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Automatic Rectangular Refinement of Affine Hybrid Automata

Automatic Rectangular Refinement of Affine Hybrid Automata. Laurent Doyen ULB. Jean-François Raskin ULB. Tom Henzinger EPFL. FORMATS 2005 – Sep 27 th - Uppsala. Overview. Automatic analysis of affine hybrid systems. Overview. {. Navigation Benchmark.

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Automatic Rectangular Refinement of Affine Hybrid Automata

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  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

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