Automatic Rectangular Refinement of Affine Hybrid Automata

1. Automatic Rectangular Refinement of Affine Hybrid Automata Laurent Doyen ULB Jean-François Raskin ULB Tom Henzinger EPFL FORMATS 2005 – Sep 27th - Uppsala

2. Overview • Automatic analysis of affine hybrid systems

3. Overview { Navigation Benchmark • Automatic analysis of affine hybrid systems • Example:

4. Overview { • Automatic analysis of affinehybrid systems • Example: Two trajectories

5. Overview { Navigation Benchmark • Automatic analysis of affine hybrid systems • Example: Affine dynamics

6. Overview { • Automatic analysis of affine hybrid systems • Example: B 2 4 4 3 4 2 A 2 Discrete states Affine dynamics +

7. Reminder • Some classes of hybrid automata: • Timed automata ( ) • Rectangular automata ( ) • Linear automata ( )

8. Reminder Limit for decidability of Language Emptiness • Some classes of hybrid automata: • Timed automata ( ) • Rectangular automata ( ) • Linear automata ( )

9. Reminder Limit for decidability of Language Emptiness • Some classes of hybrid automata: • Timed automata ( ) • Rectangular automata ( ) • Linear automata ( ) • Affine automata ( ) • Polynomial automata ( ) • etc.

10. Reminder Limit for symbolic computation of Post with HyTech Limit for decidability of Language Emptiness • Some classes of hybrid automata: • Timed automata ( ) • Rectangular automata ( ) • Linear automata ( ) • Affine automata ( ) • Polynomial automata ( ) • etc.

11. Methodology • Affine automaton A and set of states Bad • Check that Reach(A)  Bad = Ø

12. Methodology • Affine automaton A and set of states Bad • Check that Reach(A)  Bad = Ø • Affine dynamics is too complex ? • Abstract it !

13. Methodology • Affine dynamics is too complex ? • Abstract it ! • Abstraction is too coarse ? • Refine it ! • Affine automaton A and set of states Bad • Check that Reach(A)  Bad = Ø HOW ?

14. Methodology • 1. Abstraction: over-approximation Affine dynamics Rectangular dynamics

15. Methodology • 1. Abstraction: over-approximation Affine dynamics Rectangular dynamics { Let Then

16. Methodology l 3 0 • 2. Refinement: split locations by a line cut Line l

17. Methodology l 3 0 • 2. Refinement: split locations by a line cut Line l

18. Methodology Abstract Reach(A’)Bad Ø ? = Original Automaton A A’ Yes Property verified

19. Methodology Abstract Reach(A’)Bad Ø ? = Original Automaton A A’ Yes (Undecidable) Property verified

20. Methodology Refine Abstract Reach(A’)Bad Ø ? = Original Automaton A • using Reach(A’) • using Pre*(Bad) A’ No Yes (Undecidable) Property verified

21. Refinement • 2. Refinement: split locations by a line cut • Which location(s) ? • Loc1 = Locations reachable in the last step • Loc2 = Reachable locations that can reach Bad • Better: replace the state space by Loc2

22. Refinement • 2. Refinement: split locations by a line cut • Which location(s) ? • Loc1 = Locations reachable in the last step • Loc2 = Reachable locations that can reach Bad • Better: replace the state space by Loc2 • Which line cut ? • The best cut for some criterion characterizing the goodness of the resulting approximation.

23. Notations

24. Notations

25. Notations l A P+ B P-

26. Notations l A P+ B P-

27. Goodness of a cut • A good cut should minimize • ?

28. Goodness of a cut • A good cut should minimize • ? • ?

29. Goodness of a cut • A good cut should minimize • ? • ? • ? • …

30. Goodness of a cut • A good cut should minimize • ? • ? • ? • … Our choice

31. Finding the optimal cut P

32. Extremal level sets of f(x,y) P

33. Extremal level sets of g(x,y) P

34. Example Assume P

35. Example Assume P Then any line separating { } and { } is better than any other line.

36. Example P

37. Example P Any line separating { } and { } is better than any other line.

38. Example P Any line separating { } and { } is better than any other line.

39. Example P Thus, for every the best line separates and

40. Example P Thus, for every the best line separates and

41. Example P Thus, for every the best line separates and

42. Example P Thus, for every the best line separates and

43. Example P When

44. Example the best line cut must separate both from and from P When

45. Example The best line cut must separate both from and from P

46. Example Intersection P When an intersection occurs… The process continues because it is still possible to separate both from and from

47. Example P

48. Example P

49. Example P

50. Example P