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Formal methods & Tools UCb CUPPAAL CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata Gerd Behrman, Ed Brinksma, Ansgar Fehnker, Thomas Hune, Kim Larsen, Paul Pettersson, Judi Romijn, Frits Vaandrager

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VHS meeting 27.11.00Kim G. Larsen UCb 2 Overview 1.Introduction 2.Linear Priced Timed Automata 3.Priced Zones and Facets 4.Operations on Priced Zones 5.Algorithm 6.First Experimental Findings 7.Conclusion

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VHS meeting 27.11.00Kim G. Larsen UCb 3 Observation Many scheduling problems can be phrased naturally as reachability problems for timed automata! INTRODUCTION

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VHS meeting 27.11.00Kim G. Larsen UCb 4 Observation Many scheduling problems can be phrased naturally as reachability problems for timed automata! UNSAFE SAFE 510 20 25 At most 2 crossing at a time Need torch At most 2 crossing at a time Need torch Mines Can they make it within 60 minutes ? Can they make it within 60 minutes ? INTRODUCTION

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VHS meeting 27.11.00Kim G. Larsen UCb 5 Observation Many scheduling problems can be phrased naturally as reachability problems for timed automata! UNSAFE SAFE 510 20 25 Mines INTRODUCTION

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VHS meeting 27.11.00Kim G. Larsen UCb 6 Steel Production Plant Machine 1 Machine 2Machine 3 Machine 4Machine 5 Buffer Continuos Casting Machine Storage Place Crane B Crane A zA. Fehnker, T. Hune, K. G. Larsen, P. Pettersson zCase study of Esprit-LTR project 26270 VHS zPhysical plant of SIDMAR located in Gent, Belgium. zPart 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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VHS meeting 27.11.00Kim G. Larsen UCb 7 Batch Processing Plant (VHS) hbrine water store mbrine heat water heater cooling water pump cooling water water salt INTRODUCTION

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VHS meeting 27.11.00Kim G. Larsen UCb 8 Earlier work zAsarin & Maler (1999) Time optimal control using backwards fixed point computation zVHS consortium (1999) Steel plant and chemical batch plant case studies zNiebert, Tripakis & Yovine (2000) Minimum-time reachability using forward reachability zBehrmann, Fehnker et all (2000) Minimum-time reachability using branch-and-bound INTRODUCTION

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VHS meeting 27.11.00Kim G. Larsen UCb 9 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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VHS meeting 27.11.00Kim G. Larsen UCb 10 More general cost function zIn scheduling theory one is not just interested in shortest schedules; also other cost functions are considered zThis leads us to introduce a model of linear priced timed automata which adds prices to locations and transitions zThe 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

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Formal methods & Tools UCb Linearly Priced Timed Automata

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VHS meeting 27.11.00Kim G. Larsen UCb 12 Example PRICED AUTOMATA

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VHS meeting 27.11.00Kim G. Larsen UCb 13 EXAMPLE : Optimal rescue plan for important persons (Presidents and Actors) UNSAFE SAFE 510 20 25 Mines GORECLINTON BUSH DIAZ 9 2 310 OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65! PRICED AUTOMATA

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VHS meeting 27.11.00Kim G. Larsen UCb 14 Definition PRICED AUTOMATA

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VHS meeting 27.11.00Kim G. Larsen UCb 15 Definition PRICED AUTOMATA

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VHS meeting 27.11.00Kim G. Larsen UCb 16 Example of execution PRICED AUTOMATA

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VHS meeting 27.11.00Kim G. Larsen UCb 17 Cost zThe cost of a finite execution is the sum of the prices of all the transitions occuring in it zThe minimal cost of a location is the infimum of the costs of the finite executions ending in the location zThe minimum-cost problem for LPTAs is the problem to compute the minimal cost of a given location of a given LPTA zIn the example below, mincost(C ) = 7 PRICED AUTOMATA ? DECIDABILITY ?

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Formal methods & Tools UCb Priced Zones

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VHS meeting 27.11.00Kim G. Larsen UCb 19 Zones Operations PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 20 Canonical Datastructure for Zones Difference Bounded Matrices x1-x2<=4 x2-x1<=10 x3-x1<=2 x2-x3<=2 x0-x1<=3 x3-x0<=5 x1-x2<=4 x2-x1<=10 x3-x1<=2 x2-x3<=2 x0-x1<=3 x3-x0<=5 x1x2 x3x0 -4 10 2 2 5 3 x1x2 x3x0 -4 4 2 2 5 33 -2 1 Shortest Path Closure O(n^3) Bellman’58, Dill’89 PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 21 New Canonical Datastructure Minimal collection of constraints x1-x2<=4 x2-x1<=10 x3-x1<=2 x2-x3<=2 x0-x1<=3 x3-x0<=5 x1-x2<=4 x2-x1<=10 x3-x1<=2 x2-x3<=2 x0-x1<=3 x3-x0<=5 x1x2 x3x0 -4 10 2 2 5 3 x1x2 x3x0 -4 4 2 2 5 3 x1x2 x3x0 -4 2 2 3 3 -2 1 Shortest Path Closure O(n^3) Shortest Path Reduction O(n^3) 3 Space worst O(n^2) practice O(n) RTSS 1997 PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 22 Priced Zone PRICED ZONES x y 4 2 Z

