As cheap as possible

1
As cheap as possible
Linearly Priced Timed Automata
Gerd Behrmann, Ed Brinksma, Ansgar Fehnker,
Thomas Hune, Kim Larsen, Paul Pettersson, Judi
Romijn, Frits Vaandrager
Brics Aalborg, Nijmegen, Twente, Uppsala, CMU,
TERMA, TUE
2
Motivation
Observation (VHS project) Many scheduling
problems can be phrased in a natural way as
reachability problems for timed automata.
3
Motivation
25min
20min
10min
5min
Can they make it within 60 minutes ?
What is the fastest schedule?
What schedule minmizes unsafe time?
What schedule minimizes bridge crossings?
Unsafe
Safe
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Outline

• Timed Automata (A review)
• Linearly Priced Timed Automata
• A basic Algorithm
• Efficient Data Structures
• Uniformly Priced Timed Automata
• More efficient Data Structures
• Improved State-Space Exploration
• Minimum-Cost Order Search, Estimates of
  Remaining Cost, Heuristics
• Results
• Bridge Problem
• Job-Shop Problems
• Aircraft Landing
• others
• Conclusion

5
Timed Automata
(UPPAAL)

• Network of Automata
• Synchronization (CCS-like)

a!
a?
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Timed Automata
(UPPAAL)

• Network of Automata
• Synchronization (CCS-like)
• Clocks in description
• Time passes uniformly
• Guard/reset on action
• Invariants on location
• Infinitely many states!

y 4
a?
y0
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Regions (review)
Alur Dill
xlt3
xlt3
ygt2
c
a
b
x0
y
y
y
3
3
3
2
2
2
1
1
1
x
x
x
1
2
3
1
2
3
1
2
3
An equivalence class (i.e. a region). In fact
there is only a finite number of regions!!
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Alur Dill
Regions (review)
xlt3
xlt3
ygt2
c
a
b
x0
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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Zones (review)
xlt3
xlt3
ygt2
c
a
b
x0
y
y
y
3
3
2
2
1
1
x
x
1
2
3
1
2
3
1
2
3
Data Structures like DBMs, CDDs g efficiency!
Convex unions of regions are called zones. Delay,
reset, transition in terms of zones
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Linearly Priced Timed Automata
b

• Timed Automata Costs on transitions and
  locations
• Cost of performing transition Transition cost
• Cost of performing delay d ( d x location cost )

Problem Finding the minimum cost of reaching
location c
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Example Aircraft Landing
Planes have to keep separation distance to avoid
turbulences caused by preceding planes
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Example Aircraft Landing
x lt 5
x gt 4
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
x5
land!
cost2
x lt 5
x lt 9
cost3
cost1
x5
land!
Planes have to keep separation distance to avoid
turbulences caused by preceding planes
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Priced Regions
cost
5
4
3
2
1
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Priced Regions
cost
5
4
3
2
1
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Priced Regions
cost
y
5
4
5
3
2
3
2
2
1
1
x
1
2
3
costs
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An Algorithm

• State-Space Exploration Use of global variable
  Cost
• Updated Cost whenever goal state with min( C )
  ltCost is found
• Terminates when entire state-space is explored

Cost?
Cost80
80
60
Cost60
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An Algorithm

• Cost?, Pass , Wait (l0,C0), Goal?
• while Wait ? do
• select (l,C) from Wait
• if (l,C) ? and mincost(C)ltCost then
  Costmincost(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

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An Algorithm

• Cost?, Pass , Wait (l0,C0), Goal?
• while Wait ? do
• select (l,C) from Wait
• if (l,C) ? and mincost(C)ltCost then
  Costmincost(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

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An Algorithm

• Cost?, Pass , Wait (l0,C0), Goal?
• while Wait ? do
• select (l,C) from Wait
• if (l,C) ? and mincost(C)ltCost then
  Costmincost(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

preorder that defines better
cost zones.
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An Algorithm

• Cost?, Pass , Wait (l0,C0), Goal?
• while Wait ? do
• select (l,C) from Wait
• if (l,C) ? and mincost(C)ltCost then
  Costmincost(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

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An Algorithm
Theorem
When the algorithm terminates, the value of COST
equals mincost(?)
Theorem
The algorithm terminates
Can it be done efficiently?
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Outline

• Timed Automata. (A review
• Linearly Priced Timed Automata
• A basic Algorithm
• Efficient Data Structures
• Uniformly Priced timed Automata
• More efficient Data Structures
• Improved State-Space Exploration
• Minimum-Cost Order Search, Estimates of Remaining
  Cost, Heuristics
• Results
• Bridge Problem
• Job-Shop Problems
• Aircraft Landing
• others
• Conclusion

