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Automatic Rectangular Refinement of Affine Hybrid Automata

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Automatic Rectangular Refinement of Affine Hybrid Automata. Tom Henzinger. EPFL. Laurent Doyen ... Automatic analysis of affine hybrid systems. Overview ... – PowerPoint PPT presentation

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Title: 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
  • Automatic analysis of affine hybrid systems
  • Example

4
Overview
  • Automatic analysis of affine hybrid systems
  • Example

Two trajectories
5
Overview
  • Automatic analysis of affine hybrid systems
  • Example

Affine dynamics
6
Overview
  • Automatic analysis of affine hybrid systems
  • Example

B
2
4
4
4
3
A
2
2
Affine dynamics
Discrete states

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

8
Reminder
  • Some classes of hybrid automata
  • Timed automata ( )
  • Rectangular automata ( )
  • Linear automata ( )

9
Reminder
  • Some classes of hybrid automata
  • Timed automata ( )
  • Rectangular automata ( )
  • Linear automata ( )
  • Affine automata ( )
  • Polynomial automata ( )
  • etc.

10
Reminder
  • 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 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
  • 2. Refinement split locations by a line cut

Line l ?
17
Methodology
  • 2. Refinement split locations by a line cut

Line l ?
18
Methodology
Original Automaton
A
A
Yes
Property verified
19
Methodology
Original Automaton
A
A
Yes
(Undecidable)
Property verified
20
Methodology
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
26
Notations
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
P
35
Example
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
P
When
45
Example
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
51
Example
P
Intersection
When a second intersection occurs
52
Example
P
Intersection
In this case, we have reached the "limit of
separability"
53
Example
P
An optimal cut
54
How to compute the intersection ?
P
55
How to compute the intersection ?
We have to find the minimal ? such that
P
(u,v)
56
How to compute the intersection ?
We have to find the minimal ? such that
P
(u,v)
This is a linear program !
57
The algorithm
  • Applies in the plane (2D)
  • Several particular cases

58
The algorithm
  • Applies in the plane (2D)
  • Several particular cases
  • What for higher dimension ?
  • An option discretize the problem using a grid
  • Apply a (more) discrete algorithm
  • The exact solution can be arbitrarily closely
    approximated

59
The algorithm
  • Applies in the plane (2D)
  • Several particular cases
  • What for higher dimension ?
  • An option discretize the problem using a grid
  • Apply a (more) discrete algorithm
  • The exact solution can be arbitrarily closely
    approximated

60
The algorithm
  • Applies in the plane (2D)
  • Several particular cases
  • What for higher dimension ?
  • An option discretize the problem using a grid
  • Apply a (more) discrete algorithm
  • The exact solution can be arbitrarily closely
    approximated

61
Navigation benchmark
  • In each location, the dynamics has the form
  • We cut in the plane v1-v2

62
Navigation benchmark
  • In each location, the dynamics has the form
  • We cut in the plane v1-v2

63
Results
  • NAV 07
  • NAV 04

64
Results NAV 04
  • Forward
  • Backward

65
Results NAV 04
  • Forward
  • Forward

66
Results NAV 07
  • Backward

67
Conclusion
  • Approximations
  • Rectangular
  • Over-approximations

68
Conclusion
  • Approximations
  • Rectangular
  • Over-approximations
  • Refinements
  • Automatic
  • Optimal split for some criterion (at least in 2D)

69
Conclusion
  • Approximations
  • Rectangular
  • Over-approximations
  • Refinements
  • Automatic
  • Optimal split for some criterion (at least in 2D)
  • Possible future work
  • Under-approximations
  • Optimal split for some other criterion
  • Combine with other approaches (barrier
    certificates, ellipsoïds, )

70
References
  • FI04 A. Fehnker and F. Ivancic. Benchmarks for
    hybrid systems verification. In HSCC 2004, LNCS
    2993, pp 326-341.
  • Fre05 G. Frehse. Phaver Algorithmic
    verification of hybrid systems past hytech. In
    HSCC 2005, LNCS 3414, pp 258-273.
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