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Reachability, Schedulability and Optimality

Reachability, Schedulability and Optimality. Ansgar Fehnker. June 3. Outline. Timed automata a la Uppaal From Reachability to Schedulability LPTAs Priced regions and operations Algorithm Termination Priced Zones Verification vs. Optimization Guiding and Bounding examples

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Reachability, Schedulability and Optimality

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  1. Reachability, Schedulability and Optimality Ansgar Fehnker June 3

  2. Outline • Timed automata a la Uppaal • From Reachability to Schedulability • LPTAs • Priced regions and operations • Algorithm • Termination • Priced Zones • Verification vs. Optimization • Guiding and Bounding • examples • examples

  3. Timed Automata (UPPAAL) • Network of Automata • Synchronization (CCS-like) a! a?

  4. Timed Automata (UPPAAL) • Network of Automata • Synchronization (CCS-like) • Clocks in description • Time passes uniformly • Guard/reset on action • Invariants on location x  7 3  x  7 y > 4 a! a? y:=0 Uppaal is a modelchecker forTimed Automata with emphasis on reachability properties

  5. 25min 20min 10min 5min Can they make it within 60 minutes ? Unsafe Safe Motivation Observation Many scheduling problems can be phrased in a natural way as reachability problems for timed automata!

  6. unsafe unsafe unsafe take? take? L==0 L==0 L==0 unsafe take! y:=0 take! y:=0 take! y:=0 y>=20 y>=10 y>=5 release? release? L==0 release! release! release! release! release! release! L:=1-L L==1 L==1 L==1 y>=25 take! y:=0 y>=25 y>=25 y>=25 take! y:=0 take! y:=0 take! y:=0 safe safe safe release! release! L==1 y>=25 take! y:=0 safe Motivation 25min 20min 10min 5min What schedule minimizes crossings? What is the fastest schedule? Can they make it within 60 minutes ? What schedule mini-mizes unsafe time? Unsafe Safe

  7. cost’=1 cost’=2 cost’=0 x<5 x<3 cost+=4 y>2 c a b y:=0 (2.5) (a,x=y=0) (b,x=y=0) (b,x=y=2) (a,x=0,y=2) 4 2.5 x 2 0 Linearly Priced Timed Automata • Timed Automata + Costs on transitions and locations. • Cost of performing transition: Transition cost. • Cost of performing delay d: ( d x location cost ). • Cost of Execution Trace: Sum of costs: 4 + 5 + 0 = 9

  8. cost E earliest landing time T target time L latest time ecost rate for being early l cost rate for being late dfixed cost for being late d+l*(t-T) e*(T-t) t E T L Example: Aircraft Landing Planes have to keep separation distance to avoid turbulences caused by preceding planes Runway

  9. Example: Aircraft Landing x <= 5 x=5 x >= 4 4 earliest landing time 5 target time 9 latest time 3cost rate for being early 1 cost rate for being late 2fixed cost for being late land! cost+=2 x <= 5 x <= 9 cost’=3 cost’=1 x=5 land! Planes have to keep separation distance to avoid turbulences caused by preceding planes Runway

  10. Symbolic semantics of Linearly Priced Timed Automata

  11. delay 1  y 0  x -2  x-y 0 Zones Basic idea: Define a delay and reset over zones 1  y  4 0  x  3 -2  x-y 0 y x x<3 x<3 y>2 c a b y:=0

  12. reset y 0  y  0 0  x  3 Zones Basic idea: Define a delay and reset over zones 1  y  4 0  x  3 -2  x-y 0 y x x<3 x<3 y>2 c a b y:=0

  13. cost’=1 cost’=2 cost’=0 cost+=4 3 -1 0 delay 2 cost=c’’ -1 x + 3 y cost=c’+ 0 x + 2 y cost=c - 1 x + 2 y Priced Zones Basic idea: Define a linear cost function on zones cost = c - 1 x + 2 y y 2 -1 x x<5 x<3 y>2 c a b y:=0

