[PPT] - Diverse and Additive Cartesian Abstraction Heuristics Jendrik Seipp PowerPoint Presentation

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Diverse and Additive Cartesian Abstraction Heuristics

Jendrik Seipp Malte Helmert

University of Basel, Switzerland

June 2014

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Setting:

Cost-optimal classical planning
Admissible heuristic for A∗

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Overview

Single Cartesian abstraction
Additive abstractions
Diversification strategies

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

CEGAR

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Counter-example guided abstraction refinement (CEGAR)

CEGAR algorithm Start with coarsest abstraction Until concrete solution is found or time runs out:

Find abstract solution
Check if and why it fails in the real world
Refine abstraction

Drawbacks:

Diminishing returns
Goal facts are considered one after another

→ Multiple abstractions

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Counter-example guided abstraction refinement (CEGAR)

CEGAR algorithm Start with coarsest abstraction Until concrete solution is found or time runs out:

Find abstract solution
Check if and why it fails in the real world
Refine abstraction

Drawbacks:

Diminishing returns
Goal facts are considered one after another

→ Multiple abstractions

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

SLIDE 7

CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Counter-example guided abstraction refinement (CEGAR)

CEGAR algorithm Start with coarsest abstraction Until concrete solution is found or time runs out:

Find abstract solution
Check if and why it fails in the real world
Refine abstraction

Drawbacks:

Diminishing returns
Goal facts are considered one after another

→ Multiple abstractions

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

SLIDE 8

CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Additive abstractions

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Multiple abstractions

h = 4 h = 3 h = 1 h = 0 3

2

5 o1 4 o4 2

5

1 o3 5

1

h = 5 h = 0 5

1
2, o3, o4, o5

How to combine heuristic estimates?

Maximum: h(s0) = max(4, 5) = 5
Cost partitioning: h(s0) = 0 + 5 = 5

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Multiple abstractions

h = 4 h = 3 h = 1 h = 0 3

2

5 o1 4 o4 2

5

1 o3 5

1

h = 5 h = 0 5

1
2, o3, o4, o5

How to combine heuristic estimates?

Maximum: h(s0) = max(4, 5) = 5
Cost partitioning: h(s0) = 0 + 5 = 5

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Multiple abstractions

h = 4 h = 3 h = 1 h = 0 3

2

5 o1 4 o4 2

5

1 o3 5

1

h = 5 h = 0 5

1
2, o3, o4, o5

How to combine heuristic estimates?

Maximum: h(s0) = max(4, 5) = 5
Cost partitioning: h(s0) = 0 + 5 = 5

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Saturated cost partitioning

Saturated cost function

h = 4 h = 3 h = 1 h = 0 3

2

5 o1 4 o4 2

5

1 o3 5

1

h = 5 h = 0 5

1
2, o3, o4, o5
h(s0) = 4 + 2 = 6

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Saturated cost partitioning

Saturated cost function

h = 4 h = 3 h = 1 h = 0 3

2

✁

5 3 o1

✁

4 0 o4 2

5

1 o3

✁

5 3

1

h = ✁ 5 2 h = 0

✁

5 2

1
2, o3, o4, o5
h(s0) = 4 + 2 = 6

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Saturated cost partitioning

Saturated cost function

h = 4 h = 3 h = 1 h = 0 3

2

✁

5 3 o1

✁

4 0 o4 2

5

1 o3

✁

5 3

1

h = ✁ 5 2 h = 0

✁

5 2

1
2, o3, o4, o5
h(s0) = 4 + 2 = 6

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Additive CEGAR abstractions

Build n abstractions
No changes to the CEGAR algorithm

Experiment settings:

30 minutes, 2 GB
15 minutes refinement

Results Abstractions Coverage 1 2 5 10 20 50 Sum (1396) 562 559 564 566 566 562

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Additive CEGAR abstractions

Build n abstractions
No changes to the CEGAR algorithm

Experiment settings:

30 minutes, 2 GB
15 minutes refinement

Results Abstractions Coverage 1 2 5 10 20 50 Sum (1396) 562 559 564 566 566 562

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Additive CEGAR abstractions

Build n abstractions
No changes to the CEGAR algorithm

Experiment settings:

30 minutes, 2 GB
15 minutes refinement

Results Abstractions Coverage 1 2 5 10 20 50 Sum (1396) 562 559 564 566 566 562

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Diversification strategies

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Abstraction by goals

Build an abstraction for each goal fact
Focus on different subproblems
Problem: tasks with single goal fact

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Abstraction by landmarks

Compute fact landmarks
Build an abstraction for each fact landmark l
Problem: landmarks as goals not admissible
Solution: hl(s) = 0 if l might have been achieved
Path-dependent landmark heuristics → state-based criterion

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Abstraction by landmarks

Modified task for landmark l:

Compute possibly-before set pb(l)
Facts: pb(l) ∪ {l}
Goal: l
Operators:
discard operators with preconditions not in pb(l)
let operators achieving l achieve only l
Initial state: unmodified

hl(s) = 0 if s pb(l) ∪ {l}

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Abstraction by landmarks: improved

x = 0 x = 1 x = 2

1
2

Solution:

Compute landmark orderings
Combine facts that have probably already been achieved

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Abstraction by landmarks: improved

x = 0 x = 1 x = 2

1
2

Solution:

Compute landmark orderings
Combine facts that have probably already been achieved

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Abstraction by landmarks: improved

Example

x = 0 x = 1 x = 2 y = 0 y = 1 y = 2

x = 1: {y = 0, y = 1}
x = 2: {y = 0, y = 1, y = 2}, {x = 0, x = 1}

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Experiments

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Comparison to other abstraction heuristics

i P D B m a s 1 m a s 2 s i n g l e g

a

l s L M L M + L M + g

a

l s h a d d

u

p h a d d

d
w

n 200 400 600 800 600 475 578 562 627 625 635 653 667 685 Coverage

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

hCEGAR vs. hCEGAR

LM+s⋆

200 400 600 200 400 600 hCEGAR

LM+s⋆ with hadd ↓

hCEGAR h(s0)

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

hCEGAR vs. hCEGAR

LM+s⋆

10 −210 −1 10 0 10 1 10 2 10 3 10 −2 10 −1 10 0 10 1 10 2 10 3 hCEGAR

LM+s⋆ with hadd ↓

hCEGAR Time for Computing Abstractions (secs)

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Conclusion

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Future work

Investigate impact of fact ordering
Use saturated cost partitioning for other abstraction heuristics

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics

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CEGAR Additive abstractions Diversification strategies Experiments Conclusion

Summary

New cost partitioning algorithm
Several diversification strategies
Competitive performance

Jendrik Seipp, Malte Helmert (Basel) Diverse and Additive Cartesian Abstraction Heuristics