1. Model Checking Anders P. Ravn Department of Computer Science, Aalborg University, Denmark Hybrid Systems – PhD School Aalborg University January 2007

2. A Logic Syntax p  L Model Theory (semantics) M |= p - meaning, mathematical objects; [p] = {M | M |=p} Proof Theory (axioms, deduction rules) - |- p (axiom) - p 1  p 2 (rewriting rule) - If |-p 1,…,|-p n then |- p c A deduction: p 1,…,p n |- p c

3. Symbolic Analysis Q - states Q 0 – initial states, … A - labels, …  - transition relation, A  Q  Q a Theory: T = {p 1, … p n … }, p is a predicate, e.g. pred(X  V) Meaning of p: [p]  Q q 1  q 2 iff p(q 1 ) = r(q 2 ) for all p, r  T

4. Verification Tasks Reachability of (v,x) – finitary, time-abstract trace inclusion Emptiness – time-abstract trace inclusion Trace (finitary) inclusion Time-abstract (finitary) trace inclusion

5. Exercises Work some examples Check the theorems and remarks Experiment with tools Investigate links with equivalences generated by Rafael’s homotopy (di-paths) Compositionality, remarks on p. 7, 10, 17 – compositional model checking, abstraction-refinement Build your own HA Tool!