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Landmarks, the Universe, and Everything Julie Porteous Laura - - PowerPoint PPT Presentation

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References Landmarks, the Universe, and Everything Julie Porteous Laura Sebastia J org Hoffmann Teesside University, UK Universidad Polit ecnica de Valencia, Spain Saarland University,

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Landmarks, the Universe, and Everything

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Teesside University, UK Universidad Polit´ ecnica de Valencia, Spain Saarland University, Germany

June 13, 2013

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 1/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 1

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 1

Imagine there’s no Landmarks It’s easy if you try No benchmarks below us Above us only Blai Imagine all the planners Planning for real

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 1

Imagine there’s no Landmarks It’s easy if you try No benchmarks below us Above us only Blai Imagine all the planners Planning for real

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning,

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning,

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS!

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations,

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations, Truth ramifications,

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations, Truth ramifications, Wishing FF was by my side!

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations, Truth ramifications, Wishing FF was by my side! My soft goals they are kissin’

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations, Truth ramifications, Wishing FF was by my side! My soft goals they are kissin’ My landmarks have gone missin’

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations, Truth ramifications, Wishing FF was by my side! My soft goals they are kissin’ My landmarks have gone missin’ My stubborn set has turned off the light.

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SLIDE 15

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Song # 2

Planning, planning, planning, P-D-D-L scanning, Keep ’em planners planning, ICAPS! Uncertain durations, Truth ramifications, Wishing FF was by my side! My soft goals they are kissin’ My landmarks have gone missin’ My stubborn set has turned off the light.

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 0 (The Dark Ages)

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Once Upon a Time, There Was a Landmark . . .

Verbatim from [Porteous et al. (2001)]:

A C D B D C B A

initial state goal

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 2/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1. Landmarks:

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1. Landmarks:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1. Landmarks:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open,

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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SLIDE 23

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1. Landmarks:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A,

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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SLIDE 24

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1. Landmarks:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A, Have-key-B,

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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SLIDE 25

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

What Are Landmarks?

Problem: Bring key B to position 1. Landmarks:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A, Have-key-B, . . .

→ A landmark is a fact that is true at some point on every solution plan. Find landmarks in a pre-process to planning. Can also find landmark orderings L ≤ L′.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 3/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

And Now?

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 4/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

And Now?

Well, some guy (me, that is) proposed to use this for problem decomposition, but never mind that.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 4/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

And Now?

Well, some guy (me, that is) proposed to use this for problem decomposition, but never mind that.

  • ps. Actually, see [Vernhes et al. (2013)] for an interesting modernized version!

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 4/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 1 (Preparing for Take-Off)

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 1 (Preparing for Take-Off)

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Use Landmarks!

Problem: Bring key B to position 1. Landmarks set {LM}:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A, Have-key-B, . . .

→ h(s) := |{LM} \ s|. (”Number of open items on the to-do list”)

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 6/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Use Landmarks!

Problem: Bring key B to position 1. Landmarks set {LM}:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A, Have-key-B, . . .

→ h(s) := |{LM} \ s|. (”Number of open items on the to-do list”) We can analyze orders and interferences to “put an item back on”.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 6/13

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SLIDE 33

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Use Landmarks!

Problem: Bring key B to position 1. Landmarks set {LM}:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A, Have-key-B, . . .

→ h(s) := |{LM} \ s|. (”Number of open items on the to-do list”) We can analyze orders and interferences to “put an item back on”. LAMA combines this with relaxed plans, iterated WA∗, . . . [Richter et al. (2008); Richter and Westphal (2010)]

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 6/13

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SLIDE 34

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Use Landmarks!

Problem: Bring key B to position 1. Landmarks set {LM}:

robot-at-2, robot-at-3, robot-at-4, robot-at-5, robot-at-6, robot-at-7. Lock-open, Have-key-A, Have-key-B, . . .

→ h(s) := |{LM} \ s|. (”Number of open items on the to-do list”) We can analyze orders and interferences to “put an item back on”. LAMA combines this with relaxed plans, iterated WA∗, . . . [Richter et al. (2008); Richter and Westphal (2010)] Credits to [Zhu and Givan (2003)] for their “forgotten work” . . . !

