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CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata

Gerd Behrman , Ed Brinksma, Ansgar Fehnker , Tho ma s Hune , Kim Lars en , Paul Pet tersson , Judi Romijn , Frits Vaandrager. CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata. Overview . Introduction Linear Priced Timed Automata Priced Zones and Facets

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CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata

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  1. Gerd Behrman,Ed Brinksma, Ansgar Fehnker,Thomas Hune, Kim Larsen, PaulPettersson, Judi Romijn, Frits Vaandrager CUPPAALEfficient Minimum-Cost Reachabilityfor Linearly Priced Timed Automata

  2. Overview • Introduction • Linear Priced Timed Automata • Priced Zones and Facets • Operations on Priced Zones • Algorithm • First Experimental Findings • Conclusion

  3. INTRODUCTION Observation Many scheduling problems can be phrased naturally as reachability problems for timed automata!

  4. INTRODUCTION UNSAFE SAFE Mines 5 10 25 20 At most 2 crossing at a time Need torch Can they make it within 60 minutes ? Observation Many scheduling problems can be phrased naturally as reachability problems for timed automata!

  5. INTRODUCTION UNSAFE SAFE Mines 5 10 25 20 Observation Many scheduling problems can be phrased naturally as reachability problems for timed automata!

  6. INTRODUCTION Steel Production Plant 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. Machine 2 Machine 3 Machine 1 Lane 1 Machine 4 Machine 5 Lane 2 Buffer Crane B Storage Place Continuos Casting Machine

  7. INTRODUCTION hbrine water water mbrine salt store cooling water heat water water heater hbrine cooling water pump pump Batch Processing Plant (VHS)

  8. INTRODUCTION 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

  9. INTRODUCTION • 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.

  10. INTRODUCTION 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.

  11. Linearly Priced Timed Automata

  12. PRICED AUTOMATA Example

  13. PRICED AUTOMATA EXAMPLE: Optimal rescue plan for important persons (Presidents and Actors) UNSAFE GORE CLINTON SAFE Mines 9 2 5 10 25 20 BUSH DIAZ 3 10 OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65!

  14. PRICED AUTOMATA Definition

  15. PRICED AUTOMATA Definition

  16. PRICED AUTOMATA Example of execution

  17. PRICED AUTOMATA 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 ? DECIDABILITY ?

  18. Priced Zones

  19. PRICED ZONES Zones Operations

  20. PRICED ZONES Canonical Datastructure for ZonesDifference Bounded Matrices Bellman’58, Dill’89 -4 -4 x1-x2<=4 x2-x1<=10 x3-x1<=2 x2-x3<=2 x0-x1<=3 x3-x0<=5 Shortest Path Closure O(n^3) x1 x2 x1 x2 4 10 3 3 2 3 2 -2 -2 2 2 x0 x3 x0 x3 1 5 5

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

  22. PRICED ZONES Priced Zone y Z 2 -1 4 x

  23. PRICED ZONES Reset Z y 2 -1 4 x

  24. PRICED ZONES Reset Z y 2 -1 4 x {y}Z

  25. PRICED ZONES Reset Z y 2 -1 4 x 4 {y}Z

  26. PRICED ZONES Reset Z y 2 -1 4 -1 1 x 4 2 {y}Z 4 A split of {y}Z

  27. PRICED ZONES FacetsThe solution

  28. OPERATIONS ON PZONES

  29. PRICED ZONES Delay y Z 3 -1 4 x

  30. PRICED ZONES Delay Delay in a location with cost-rate 3 3 y Z 2 3 -1 4 x

  31. PRICED ZONES Delay 4 -1 y 0 Z A split of 3 3 -1 4 x

  32. PRICED ZONES FacetsThe solution

  33. OPERATIONS ON PZONES

  34. PRICED ZONES Optimal Forward ReachabilityExample 8 6 10 4 10 2 0 0 10 10 10 2 4 6 8 10 10 10 1 1 1 1 1 8 6 4 2 8 10 10 6 4 2 10 10

  35. OPERATIONS ON PZONES

  36. OPERATIONS ON PZONES

  37. Algorithm

  38. ALGORITHM Branch & Bound Algorithm

  39. ALGORITHM

  40. ALGORITHM

  41. Experiments

  42. EXPERIMENTS EXAMPLE: Optimal rescue plan for important persons (Presidents and Actors) UNSAFE GORE CLINTON SAFE Mines 9 2 5 10 25 20 BUSH DIAZ 3 10 OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65!

  43. EXPERIMENTS Experiments MC Order

  44. EXPERIMENTS Optimal Broadcast Router2 Router1 k=1 k=0 costA1, costB1 costA2, costB2 B 3 sec Basecost 5 sec A costA4, costB4 costA3, costB3 k=0 k=0 costB1 costA1 Router4 Router3 Given particular subscriptions, what is the cheapest schedule for broadcasting k?

  45. EXPERIMENTS Experimental Results

  46. EXPERIMENTS Scaling Up ? • # Schedules • 4 routers: 120 • 5 routers: 83.712 • 6 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.

  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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