automatic rectangular refinement of affine hybrid automata tom henzinger epfl laurent doyen ulb...
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![Page 1: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/1.jpg)
Automatic Rectangular Automatic Rectangular Refinement of Affine Hybrid Refinement of Affine Hybrid
AutomataAutomata
Tom HenzingerTom Henzinger
EPFLEPFL
Laurent DoyenLaurent Doyen
ULBULB
Jean-François RaskinJean-François Raskin
ULBULB
FORMATS 2005 – Sep 27FORMATS 2005 – Sep 27thth - Uppsala - Uppsala
![Page 2: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/2.jpg)
OverviewOverview
• Automatic analysis of affine hybrid systems
![Page 3: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/3.jpg)
OverviewOverview
• Automatic analysis of affine hybrid systems• Example:
Navigation Benchmark
vx
)( dvvAv {
![Page 4: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/4.jpg)
OverviewOverview
• Automatic analysis of affine hybrid systems• Example:
Two trajectories
vx
)( dvvAv {
![Page 5: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/5.jpg)
OverviewOverview
• Automatic analysis of affine hybrid systems• Example:
Navigation Benchmark
vx
)( dvvAv {Affine dynamics
![Page 6: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/6.jpg)
OverviewOverview
• Automatic analysis of affine hybrid systems• Example:
vx
)( dvvAv {Affine dynamicsDiscrete states
B 2
+
A2 2
4
44 3
![Page 7: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/7.jpg)
• Some classes of hybrid automata:– Timed automata ( )– Rectangular automata ( )– Linear automata ( )
ReminderReminder
1x][ ,bax
bxa ii ~
![Page 8: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/8.jpg)
• Some classes of hybrid automata:– Timed automata ( )– Rectangular automata ( )– Linear automata ( )
ReminderReminder
1x][ ,bax
bxa ii ~
Limit for decidability of Language Emptiness
![Page 9: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/9.jpg)
ReminderReminder
• Some classes of hybrid automata:– Timed automata ( )– Rectangular automata ( )– Linear automata ( )– Affine automata ( )– Polynomial automata ( )– etc.
1x][ ,bax
bxa ii ~ cxbxa iiii ~
cxxp ii ~),(
Limit for decidability of Language Emptiness
![Page 10: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/10.jpg)
ReminderReminder
• Some classes of hybrid automata:– Timed automata ( )– Rectangular automata ( )– Linear automata ( )– Affine automata ( )– Polynomial automata ( )– etc.
1x][ ,bax
bxa ii ~ cxbxa iiii ~
cxxp ii ~),(
Limit for symbolic computation of Post with HyTech
Limit for decidability of Language Emptiness
![Page 11: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/11.jpg)
MethodologyMethodology
• Affine automaton A and set of states Bad
• Check that Reach(A) Bad = Ø
![Page 12: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/12.jpg)
MethodologyMethodology
• Affine dynamics is too complex ?Abstract it !
• Affine automaton A and set of states Bad
• Check that Reach(A) Bad = Ø
![Page 13: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/13.jpg)
MethodologyMethodology
• Affine dynamics is too complex ?Abstract it !
• Abstraction is too coarse ? Refine it !
HOW ?
• Affine automaton A and set of states Bad
• Check that Reach(A) Bad = Ø
![Page 14: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/14.jpg)
MethodologyMethodology
Affine dynamics Rectangular dynamics
xx 2
30 x
• 1. Abstraction: over-approximation
![Page 15: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/15.jpg)
Let Then
MethodologyMethodology
Affine dynamics Rectangular dynamics
xx 2
30 x
]2,1[x
30 x
• 1. Abstraction: over-approximation
30Inv x)](max),([min InvInv]2,1[ xfxf xx
-x f(x) 2{
![Page 16: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/16.jpg)
xx 2
30 x
MethodologyMethodology
Line l
• 2. Refinement: split locations by a line cut
23x
l
0 3
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xx 2
30 x
323 x
MethodologyMethodology
Line l
230 x
• 2. Refinement: split locations by a line cut
23x xx 2
xx 2l
0 3
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MethodologyMethodology
Abstract
Reach(A’)Bad Ø?=
A’
A
Yes
Original Automaton
Property verified
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MethodologyMethodology
Abstract
Reach(A’)Bad Ø?=
A’
A
Yes
Original Automaton
(Undecidable)Property verified
![Page 20: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/20.jpg)
MethodologyMethodology
Abstract Refine
Reach(A’)Bad Ø?=
A’
A
No
Yes
Original Automaton
(Undecidable)Property verified
•using Reach(A’)•using Pre*(Bad)
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RefinementRefinement
• 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
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RefinementRefinement
• 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.
