CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata
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Transcript of CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata
Formalmethods& Tools
UCb
CUPPAALCUPPAALEfficient Minimum-Cost Reachabilityfor Linearly Priced Timed Automata
Gerd Behrman, Ed Brinksma, Ansgar Fehnker, Thomas Hune, Kim Larsen, Paul Pettersson,
Judi Romijn, Frits Vaandrager
2VHS meeting 27.11.00 Kim G. Larsen
UCb
Overview
1. Introduction2. Linear Priced Timed Automata3. Priced Zones and Facets4. Operations on Priced Zones5. Algorithm6. First Experimental Findings7. Conclusion
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UCb Observation
Many scheduling problems can be phrased naturally asreachability problems for timed automata!
INTRODUCTION
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UCb Observation
Many scheduling problems can be phrased naturally asreachability problems for timed automata!
UNSAFE SAFE
5 10 20 25
At most 2crossing at a timeNeed torch
At most 2crossing at a timeNeed torch
Mines
Can they makeit within 60 minutes ?
Can they makeit within 60 minutes ?
INTRODUCTION
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UCb Observation
Many scheduling problems can be phrased naturally asreachability problems for timed automata!
UNSAFE SAFE
5 10 20 25
Mines
INTRODUCTION
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Steel Production PlantMachine 1 Machine 2 Machine 3
Machine 4 Machine 5
Buffer
Continuos Casting Machine
Storage Place
Crane B
Crane A
A. Fehnker, T. Hune, K. G. Larsen, P. Pettersson
Case study of Esprit-LTRproject 26270 VHS
Physical plant of SIDMARlocated in Gent, Belgium.
Part between blast furnace and hot rolling mill.
Objective: model the plant, obtain schedule and control program for plant.
Lane 1
Lane 2
INTRODUCTION
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Batch Processing Plant (VHS)
hbrine
hbrine
water
store
mbrine
heat
water
waterheater
cooling water
pump pump cooling water
watersalt
INTRODUCTION
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Earlier work
Asarin & Maler (1999)Time optimal control using backwards fixed point computation
VHS consortium (1999)Steel plant and chemical batch plant case studies
Niebert, Tripakis & Yovine (2000)Minimum-time reachability using forward reachability
Behrmann, Fehnker et all (2000)Minimum-time reachability using branch-and-bound
INTRODUCTION
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Advantages• Easy and flexible modeling of systems• whole range of verification techniques becomes available• Controller/Program synthesis
Disadvantages• Existing scheduling approaches perform somewhat better
Our goal• See how far we get;• Integrate model checking and scheduling theory.
INTRODUCTION
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More general cost function
In scheduling theory one is not just interested in shortest schedules; also other cost functions are considered
This leads us to introduce a model of linear priced timed automata which adds prices to locations and transitions
The price of a transition gives the cost of taking it, and the price of a location specifies the cost per time unit of staying there.
INTRODUCTION
Formalmethods& Tools
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Linearly Priced Timed Automata
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Example
PRICED AUTOMATA
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UCb EXAMPLE: Optimal rescue plan for important persons (Presidents and Actors)
UNSAFE
SAFE
5 10 20 25
Mines
GORE CLINTON
BUSH DIAZ
9 2
3 10
OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65!
PRICED AUTOMATA
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Definition
PRICED AUTOMATA
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Definition
PRICED AUTOMATA
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Example of execution
PRICED AUTOMATA
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Cost The cost of a finite execution is the sum of the prices of all the transitions
occuring in it
The minimal cost of a location is the infimum of the costs of the finite executions ending in the location
The minimum-cost problem for LPTAs is the problem to compute the minimal cost of a given location of a given LPTA
In the example below, mincost(C ) = 7
PRICED AUTOMATA
? DECIDABILITY ?
