CUPPAAL Efficient Minimum-Cost Reachability for Linearly Priced Timed Automata

1
CUPPAALEfficient Minimum-Cost Reachabilityfor
Linearly Priced Timed Automata

• Gerd Behrman, Ed Brinksma, Ansgar Fehnker, Thomas
  Hune, Kim Larsen, Paul Pettersson,
• Judi Romijn, Frits Vaandrager

2
Overview

1. Introduction
2. Linear Priced Timed Automata
3. Priced Zones and Facets
4. Operations on Priced Zones
5. Algorithm
6. First Experimental Findings
7. Conclusion

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INTRODUCTION
Observation Many scheduling problems can be
phrased naturally as reachability problems for
timed automata!
4
INTRODUCTION
Observation Many scheduling problems can be
phrased naturally as reachability problems for
timed automata!
5
INTRODUCTION
Observation Many scheduling problems can be
phrased naturally as reachability problems for
timed automata!
6
Steel Production Plant
INTRODUCTION
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
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Batch Processing Plant (VHS)
INTRODUCTION
8
Earlier work
INTRODUCTION

• 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

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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.

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More general cost function
INTRODUCTION

• 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.

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Linearly Priced Timed Automata
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Example
PRICED AUTOMATA
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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 COST195 and TOTAL
TIME65!
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Definition
PRICED AUTOMATA
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Definition
PRICED AUTOMATA
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Example of execution
PRICED AUTOMATA
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Cost
PRICED AUTOMATA

• 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
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Zones
PRICED ZONES
Operations
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Canonical Datastructure for Zones Difference
Bounded Matrices
PRICED ZONES
Bellman58, Dill89
-4
-4
x1-x2lt4 x2-x1lt10 x3-x1lt2 x2-x3lt2 x0-x1lt3 x3-x
0lt5
x1
x2
Shortest Path Closure O(n3)
x1
x2
4
10
2
3
3
2
3
-2
-2
2
2
x3
x0
x3
x0
1
5
5
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New Canonical Datastructure Minimal
collection of constraints
PRICED ZONES
RTSS 1997
-4
-4
x1-x2lt4 x2-x1lt10 x3-x1lt2 x2-x3lt2 x0-x1lt3 x3-x
0lt5
x1
x2
Shortest Path Closure O(n3)
x1
x2
4
10
2
3
3
2
3
-2
-2
2
2
x3
x0
x3
x0
1
5
5
-4
Shortest Path Reduction O(n3)
x1
x2
Space worst O(n2) practice O(n)
3
2
3
2
x3
x0
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Priced Zone
PRICED ZONES
y
Z
2
-1
4
x
23
Reset
PRICED ZONES
Z
y
2
-1
4
x
24
Reset
PRICED ZONES
Z
y
2
-1
4
x
yZ
25
Reset
PRICED ZONES
Z
y
2
-1
4
x
yZ
4
26
Reset
PRICED ZONES
Z
y
2
-1
4
-1
1
x
yZ
4
2
4
A split of yZ
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FacetsThe solution
PRICED ZONES
28
OPERATIONS ON PZONES
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Delay
PRICED ZONES
y
Z
3
-1
4
x
30
Delay
PRICED ZONES
Delay in a location with cost-rate 3
3
y
Z
2
3
-1
4
x
31
Delay
PRICED ZONES
4
-1
y
0
Z
A split of
3
3
-1
4
x
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FacetsThe solution
PRICED ZONES
33
OPERATIONS ON PZONES
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Optimal Forward ReachabilityExample
PRICED ZONES
8
6
10
4
10
2
0
0
10
10
10
2
4
6
8
10
10
10
1
1
1
1
1
4
6
8
2
8
10
10
6
4
2
10
10
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OPERATIONS ON PZONES
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OPERATIONS ON PZONES
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Algorithm
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Branch Bound Algorithm
ALGORITHM
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ALGORITHM
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ALGORITHM
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Experiments
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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 COST195 and TOTAL
TIME65!
43
Experiments MC Order
EXPERIMENTS
COST-rates COST-rates COST-rates COST-rates SCHEDULE COST TIME Expl Popd
G5 C10 B20 D25 SCHEDULE COST TIME Expl Popd
Min Time Min Time Min Time Min Time CGgt Glt BDgt Clt CGgt 60 1762 1538 2638
1 1 1 1 CGgt Glt BGgt Glt GDgt 55 65 252 378
9 2 3 10 GDgt Glt CGgt Glt BGgt 195 65 149 233
1 2 3 4 CGgt Glt BDgt Clt CGgt 140 60 232 350
1 2 3 10 CDgt Clt CBgt Clt CGgt 170 65 263 408
1 20 30 40 BDgt Blt CBgt Clt CGgt 975 1085 85 timelt85 - -
0 0 0 0 - 0 - 406 447
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Optimal Broadcast
EXPERIMENTS
Router2
Router1
k1
k0
costA1, costB1
costA2, costB2
B
3 sec
Basecost
5 sec
A
costA4, costB4
costA3, costB3
k0
k0
costB1
Router4
costA1
Router3
Given particular subscriptions, what is the
cheapest schedule for broadcasting k?
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Experimental Results
EXPERIMENTS
COST-rates COST-rates COST-rates COST-rates COST-rates SCHEDULE COST TIME Expl
BC R1 R2 R3 R4 SCHEDULE COST TIME Expl
Min Time Min Time Min Time Min Time Min Time 1gt3(B) ( 3gt4(B) 1gt2(A) ) 8 1016
0 13 13 13 13 1gt4(A) 3gt4(A) 4gt2(A) 15 15 2982
3 13 13 13 13 1gt3(B) ( 3gt4(B) 1gt2(A) ) 47 8 1794
0 10 30 5 15 13 62 1gt3(A) 3gt2(A) 3gt4(A) 60 15 665
3 10 30 5 15 13 62 1gt4(A) 4gt3(B) 4gt2(B) 95 11 571
100 10 30 5 15 13 62 1gt4(B) ( 1gt3(A) 4gt2(B) ) 946 8 1471
0 tlt10 10 30 5 15 13 62 1gt4(B) 4gt2(B) 4gt3(B) 102 9 1167
0 tlt8 10 30 5 15 13 62 1gt4(B) ( 1gt3(A) 4gt2(B) ) 146 8 1688
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Scaling Up ?
EXPERIMENTS

• 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 (BC1,
  all other cost0) time optimal schedule
  takes 12 seconds.

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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