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

2. Temporal Logic
An extension of propositional logic to reason about sequences (executions)
Derived from modal logic
“linguistic modality: expression of how the world might or should be. This includes expressions of necessity, permissibility and probability, and negations of these” (Wikipedia)
avoids the use of (logical) variables and quantification: “necessarily p” instead of “under all circumstances x, p holds at x”
Temporal modalities “always” and “sometime”
“always p” instead of “for all future time points x, p holds at x”
“sometime p” instead of “for some future time points x, p holds at x”
always p  not sometime not p not always p  sometime not p

3. Linear versus Branching Time
Several variants of temporal logic have been proposed
state-, transition-, or interval-based
linear or branching time
modal, star-free, or regular
propositional or first order
...
Standardized notations like IEEE 1850 PSL exist
Linear vs. branching: two ways to consider time in a computation
the set of executions versus
the execution tree of a program

4. Recap: Syntax & Semantics of FOL

5. Syntax & Semantics of Modal Logic

6. Special Relations

7. Until and Unless

8. Since

9. LTL Expressively equivalent to FOL (<) on linear orders
Arguably more intuitive to use

10. CTL

12. branching time – possibility properties
Kripke models = LTS
branching time – possibility properties
linear time – fairness properties
CTL* subsumes both CTL and LTL