  14. reset y cost = c - 1 x cost = c’+ 1 x -1 1 Priced Zones Basic idea: Define a delay and reset over zones cost = c - 1 x + 2 y y 2 -1 x x<3 x<3 y>2 c a b y:=0

  15. State-Space Exploration Algorithm

  16. Cost=60 An Algorithm • State-Space Exploration + Use of global variable Cost. • Updated Cost whenever goal state with min( C ) <Cost is found: Cost= Cost=80 80 60

  17. An Algorithm Cost:=, Pass := {}, Wait := {(l0,C0)}, Goal= while Wait  {} do select (l,C) from Wait if (l,C)= and mincost(C)<Cost then Cost:=mincost(C) if forall (l’,C’) in Pass: C’ C then add (l,C) to Pass forall (m,D) such that (l,C) (m,D): add (m,D) to Wait Return Cost

  18. An Algorithm Cost:=, Pass := {}, Wait := {(l0,C0)}, Goal= while Wait  {} do select (l,C) from Wait if (l,C)= and mincost(C)<Cost then Cost:=mincost(C) if forall (l’,C’) in Pass: C’ C then add (l,C) to Pass forall (m,D) such that (l,C) (m,D): add (m,D) to Wait Return Cost Performs: symbolic operations Delay, Conjun-ction, and Reset of clocks.

  19. C C’ C’ isbigger & cheaper than C is a well-quasi ordering which guarantees termination! An Algorithm Cost:=, Pass := {}, Wait := {(l0,C0)}, Goal= while Wait  {} do select (l,C) from Wait if (l,C)= and mincost(C)<Cost then Cost:=mincost(C) if forall (l’,C’) in Pass: C’ C then add (l,C) to Pass forall (m,D) such that (l,C) (m,D): add (m,D) to Wait Return Cost .

  20. Theorem When the algorithm terminates, the value of COSTequals mincost(). An Algorithm Cost:=, Pass := {}, Wait := {(l0,C0)}, Goal= while Wait  {} do select (l,C) from Wait if (l,C)= and mincost(C)<Cost then Cost:=mincost(C) if forall (l’,C’) in Pass: C’ C then add (l,C) to Pass forall (m,D) such that (l,C) (m,D): add (m,D) to Wait Return Cost

  21. Efficient Reachability of LPTAs

  22. Verification vs. Optimization Safe side reachable? • Verification Algorithms: • Checks a logical property for the entire state-space • Efficient blind search. • Optimization Algorithms: • Finds (near) optimal solutions. • Uses techniques to avoid non-optimal parts of the state-space (e.g. Branch and Bound). • Objective: • Bridge the gap between these two. • New techniques and applications in UPPAAL. 80 Min time of reaching safe side? 60

  23. Minimum-Cost Order • The basic algorithm finds the minimum cost trace. • Breadth or Depth-first search-order. • Problem: Searches the entire state-space. • Minimum-Cost Search Order: Always explore state with smallest minimum cost first.

  24. Minimum-Cost Order Fact: First found goal state is optimal. • Cost grows along all paths. • The search can terminate when first goal state found. • Like Dijkstra’s shortest path algorithm. • Simpler algorithm: variable Cost no longer needed.

  25. Estimates of Remaining Cost • Often a conservative estimate of the remaining cost can be found. • REM( l, C ) = conservative estimate of remaining cost. • Bridge example: REM( l, C ) = time of slowest person on Unsafe side. At least 25 mins needed to complete schedule.

  26. Estimates of Remaining Cost • Basic Algorithm + Estimate of remaining cost:Only states with (min(C) + REM(l, C)) < Cost are further explored. Cost=80 min( C ) + REM( l, C )  80

  27. Estimates of Remaining Cost • Basic Algorithm + Estimate of remaining cost:Only states with (min(C) + REM(l, C)) < Cost are further explored. • Minimum Cost + Estimate of remaining cost:Explore states with smallest ( min(C) + REM( l, C ) ) first. Cost=80 min( C ) + REM( l, C )  80

  28. Using Heuristics • Allows the users to control the search order according to heuristics. • Symbolic states extended to (l, C, h), whereh is the priority of a state. • Transitions are annotated with assignments to h. • Flexible! Basic Algorithm + Heuristics:State with highest h is explored first.