![Page 23: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/23.jpg)
NotationsNotations
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NotationsNotations
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NotationsNotations
l
P+
P-A
B
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NotationsNotations
l
P+
P-A
B
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• ?
Goodness of a cutGoodness of a cut
• A good cut should minimize
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• ?
• ?
Goodness of a cutGoodness of a cut
• A good cut should minimize
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• ?
• ?
• ?
• …
Goodness of a cutGoodness of a cut
• A good cut should minimize
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• ?
• ?
• ?
• …
Goodness of a cutGoodness of a cut
Our choice
• A good cut should minimize
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Finding the optimal cutFinding the optimal cut
P
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Extremal level sets of f(x,y)Extremal level sets of f(x,y)
P
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Extremal level sets of g(x,y)Extremal level sets of g(x,y)
P
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ExampleExample
P
Assume
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Then any line separating { } and { }
is better than any other line.
ExampleExample
P
Assume
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ExampleExample
P
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ExampleExample
Any line separating { } and { }
is better than any other line.
P
![Page 38: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/38.jpg)
ExampleExample
Any line separating { } and { }
is better than any other line.
P
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ExampleExample
Thus, for every
the best line separates and
P
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ExampleExample
Thus, for every
the best line separates and
P
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ExampleExample
Thus, for every
the best line separates and
P
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ExampleExample
Thus, for every
the best line separates and
P
![Page 43: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/43.jpg)
ExampleExample
P
When
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ExampleExample
P
When the best line cut must separate both
from and from
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ExampleExample
P
The best line cut must separate both
from and from
![Page 46: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/46.jpg)
ExampleExample
Intersection
P
The process continues because it is still possible to separate
both from and from
When an intersection occurs…
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ExampleExample
P
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ExampleExample
P
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ExampleExample
P
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ExampleExample
P
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ExampleExample
Intersection
P
When a second intersection occurs…
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ExampleExample
Intersection
P
In this case, we have reached the "limit of separability"
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ExampleExample
An optimal cut
P
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How to compute the How to compute the intersection ?intersection ?
P
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How to compute the How to compute the intersection ?intersection ?
P
We have to find the minimal such that:
(u,v)
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How to compute the How to compute the intersection ?intersection ?
P
We have to find the minimal such that:
This is a linear program !
(u,v)
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The algorithmThe algorithm
• Applies in the plane (2D)– Several particular cases
![Page 58: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/58.jpg)
The algorithmThe 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
![Page 59: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/59.jpg)
The algorithmThe 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
N.B.: it is possible to define a general algorithm in nD, but it requires to solve difficult
geometrical problems (parametric convex hulls).
![Page 60: Automatic Rectangular Refinement of Affine Hybrid Automata Tom Henzinger EPFL Laurent Doyen ULB Jean-François Raskin ULB FORMATS 2005 – Sep 27 th - Uppsala](https://reader038.vdocuments.site/reader038/viewer/2022110207/56649d4c5503460f94a2a707/html5/thumbnails/60.jpg)
The algorithmThe 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
N.B.: it is possible to define a general algorithm in nD, but it requires to solve difficult
geometrical problems (parametric convex hulls).
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Navigation benchmarkNavigation benchmark
In each location, the dynamics has the form:
We cut in the plane v1-v2
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Navigation benchmarkNavigation benchmark
In each location, the dynamics has the form:
We cut in the plane v1-v2
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ResultsResults
NAV 04 NAV 07
Initial states Bad states Good states
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Results: NAV 04Results: NAV 04
Forward Backward
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Results: NAV 04Results: NAV 04
Forward Forward
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Results: NAV 07Results: NAV 07
Backward
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ConclusionConclusion
• Approximations• Rectangular• Over-approximations
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ConclusionConclusion
• Approximations• Rectangular• Over-approximations
• Refinements• Automatic• Optimal split for some criterion (at least in 2D)
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ConclusionConclusion
• 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, …)
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ReferencesReferences
• [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.