL-formula over a set of atomic propositions AP . Prove the following property:
For all elementary sets B ⊆ closure(ϕ) and for all B ′ ∈ δ(B, B ∩ AP ), it holds:
¬ψ ∈ B ⇐⇒ ψ 6∈ B ′ .
Exercise 4
(4 points)
Consider the following LTL-formula ϕ = aUa over the set of atomic propositions AP = {a}. Construct
an equivalent GNBA G (i.e. Lω (G) = W ords(ϕ)) according to the algorithm given in the proof of Theorem
5.37 of the lecture.