Optimal Conditional Reachability for Priced Timed Automata

1
Optimal Conditional Reachability for Priced
Timed Automata

• Kim G. Larsen
• Jacob I. Rasmussen

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Real Time Scheduling

• Only 1 Pass
• Cheat is possible (drive close to car with
  Pass)

UNSAFE
Crossing Times
5
10
Pass
20
25
SAFE
CAN THEY MAKE IT TO SAFEWITHIN 70 MINUTES ???
The Car Bridge Problem
3
Real Time Scheduling
Solve Scheduling Problem using UPPAAL
SAFE
4
Optimal Reachability forPriced Timed Automata

• A short review

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(No Transcript)
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EXAMPLE Optimal rescue plan for cars with
different subscription rates for
city driving !
Golf
Citroen
SAFE
9
2
BMW
Datsun
3
10
OPTIMAL PLAN HAS ACCUMULATED COST195 and TOTAL
TIME65!
7
Priced Timed Automata
Behrmann, Fehnker, et all (HSCC01)
Timed Automata COST variable
Alur, Torre, Pappas (HSCC01)
l2
l1
l3
x 2 3 y
0 y 4
c4
c2
?
x0 c1
c4
cost rate
cost update
y 4
x0
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Priced Timed Automata
Behrmann, Fehnker, et all (HSCC01)
Timed Automata COST variable
Alur, Torre, Pappas (HSCC01)
l2
l1
l3
x 2 3 y
0 y 4
c4
c2
?
x0 c1
c4
cost rate
cost update
y 4
x0
TRACES
e(3)
(l1,xy0) (l1,xy3) (l2,x0,y3)
(l3,_,_)
12
1
4
å c17
9
Priced Timed Automata
Behrmann, Fehnker, et all (HSCC01)
Timed Automata COST variable
Alur, Torre, Pappas (HSCC01)
l2
l1
l3
x 2 3 y
0 y 4
c4
c2
?
x0 c1
c4
Problem Find the minimum cost of reaching
location l3
cost rate
cost update
y 4
x0
TRACES
e(3)
(l1,xy0) (l1,xy3) (l2,x0,y3)
(l3,_,_)
12
1
4
å c17
e(2.5)
e(.5)
(l1,xy0) (l1,xy2.5) (l2,x0,y2.5)
(l2,x0.5,y3) (l3,_,_)
10
1
1
4
å c16
e(3)
(l1,xy0) (l2,x0,y0) (l2,x3,y3)
(l2,x0,y3) (l3,_,_)
1
6
0
4
å c11
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Symbolic Semantics
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Zones
y
Operations
Z
x
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Priced Zone
CAV01
y
Z
2
-1
4
x
13
Reset
Z
y
2
2
-1
4
y0
x
14
Reset
Z
y
2
2
-1
4
y0
x
yZ
15
Reset
Z
y
2
2
-1
4
y0
x
yZ
6
16
Reset
Z
y
2
2
-1
4
y0
-1
1
x
yZ
6
4
A split of yZ
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Delay
y
Z
3
-1
4
x
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Delay
y
Z
3
-1
4
x
19
Delay
3
3
y
Z
2
3
-1
4
x
20
Delay
3
4
-1
y
0
Z
3
A split of
3
-1
4
x
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Branch Bound Algorithm
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Branch Bound Algorithm
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Branch Bound Algorithm
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Branch Bound Algorithm
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Branch Bound Algorithm
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Branch Bound Algorithm
Z is bigger cheaper than Z
is a well-quasi ordering which guarantees
termination!
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Optimal Conditional Reachability
To be presented at FOSSACS05
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CONDITIONAL
EXAMPLE Optimal rescue plan for cars with
different subscription rates for
city driving !
Golf
Citroen
SAFE
9
2
BMW
Datsun
3
10
min CostMYCAR 270 time 70
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Optimal Conditional Reachability
Dual-priced TA
l2
l1
l3
x 2 y 1
d1
c 1 d 4
c 2 d 1
?
y0
y0
x 2
x 3 y 2
PROBLEM Reach l3 in a way which
minimizes c subject to d 4
SOLUTION c 11/3 ? wait 1/3 in l1
goto l2 wait 5/3 in l2 goto l3
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Discrete Trajectories
0,1
l1,0,0
l2,0,0
l2,1,0
l2,2,0
1,4
2,1
2,1
2,1
0,1
l2,3,1
l1,1,1
l2,1,1
l2,2,1
2,1
1,4
0,0
2,1
0,0
0,1
l1,2,2
l2,2,2
l2,3,2
0,0
0,0
?
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Discrete Trajectories
0,1
l1,0,0
l2,0,0
l2,1,0
l2,2,0
1,4
2,1
2,1
2,1
0,1
l2,3,1
l1,1,1
l2,1,1
l2,2,1
2,1
1,4
0,0
2,1
d
0,0
0,1
l1,2,2
l2,2,2
l2,3,2
10
8
6
4
0,0
2
0,0
?
c
4
2
6
4
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Dual Priced Zones
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Dual Priced Zones
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Dual Priced Zones
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Reset
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Reset
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Reset
38
Reset
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Key Lemma
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b1
b2
0.8
a1
0.3
a2
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b1
b2
a1
a2
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b1
b2
a1
a2
43
a2
a1
b1
b2
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a2
a1
b1
b2
45
not convex !!!
46
Exploration
l2
l1
l3
x 2 y 1
d1
c 1 d 4
c 2 d 1
?
y0
y0
x 3 y 2
x 2
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Exploration
l2
l1
l3
x 2 y 1
d1
c 1 d 4
c 2 d 1
?
y0
y0
x 3 y 2
x 2
48
Exploration
l2
l1
l3
x 2 y 1
d1
c 1 d 4
c 2 d 1
?
y0
y0
x 3 y 2
x 2
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Termination
THEOREM Optimal conditional reachability for
multi-priced TA is computable.