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

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

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

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

7. Linearly Priced Timed Automata4
2.5 x 2
cost’=1
cost+=4
cost’=0
cost’=2 b
x<5
y>2
x<3
y:=0 a c
Timed Automata + Costs on transitions and locations.
Cost of performing transition: Transition cost.
Cost of performing delay d: ( d x location cost ).
(a,x=y=0)
(b,x=y=0)
(b,x=y=2)
(2.5)
(a,x=0,y=2)
Cost of Execution Trace: Sum of costs: = 9

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

9. Example: Aircraft Landing
4 earliest landing time
5 target time
9 latest time
3 cost rate for being early
1 cost rate for being late
2 fixed 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. Zones c a b Basic idea: Define a delay and reset over zones y delay x
-2  x-y 0
delay
1  y  4
0  x  3
-2  x-y 0 y x
x<3
x<3
y>2 a c b
y:=0

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

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

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

15. State-Space Exploration Algorithm

16. An Algorithm State-Space Exploration + Use of global variable Cost.
Updated Cost whenever goal state with
min( C ) <Cost is found:
Cost=
Cost=80
Cost=60 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. An Algorithm C C’ Cost:=, Pass := {}, Wait := {(l0,C0)}, Goal=
C’ is bigger & cheaper than C
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 .
is a well-quasi ordering which guarantees termination!

20. When the algorithm terminates, the value of COST equals 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
Theorem
When the algorithm terminates, the value of COST equals mincost().

21. Efficient Reachability of LPTAs

22. Verification vs. Optimization
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.
Safe side reachable? 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. Fact: First found goal state is optimal.
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.
Cost=80
min( C )
+ REM( l, C )  80
Minimum Cost + Estimate of remaining cost: Explore states with smallest ( min(C) + REM( l, C ) ) first.

28. Basic Algorithm + Heuristics: State with highest h is explored first.
Using Heuristics
Allows the users to control the search order according to heuristics.
Symbolic states extended to (l, C, h), where h 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. Try to schedule planes in the order of their preferred landing times
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?
BF = Breadth-First, DF = Depth-First, MC = Minimum Cost Order, MC+ = MC + REM
Number of symbolic states generated with cost-extended version of UPPAAL.
Minimum Cost Order + Estimate of Remaining cost <10% of Breadth-First Search.

33. SIDMAR Steel Production Plant
Crane A
Machine 1
Machine 2
Machine 3
A. Fehnker [RTCSA99],
T. Hune, K. G. Larsen,
P. Pettersson [DSV00]
Case study of Esprit-LTR project VHS
Physical plant of SIDMAR located 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 1
Machine 2
Machine 3
@10
@20 2
@10 2 2
Lane 1
Machine 4
Machine 5 15
@10
Load follows Recipe to
obtain certain quality, e.g:
start;
end within 120.
Lane 2 16
Buffer
Crane B
=127
Storage Place
Good schedules for ten batches within 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
LEGO RCX Mindstorms.
Local controllers with control programs.
IR protocol for remote invocation of programs.
Central controller.
crane a 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 Disadvantages Our goal Future work
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

41. End of slide show

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

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

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

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