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 6/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 1

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 1

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 2 (Leaving the Atmosphere)

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 2 (Leaving the Atmosphere)

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Planning task: Goals G = {A, B}, initial state I = ∅, actions carA : ∅ → A cost 1, carB : ∅ → B cost 1, fancyCar : ∅ → A ∧ B cost 1.5.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Planning task: Goals G = {A, B}, initial state I = ∅, actions carA : ∅ → A cost 1, carB : ∅ → B cost 1, fancyCar : ∅ → A ∧ B cost 1.5. Landmarks set {LM}:

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Planning task: Goals G = {A, B}, initial state I = ∅, actions carA : ∅ → A cost 1, carB : ∅ → B cost 1, fancyCar : ∅ → A ∧ B cost 1.5. Landmarks set {LM}: {A, B}. Thus h(I) =

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 43

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Planning task: Goals G = {A, B}, initial state I = ∅, actions carA : ∅ → A cost 1, carB : ∅ → B cost 1, fancyCar : ∅ → A ∧ B cost 1.5. Landmarks set {LM}: {A, B}. Thus h(I) = 2 > h∗(I).

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 44

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Planning task: Goals G = {A, B}, initial state I = ∅, actions carA : ∅ → A cost 1, carB : ∅ → B cost 1, fancyCar : ∅ → A ∧ B cost 1.5. Landmarks set {LM}: {A, B}. Thus h(I) = 2 > h∗(I). Solution: [Karpas and Domshlak (2009)]

1

Consider disjunctive action landmarks instead: LA = {carA, fancyCar}, LB = {carB, fancyCar}. (= Achievers of each landmark) → Elementary landmark heuristic hLM

L (s) = min {c(a) | a ∈ L} if L is a disjunctive action

landmark for s, and hLM

L (s) = 0 otherwise.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 45

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

How To Admissibly Combine Landmarks!

Planning task: Goals G = {A, B}, initial state I = ∅, actions carA : ∅ → A cost 1, carB : ∅ → B cost 1, fancyCar : ∅ → A ∧ B cost 1.5. Landmarks set {LM}: {A, B}. Thus h(I) = 2 > h∗(I). Solution: [Karpas and Domshlak (2009)]

1

Consider disjunctive action landmarks instead: LA = {carA, fancyCar}, LB = {carB, fancyCar}. (= Achievers of each landmark) → Elementary landmark heuristic hLM

L (s) = min {c(a) | a ∈ L} if L is a disjunctive action

landmark for s, and hLM

L (s) = 0 otherwise.

2

Partition action costs to make

L hLM L (s) admissible!

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Cost Partitionings

Cost Partitioning: Ensemble of functions c1, . . . , cn : A → R+

0 s.t. for all a ∈ A,

n

i=1 ci(a) ≤ cost(a).

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

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SLIDE 47

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Cost Partitionings

Cost Partitioning: Ensemble of functions c1, . . . , cn : A → R+

0 s.t. for all a ∈ A,

n

i=1 ci(a) ≤ cost(a).

Admissible Sum: For heuristics h1, . . . , hn, n

i=1 hi[ci] ≤ h∗.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

slide-48
SLIDE 48

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Cost Partitionings

Cost Partitioning: Ensemble of functions c1, . . . , cn : A → R+

0 s.t. for all a ∈ A,

n

i=1 ci(a) ≤ cost(a).

Admissible Sum: For heuristics h1, . . . , hn, n

i=1 hi[ci] ≤ h∗.

→ c1, . . . , cn optimal for h1, . . . , hn and s if n

i=1 hi[ci](s) is maximal.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

slide-49
SLIDE 49

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Cost Partitionings

Cost Partitioning: Ensemble of functions c1, . . . , cn : A → R+

0 s.t. for all a ∈ A,

n

i=1 ci(a) ≤ cost(a).

Admissible Sum: For heuristics h1, . . . , hn, n

i=1 hi[ci] ≤ h∗.

→ c1, . . . , cn optimal for h1, . . . , hn and s if n

i=1 hi[ci](s) is maximal.

