1. Software Verification 2 Automated Verification
Prof. Dr. Holger Schlingloff
Institut für Informatik der Humboldt Universität
and
Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

2. Recap: LTL and CTL Kripke models = LTS
Expressively equivalent to FOL (<) on linear orders
Arguably more intuitive to use
Kripke models = LTS
branching time – possibility properties
linear time – fairness properties
CTL* subsumes both CTL and LTL

3. Safety and Liveness

4. Some Theorems
Every LTL formula can be written in one of these forms (where p is a pure past formula)
Every LTL safety property is expressible as G*φ (where φ is pure past)
Every LTL formula can be written as reactivity ⋀(G*F*φ  F*G*ψ)
All the inclusions are strict
All the non-inclusions are provable (dualities)
e.g. F*G*p cannot be expressed as G*F*φ
Conjunction and disjunction of a recurrence is a recurrence
G*F*p  G*F*q = G*F*(p  -p U-q)
G*F*p  G*F*q = G*F*(p  q)
Obligations can be expressed as recurrences and persistences
G*p = G*F*H*p, F*p = G*F*P*p , where H*p = ¬P*¬p

5. Safety and Liveness Properties
Proof of decomposition theorem:
φs={w0w1... | for every i, w0w1... wi is a prefix of φ}
φl= φ{w0w1... | for some i, w0w1... wi is not a prefix of φ}
show: φs is safety, φl is liveness, φ = φs  φl

6. Examples (p U+ q) = ((p W+ q)  F+ q) G*(p  F*q) = (G*p  G*F*q)
G*p  G*q = G*(H*p  H*q) (gilt nur initial, im Anfangspunkt!)
Total program correctness = invariance  termination
other direction does not hold