Reachability, Schedulability and Optimality

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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 gt 4
a?
a!
y0
Uppaal is a modelchecker forTimed Automata with
emphasis on reachability properties
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Motivation
Observation Many scheduling problems can be
phrased in a natural way as reachability problems
for timed automata!
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Motivation
25min
20min
10min
5min
Can they make it within 60 minutes ?
What is the fastest schedule?
What schedule mini-mizes unsafe time?
What schedule minimizes crossings?
Unsafe
Safe
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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

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Example Aircraft Landing
Planes have to keep separation distance to avoid
turbulences caused by preceding planes
Runway
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Example Aircraft Landing
x lt 5
x5
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
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
Runway
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Symbolic semantics of Linearly Priced Timed
Automata
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Zones
Basic idea Define a delay and reset over zones
y
1 ? y ? 4 0 ? x ? 3 -2 ? x-y? 0
x
xlt3
xlt3
ygt2
c
a
b
y0
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Zones
Basic idea Define a delay and reset over zones
y
1 ? y ? 4 0 ? x ? 3 -2 ? x-y? 0
x
xlt3
xlt3
ygt2
c
a
b
y0
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Priced Zones
Basic idea Define a linear cost function on zones
y
cost c - 1 x 2 y
2
-1
x
xlt5
xlt3
ygt2
c
a
b
y0
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Priced Zones
Basic idea Define a delay and reset over zones
y
cost c - 1 x 2 y
2
-1
x
xlt3
xlt3
ygt2
c
a
b
y0
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State-Space Exploration Algorithm
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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

Cost?
Cost80
80
60
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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

Performs symbolic operations Delay,
Conjun-ction, and Reset of clocks.
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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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Efficient Reachability of LPTAs
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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
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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.

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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 Dijkstras shortest path algorithm.
• 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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Examples
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Using Heuristics
Try to schedule planes in the order of their
preferred landing times
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Aircraft Landing Problem
runways
Benchmark by Beasley et al 2000
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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
Good schedules for ten batches within seconds,
rather than bad schedules for five batches within
almost an hour.
_at_40
Continuos Casting Machine
Output sequence of higher quality steel.
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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
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Heuristics BPM protocol
Heuristic search first for constant input 1

?
Up to 50 reduction for erroneous
instances of a simple communcation protocol.
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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 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, ...

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Related Work

• Alur, Courcourbetis, Henzinger (1993)Accumulated
  delays in Realtime Systems
• Alur, Torre, Pappas (HSCC01)Optimal Paths in
  Weighted Timed Automata
• Behrmann, Fehnker, et all (HSCC01)Minimum-Cost
  Reachability for Priced Timed Automata

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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 (TACAS2001,
  CAV01)Minimum-time reachability using
  Branch-and-Bound
• Brinksma, Maler, Fehnker(STTT02)
• Using UPPAAL en SPIN to compute optimal
  schedules.
• Abdeddaim, Maler (CAV01)Job-Shop Scheduling
  using Timed Automata
• General Trend (AAAI01) Integrating
  Scheduling/Planning and Model Checking

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End of slide show
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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 ).

• Cost of Execution Trace Sum of costs 4 5 0
  9

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Regions
xlt3
xlt3
ygt2
c
a
b
x0
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Regions
xlt3
xlt3
ygt2
c
a
b
x0
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Alur Dill
Regions
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!