  29. Examples

  30. Using Heuristics Try to schedule planes in the order of their preferred landing times

  31. Aircraft Landing Problem runways Benchmark by Beasley et al 2000

  32. Example: Bridge Problem What is the fastest schedule? • Number of symbolic states generated with cost-extended version of UPPAAL. • Minimum Cost Order + Estimate of Remaining cost<10% of Breadth-First Search. BF = Breadth-First, DF = Depth-First, MC = Minimum Cost Order, MC+ = MC + REM

  33. SIDMAR Steel Production Plant Crane A Machine 2 Machine 3 Machine 1 • A. Fehnker [RTCSA99], T. Hune, K. G. Larsen, P. Pettersson [DSV00] • 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 Machine 4 Machine 5 Lane 2 Buffer Crane B Storage Place Continuos Casting Machine

  34. SIDMAR Steel Production Plant Crane A Input: sequence of steel loads (“pigs”). Machine 2 Machine 3 Machine 1 @10 @20 2 @10 2 2 Lane 1 Machine 4 Machine 5 15 @10 Load follows Recipe to obtain certain quality, e.g: start; T1@10; T2@20; T3@10; T2@10; end within 120. Lane 2 16 Buffer Crane B =127 Storage Place Good schedules forten batcheswithin seconds, rather than bad schedules for five batches within almost an hour. @40 Continuos Casting Machine Output: sequence of higher quality steel.

  35. SIDMAR Steel Production Plant crane a • LEGO RCX Mindstorms. • Local controllers with control programs. • IR protocol for remote invocation of programs. • Central controller. m1 m2 m3 m4 m5 crane b buffer storage central controller casting Synthesis

  36. Heuristics: BPM protocol Heuristic: search first for constant input 1  Up to 50% reduction for erroneous instances of a simple communcation protocol.

  37. Conclusion • 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. • Future work • Tailoring Linear Programming to Priced Zones • Translation trace to schedule, re-use of schedules, ...

  38. Related Work • Alur, Courcourbetis, Henzinger (1993)Accumulated delays in Realtime Systems • Alur, Torre, Pappas (HSCC’01)Optimal Paths in Weighted Timed Automata • Behrmann, Fehnker, et all (HSCC’01)Minimum-Cost Reachability for Priced Timed Automata

  39. Related Work (cont) • Asarin & Maler (1999)Time optimal control using backwards fixed point computation • Niebert, Tripakis & Yovine (2000)Minimum-time reachability using forward reachability • Behrmann, Fehnker et all (TACAS’2001, CAV’01)Minimum-time reachability using Branch-and-Bound • Brinksma, Maler, Fehnker(STTT02) Using UPPAAL en SPIN to compute optimal schedules. • Abdeddaim, Maler (CAV’01)Job-Shop Scheduling using Timed Automata • General Trend (AAAI’01):Integrating Scheduling/Planning and Model Checking

  40. End of slide show

  41. cost’=1 cost’=2 cost’=0 x<3 x<3 cost+=4 y>2 c a {x:=0} (2.5) (a,x=y=0) (b,x=y=0) (b,x=y=2) (a,x=0,y=2) 4 2.5 x 2 0 Linearly Priced Timed Automata • Timed Automata + Costs on transitions and locations. • Cost of performing transition: Transition cost. • Cost of performing delay d: ( d x location cost ). b • Cost of Execution Trace: Sum of costs: 4 + 5 + 0 = 9

  42. y 5 4 3 2 1 x 0 1 2 3 4 5 Regions x<3 x<3 y>2 c a b {x:=0}

  43. y 5 4 3 2 1 x 0 1 2 3 4 5 Regions x<3 x<3 y>2 c a b {x:=0}

  44. 3 3 3 2 2 2 1 1 1 x x x Alur & Dill Regions x<3 x<3 y>2 c a b {x:=0} y y y 1 2 3 1 2 3 1 2 3 Transitions with and w/o reset and delay can be considered as transitions on regions!

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