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Priced Zones
Basic idea Define a linear cost function on zones
BUT Priced zones are not closed under delay,
transitions, resets
y
x
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Priced Zones
Basic idea Define a linear cost function on zones
BUT Priced zones are not closed under delay,
transitions, resets
y
costc2 x 1 y
x
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Priced Zones
Basic idea Define a linear cost function on zones
BUT Priced zones are not closed under delay,
transitions, resets
y
costc2 x 1 y
x
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Priced Zones
Basic idea Define a linear cost function on zones
BUT Priced zones are not closed under delay,
transitions, resets
y
costc1 x 1 y
costc2 x 2 y
x
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Priced Zones
Basic idea Define a linear cost function on zones
BUT Priced zones are not closed under delay,
transitions, resets
y
costc2 x 1 y
x
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Outline

• Timed Automata. (A review
• Linearly Priced Timed Automata
• A basic Algorithm
• Efficient Data Structures
• Uniformly Priced Timed Automata
• More efficient Data Structures
• Improved State-Space Exploration
• Minimum-Cost Order Search, Estimates of Remaining
  Cost, Heuristics
• Results
• Bridge Problem
• Job-Shop Problems
• Aircraft Landing
• others
• Conclusion

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Uniformly Priced Timed Automata
UPTA are LPTA where all locations have the same
rate
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Uniformly Priced Timed Automata
UPTA are LPTA where all locations have the same
rate
Result
A small modification of the DBM-operations for
ordinary timed automata is sufficient to solve
cost (time) optimality problems
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Outline

• Timed Automata. (A review
• Linearly Priced Timed Automata
• A basic Algorithm
• Efficient Data Structures
• Uniformly Priced Timed Automata
• More efficient Data Structures
• Improved State-Space Exploration
• Minimum-Cost Order Search, Estimates of Remaining
  Cost, Heuristics
• Results
• Bridge Problem
• Job-Shop Problems
• Aircraft Landing
• others
• Conclusion

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Verification vs. Optimization

• Verification Algorithms
• Check a logical property of the entire
  state-space of a model
• Efficient blind search
• Optimization Algorithms
• Find (near) optimal solutions
• Use 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
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Minimum-Cost Order

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

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Minimum-Cost Order

• Fact 1 First goal state found is optimal
• Cost grows along all paths
• The search can terminate when first goal state
  found
• Like Dijkstras shortest path algorithm
• Fact 2 No other search order explores fewer
  states
• Simpler algorithm variable Cost no longer needed

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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
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Estimates of Remaining Cost

• Basic Algorithm Estimate of remaining
  costOnly states with (min(C) REM(l, C)) lt
  Cost are further explored

Cost80
min( C )
REM( l, C ) ? 80
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Estimates of Remaining Cost

• Basic Algorithm Estimate of remaining
  costOnly states with (min(C) REM(l, C)) lt
  Cost are further explored

Cost80
min( C )
REM( l, C ) ? 80

• Minimum Cost Estimate of remaining
  costExplore states with smallest ( min(C)
  REM( l, C ) ) first

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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

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

• Timed Automata. (A review
• Linearly Priced Timed Automata
• A basic Algorithm
• Efficient Data Structures
• Uniformly Priced Timed Automata
• More efficient Data Structures
• Improved State-Space Exploration
• Minimum-Cost Order Search, Estimates of Remaining
  Cost, Heuristics
• Results
• Bridge Problem
• Sidmar
• Aircraft Landing
• others
• Conclusion

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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
  costlt10 of Breadth-First Search

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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
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SIDMAR Steel Production Plant
Crane A
Machine 2
Machine 3
Input sequence of steel loads (pigs)
Machine 1
_at_10
_at_20
_at_10
2
2
2
Lane 1
Machine 4
Machine 5
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_at_10
Load follows Recipe to obtain certain quality,
e.g start T1_at_10 T2_at_20 T3_at_10 T2_at_10 end
within 120
Lane 2
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Buffer
Crane B
?127
Storage Place
Optimal schedules for ten batches using guiding
with priorities. Only for two batches without
_at_40
Continuos Casting Machine
Output sequence of higher quality steel.
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Aircraft Landing Problem
runways
Benchmark by Beasley et al 2000
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Conclusion

• Advantages
• Easy and flexible modeling of systems
• Whole range of verification techniques becomes
  available
• Controller/Program synthesis
• Disadvantages
• Existing scheduling approaches (still) perform
  somewhat better
• Our goal
• See how far we get
• Integrate model checking and scheduling theory
• New discipline of Timing Technology?
• EU IST project Ametist

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Conclusion

• Papers
• Efficient Guiding Towards Cost-Optimality in
  UPPAAL TACAS01
• Minimum Cost-Reachability for Priced Timed
  Automata HSCC01
• As Cheap as Possible Efficient Cost-Optimal
  Reachability for Priced Timed Automata CAV01
• Citius, Vilius, Melius Guiding and
  Cost-Optimality in Model Checking of Timed and
  Hybrid Systems, PhD Thesis Ansgar Fehnker,
  University of Nijmegen, April 2002

• Tool

• UPPAAL CORA!!

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End of slide show
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THE END