Formalmethods& Tools
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Priced Zones
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UCb Zones
Operations
PRICED ZONES
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UCb Canonical Datastructure for Zones
Difference Bounded Matrices
x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1 x2
x3x0
-4
10
22
5
3
x1 x2
x3x0
-4
4
22
5
3 3 -2 -2
1
ShortestPath
ClosureO(n^3)
Bellman’58, Dill’89
PRICED ZONES
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UCb New Canonical Datastructure Minimal collection of constraints
x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1 x2
x3x0
-4
10
22
5
3
x1 x2
x3x0
-4
4
22
5
3
x1 x2
x3x0
-4
22
3
3 -2 -2
1
ShortestPath
ClosureO(n^3)
ShortestPath
ReductionO(n^3) 3 Space worst O(n^2)
practice O(n)
RTSS 1997
PRICED ZONES
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UCb Priced Zone
PRICED ZONES
x
y
4
2-1
Z
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Reset
x
y
4
2-1
Z
PRICED ZONES
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Reset
x
y
4
2-1
Z
{y}Z
PRICED ZONES
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Reset
x
y
4
2-1
Z
{y}Z4
PRICED ZONES
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Reset
x
y
4
2-1
Z
{y}Z4
-1 1
PRICED ZONES
2
A split of {y}Z
4
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UCb FacetsThe solution
PRICED ZONES
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UCb OPERATIONS ON PZONES
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Delay
x
y
4
3-1
Z
Z
PRICED ZONES
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Delay
x
y
4
3-1
Z
Z
Delay in alocation withcost-rate 3
3
2
PRICED ZONES
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Delay
x
y
4
3-1
Z 3
4
-10
PRICED ZONES
A split of
Z
Z
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UCb FacetsThe solution
PRICED ZONES
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UCb OPERATIONS ON PZONES
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UCb Optimal Forward ReachabilityExample
PRICED ZONES
10
10
0
10
10
0
2
4
6
8
10
10
2 4 6 8
10
10
10
10 10
10
24
68
468 2
1 1 1 1 1
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UCb OPERATIONS ON PZONES
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UCb OPERATIONS ON PZONES
Formalmethods& Tools
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Algorithm
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UCb
Branch & Bound Algorithm
ALGORITHM
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UCb ALGORITHM
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UCb ALGORITHM
Formalmethods& Tools
UCb
Experiments
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UCb EXAMPLE: Optimal rescue plan for important persons (Presidents and Actors)
UNSAFE
SAFE
5 10 20 25
Mines
GORE CLINTON
BUSH DIAZ
9 2
3 10
OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65!
EXPERIMENTS
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UCb Experiments MC Order
COST-rates
SCHEDULE COST
TIME #Expl#Pop’
dG5 C10 B20 D25
Min Time CG> G< BD> C< CG> 60 1762
15382638
1 1 1 1 CG> G< BG> G< GD> 55 65 252 378
9 2 3 10 GD> G< CG> G< BG> 195 65 149 233
1 2 3 4 CG> G< BD> C< CG> 140 60 232 350
1 2 3 10 CD> C< CB> C< CG> 170 65 263 408
1 20 30 40 BD> B< CB> C< CG>
9751085
85time<85
- -
0 0 0 0 - 0 - 406 447
EXPERIMENTS
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UCb Optimal Broadcast
Router1 Router2
Router3 Router4
A
B
Given particular subscriptions, what is the cheapestschedule for broadcasting k?
Given particular subscriptions, what is the cheapestschedule for broadcasting k?
k=1 k=0
k=0 k=0
costA1, costB1 costA2, costB2
costA3, costB3costA4, costB4
Basecost
EXPERIMENTS
costB1costA1
3 sec
5 sec
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UCb Experimental ResultsCOST-rates
SCHEDULE COST TIME #ExplBC R1 R2 R3 R4
Min Time 1>3(B) ; ( 3>4(B) | 1>2(A) ) 8 1016
01:3
1:3
1:3
1:3
1>4(A) ; 3>4(A) ; 4>2(A) 15 15 2982
3 1>3(B) ; ( 3>4(B) | 1>2(A) ) 47 8 1794
0
10 :30
5 :15
1:3
6:2
1>3(A) ; 3>2(A) ; 3>4(A) 60 15 665
3 1>4(A) ; 4>3(B) ; 4>2(B) 95 11 571
100 1>4(B) ; ( 1>3(A) | 4>2(B) ) 946 8 1471
0t<=1
0
1>4(B) ; 4>2(B) ; 4>3(B) 102 9 1167
0t<=8
1>4(B) ; ( 1>3(A) | 4>2(B) ) 146 8 1688
EXPERIMENTS
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Scaling Up ?
# Schedules4 routers: 1205 routers: 83.7126 routers: ??????????
Finding Feasible Schedule using UPPAAL (6 routers)
16.490 expl. symb. st. (with Active Clock Reduction)
Minimum Time Schedule (6 routers)96.417 using Minimum Time Reachability (Ansgar)106.628 using Minimum Cost Reachability (BC=1, all other
cost=0) time optimal schedule takes 12 seconds.
EXPERIMENTS
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UCb Current & Future Work
IMPLEMENTATION – thorough analysis Applications – (Gossing Girls, Production Plant) Generalization
Minimum Cost Reachability under timing constraints avoiding certain states
Minimum Time Reachability under cost constraints Maximum Cost between two types of states
Relationships to Reward Models
Parameterized Extension Extensions to Optimal Controllability