advertisement

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.