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VHS meeting 27.11.00Kim G. Larsen UCb 23 Reset x y 4 2 Z PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 24 Reset x y 4 2 Z {y}Z PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 25 Reset x y 4 2 Z {y}Z4 PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 26 Reset x y 4 2 Z {y}Z4 1 PRICED ZONES 2 A split of {y}Z 4

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VHS meeting 27.11.00Kim G. Larsen UCb 27 Facets The solution PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 28 OPERATIONS ON PZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 29 Delay x y 4 3 Z PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 30 Delay x y 4 3 Z Delay in a location with cost-rate 3 3 2 PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 31 Delay x y 4 3 Z 3 4 0 PRICED ZONES A split of

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VHS meeting 27.11.00Kim G. Larsen UCb 32 Facets The solution PRICED ZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 33 OPERATIONS ON PZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 34 Optimal Forward Reachability Example PRICED ZONES 10 0 0 2 4 6 8 2468 2 4 6 8 4682 11111

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VHS meeting 27.11.00Kim G. Larsen UCb 35 OPERATIONS ON PZONES

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VHS meeting 27.11.00Kim G. Larsen UCb 36 OPERATIONS ON PZONES

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Formal methods & Tools UCb Algorithm

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VHS meeting 27.11.00Kim G. Larsen UCb 38 Branch & Bound Algorithm ALGORITHM

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VHS meeting 27.11.00Kim G. Larsen UCb 39 ALGORITHM

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VHS meeting 27.11.00Kim G. Larsen UCb 40 ALGORITHM

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Formal methods & Tools UCb Experiments

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VHS meeting 27.11.00Kim G. Larsen UCb 42 EXAMPLE : Optimal rescue plan for important persons (Presidents and Actors) UNSAFE SAFE 510 20 25 Mines GORECLINTON BUSH DIAZ 9 2 310 OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65! EXPERIMENTS

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VHS meeting 27.11.00Kim G. Larsen UCb 43 Experiments MC Order COST-rates SCHEDULE COSTTIME #Expl#Pop’d G5G5 C 10 B 20 D 25 Min Time CG> G C 60 1762 1538 2638 1111 CG> G G 5565252378 92310 GD> G G 19565149233 1234 CG> G C 14060232350 12310 CD> C C 17065263408 1203040 BD> B C 975 1085 85 time<85 -- 0000 - 0-406447 EXPERIMENTS

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VHS meeting 27.11.00Kim G. Larsen UCb 44 Optimal Broadcast Router1 Router2 Router3 Router4 A B Given particular subscriptions, what is the cheapest schedule for broadcasting k? Given particular subscriptions, what is the cheapest schedule for broadcasting k? k=1k=0 costA 1, costB 1 costA 2, costB 2 costA 3, costB 3 costA 4, costB 4 Basecost EXPERIMENTS costB 1 costA 1 3 sec 5 sec

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VHS meeting 27.11.00Kim G. Larsen UCb 45 Experimental Results COST-rates SCHEDULE COSTTIME #Expl BCR1R1 R2R2 R3R3 R4R4 Min Time 1>3(B) ; ( 3>4(B) | 1>2(A) ) 81016 0 1:31:31:31:31:31:31:31:3 1>4(A) ; 3>4(A) ; 4>2(A) 15 2982 3 1>3(B) ; ( 3>4(B) | 1>2(A) ) 4781794 0 10 :30 5 :15 1:31:36:26:2 1>3(A) ; 3>2(A) ; 3>4(A) 6015665 3 1>4(A) ; 4>3(B) ; 4>2(B) 9511571 100 1>4(B) ; ( 1>3(A) | 4>2(B) ) 94681471 0 t<=10 1>4(B) ; 4>2(B) ; 4>3(B) 10291167 0 t<=8 1>4(B) ; ( 1>3(A) | 4>2(B) ) 14681688 EXPERIMENTS

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VHS meeting 27.11.00Kim G. Larsen UCb 46 Scaling Up ? z# Schedules x4 routers: 120 x5 routers: 83.712 x6 routers: ?????????? zFinding Feasible Schedule using UPPAAL (6 routers) x16.490 expl. symb. st. (with Active Clock Reduction) zMinimum Time Schedule (6 routers) x96.417 using Minimum Time Reachability (Ansgar) x106.628 using Minimum Cost Reachability (BC=1, all other cost=0) time optimal schedule takes 12 seconds. EXPERIMENTS

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VHS meeting 27.11.00Kim G. Larsen UCb 47 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

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