1. Discrete Abstractions of Hybrid Systems Rajeev Alur, Thomas A. Henzinger, Gerardo Lafferriere and George J. Pappas

2. Overview Introduction Decidability Abstractions Questions

3. Introduction Abstract HS to purely discrete systems, while preserving all properties that are definable in temporal logic many safety critical applications formal analysis is important

4. Introduction Computational procedure (verifies in a finite number of steps whether the system satisfies the specification or not) Given:Desired: Property Hybrid System

5. Terminology Transition system T: –graph with possibly infinite number of nodes (> states) and edges (> transitions) Reachability problem: –given a transition system T and a property p, does the set of reachable states of T contain any states that satisfy p?

6. Undecidability obstacles Checking reachability is undecidable for a very simple class of HS –> more general classes cannot have finite bisimulation or language equivalent quotients –> continuous behaviour must be restricted –> discrete behaviour must be restricted

7. Abstraction properties about the behavior of a system over time are naturally expressible in temporal logics linear temporal logic (LTL) computation tree logic (CTL)

8. Linear temporal logic (LTL) Preserving LTL-properties leads to special partitions of the state space given by language equivalence relations T satisfies an LTL formula f T/ ~L satisfies f

9. Computation tree logic (CTL) CTL-properties are abstracted by bisimulations T satisfies an CTL formula f T/ ~B satisfies f

10. Undecidability barriers initialization is necessary variables must be decoupled consider HS with either: - simpler discrete dynamics or - simpler continuous dynamics

11. Restricted continuous dynamics A. Classes that admit finite bisimulation quotients B. Classes that admit finite language- equivalence quotients Timed automata Initialized multirate automata Rectangular automata

12. Restricted discrete dynamics Crucial to have FINITE partitions Restriction to classes with global finiteness properties -> o-minimal structures

13. O-minimal theories a theory of the reals is called o- minimal if every definable subset of the reals is a FINITE union of points and intervals –cell decomposition theorem: every definable set has a finite, definable partition of cells

14. O-minimal HS the continuous state lives in Rn for each discrete state, the flow of the vector field is complete for each discrete state, all relevant sets and the flow of the vector field are definable in the same o-minimal theory

15. O-minimal HS main theorem: –every o-minimal hybrid system admits a FINITE BISIMULATION –> bisimulation algorithm terminates for o-minimal hybrid systems