Post on 21-Jan-2016
description
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
3VHS meeting 27.11.00 Kim G. Larsen
UCb Observation
Many scheduling problems can be phrased naturally asreachability problems for timed automata!
INTRODUCTION
4VHS meeting 27.11.00 Kim G. Larsen
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
5VHS meeting 27.11.00 Kim G. Larsen
UCb Observation
Many scheduling problems can be phrased naturally asreachability problems for timed automata!
UNSAFE SAFE
5 10 20 25
Mines
INTRODUCTION
6VHS meeting 27.11.00 Kim G. Larsen
UCb
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
7VHS meeting 27.11.00 Kim G. Larsen
UCb
Batch Processing Plant (VHS)
hbrine
hbrine
water
store
mbrine
heat
water
waterheater
cooling water
pump pump cooling water
watersalt
INTRODUCTION
8VHS meeting 27.11.00 Kim G. Larsen
UCb
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
9VHS meeting 27.11.00 Kim G. Larsen
UCb
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
10VHS meeting 27.11.00 Kim G. Larsen
UCb
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
UCb
Linearly Priced Timed Automata
12VHS meeting 27.11.00 Kim G. Larsen
UCb
Example
PRICED AUTOMATA
13VHS meeting 27.11.00 Kim G. Larsen
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
14VHS meeting 27.11.00 Kim G. Larsen
UCb
Definition
PRICED AUTOMATA
15VHS meeting 27.11.00 Kim G. Larsen
UCb
Definition
PRICED AUTOMATA
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UCb
Example of execution
PRICED AUTOMATA
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UCb
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
UCb
Priced Zones
19VHS meeting 27.11.00 Kim G. Larsen
UCb Zones
Operations
PRICED ZONES
20VHS meeting 27.11.00 Kim G. Larsen
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
21VHS meeting 27.11.00 Kim G. Larsen
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
22VHS meeting 27.11.00 Kim G. Larsen
UCb Priced Zone
PRICED ZONES
x
y
4
2-1
Z
23VHS meeting 27.11.00 Kim G. Larsen
UCb
Reset
x
y
4
2-1
Z
PRICED ZONES
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UCb
Reset
x
y
4
2-1
Z
{y}Z
PRICED ZONES
25VHS meeting 27.11.00 Kim G. Larsen
UCb
Reset
x
y
4
2-1
Z
{y}Z4
PRICED ZONES
26VHS meeting 27.11.00 Kim G. Larsen
UCb
Reset
x
y
4
2-1
Z
{y}Z4
-1 1
PRICED ZONES
2
A split of {y}Z
4
27VHS meeting 27.11.00 Kim G. Larsen
UCb FacetsThe solution
PRICED ZONES
28VHS meeting 27.11.00 Kim G. Larsen
UCb OPERATIONS ON PZONES
29VHS meeting 27.11.00 Kim G. Larsen
UCb
Delay
x
y
4
3-1
Z
Z
PRICED ZONES
30VHS meeting 27.11.00 Kim G. Larsen
UCb
Delay
x
y
4
3-1
Z
Z
Delay in alocation withcost-rate 3
3
2
PRICED ZONES
31VHS meeting 27.11.00 Kim G. Larsen
UCb
Delay
x
y
4
3-1
Z 3
4
-10
PRICED ZONES
A split of
Z
Z
32VHS meeting 27.11.00 Kim G. Larsen
UCb FacetsThe solution
PRICED ZONES
33VHS meeting 27.11.00 Kim G. Larsen
UCb OPERATIONS ON PZONES
34VHS meeting 27.11.00 Kim G. Larsen
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
35VHS meeting 27.11.00 Kim G. Larsen
UCb OPERATIONS ON PZONES
36VHS meeting 27.11.00 Kim G. Larsen
UCb OPERATIONS ON PZONES
Formalmethods& Tools
UCb
Algorithm
38VHS meeting 27.11.00 Kim G. Larsen
UCb
Branch & Bound Algorithm
ALGORITHM
39VHS meeting 27.11.00 Kim G. Larsen
UCb ALGORITHM
40VHS meeting 27.11.00 Kim G. Larsen
UCb ALGORITHM
Formalmethods& Tools
UCb
Experiments
42VHS meeting 27.11.00 Kim G. Larsen
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
43VHS meeting 27.11.00 Kim G. Larsen
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
44VHS meeting 27.11.00 Kim G. Larsen
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
45VHS meeting 27.11.00 Kim G. Larsen
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
46VHS meeting 27.11.00 Kim G. Larsen
UCb
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
47VHS meeting 27.11.00 Kim G. Larsen
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