Formal methods & Tools UCb CUPPAAL CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata Gerd Behrman, Ed Brinksma, Ansgar Fehnker,

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


UCb Observation

Many scheduling problems can be phrased naturally asreachability problems for timed automata!

INTRODUCTION


UCb Observation


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


UCb Observation


UNSAFE SAFE

5 10 20 25

Mines

INTRODUCTION


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


UCb

Batch Processing Plant (VHS)

hbrine

hbrine

water

store

mbrine

heat

water

waterheater

cooling water

pump pump cooling water

watersalt

INTRODUCTION


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


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


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


UCb

Linearly Priced Timed Automata


UCb

Example

PRICED AUTOMATA


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


UCb

Definition

PRICED AUTOMATA


UCb

Definition

PRICED AUTOMATA


UCb

Example of execution

PRICED AUTOMATA


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 ?


UCb

Priced Zones


UCb Zones

Operations

PRICED ZONES


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


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


UCb Priced Zone

PRICED ZONES

x

y

4

2-1

Z


UCb

Reset

x

y

4

2-1

Z

PRICED ZONES


UCb

Reset

x

y

4

2-1

Z

{y}Z

PRICED ZONES


UCb

Reset

x

y

4

2-1

Z

{y}Z4

PRICED ZONES


UCb

Reset

x

y

4

2-1

Z

{y}Z4

-1 1

PRICED ZONES

2

A split of {y}Z

4


UCb FacetsThe solution

PRICED ZONES


UCb OPERATIONS ON PZONES


UCb

Delay

x

y

4

3-1

Z

Z

PRICED ZONES


UCb

Delay

x

y

4

3-1

Z

Z

Delay in alocation withcost-rate 3

3

2

PRICED ZONES


UCb

Delay

x

y

4

3-1

Z 3

4

-10

PRICED ZONES

A split of

Z

Z


UCb FacetsThe solution

PRICED ZONES




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






UCb

Algorithm


UCb

Branch & Bound Algorithm

ALGORITHM


UCb ALGORITHM


UCb ALGORITHM


UCb

Experiments


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


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


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


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


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


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

Documents

Formal methods & Tools UCb CUPPAAL CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata Gerd Behrman, Ed Brinksma, Ansgar Fehnker,