  • Theorem. Let s be a state, and let L1, . . . , Ln be disjunctive action landmarks for s. Then an
  • ptimal cost partitioning for s and hLM

L1 , . . . , hLM Ln can be computed in polynomial time.

  • Proof. We can encode this optimization problem into Linear Programming.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

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SLIDE 50

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Cost Partitionings

Cost Partitioning: Ensemble of functions c1, . . . , cn : A → R+

0 s.t. for all a ∈ A,

n

i=1 ci(a) ≤ cost(a).

Admissible Sum: For heuristics h1, . . . , hn, n

i=1 hi[ci] ≤ h∗.

→ c1, . . . , cn optimal for h1, . . . , hn and s if n

i=1 hi[ci](s) is maximal.

  • Theorem. Let s be a state, and let L1, . . . , Ln be disjunctive action landmarks for s. Then an
  • ptimal cost partitioning for s and hLM

L1 , . . . , hLM Ln can be computed in polynomial time.

  • Proof. We can encode this optimization problem into Linear Programming.

Example: LA = {carA, fancyCar}, LB = {carB, fancyCar}. carA : hLA ≤ 1 carB : hLB ≤ 1 fancyCar : hLA + hLB ≤ 1.5 → Maximizing hLA + hLB yields h(I) = 1.5.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

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SLIDE 51

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Cost Partitionings

Cost Partitioning: Ensemble of functions c1, . . . , cn : A → R+

0 s.t. for all a ∈ A,

n

i=1 ci(a) ≤ cost(a).

Admissible Sum: For heuristics h1, . . . , hn, n

i=1 hi[ci] ≤ h∗.

→ c1, . . . , cn optimal for h1, . . . , hn and s if n

i=1 hi[ci](s) is maximal.

  • Theorem. Let s be a state, and let L1, . . . , Ln be disjunctive action landmarks for s. Then an
  • ptimal cost partitioning for s and hLM

L1 , . . . , hLM Ln can be computed in polynomial time.

  • Proof. We can encode this optimization problem into Linear Programming.

Example: LA = {carA, fancyCar}, LB = {carB, fancyCar}. carA : hLA ≤ 1 carB : hLB ≤ 1 fancyCar : hLA + hLB ≤ 1.5 → Maximizing hLA + hLB yields h(I) = 1.5. Note: First done for abstraction heuristics [Katz and Domshlak (2008)].

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 2

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 2

→ For those of you who don’t remember that scene: It didn’t happen.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

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SLIDE 54

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 2

→ For those of you who don’t remember that scene: It didn’t happen. Karpas and Domshlak (2009)’s heuristic was part of Fast Downward Stone Soup and Selective Max in IPC’11.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 3 (Wasn’t That Mars We Just Passed?)

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 3 (Wasn’t That Mars We Just Passed?)

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SLIDE 57

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks:

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 58

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-59
SLIDE 59

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 0

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-60
SLIDE 60

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 1

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-61
SLIDE 61

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 1

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-62
SLIDE 62

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 2

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-63
SLIDE 63

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 2

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-64
SLIDE 64

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 3

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-65
SLIDE 65

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 3

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-66
SLIDE 66

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 4

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-67
SLIDE 67

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 4

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-68
SLIDE 68

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 5

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-69
SLIDE 69

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 5

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 70

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 6

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-71
SLIDE 71

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 6

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-72
SLIDE 72

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Many Disjunctive Action Landmarks!

Pre-Eff Structure: Actions get(X, Y ); init A, goal E. A B C1 D1 C2 D2 C3 D3 C4 D4 E Fact landmarks: {B, E}, yielding h(I) = 2. And now, let’s pass Mars: LM-cut! [Helmert and Domshlak (2009)] A B C1 D1 C2 D2 C3 D3 C4 D4 E

→ h(I) = 6 = h∗(I).

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 73

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 3

IPC 2008: Best optimal planner in the competition.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

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SLIDE 74

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 4 (Off to the Milky Way!!)

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SLIDE 75

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 4 (Off to the Milky Way!!)

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SLIDE 76

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Agenda: Stage 4 (Off to the Milky Way!!)

slide-77
SLIDE 77

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 78

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Precondition-Choice Functions Landmarks:

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 79

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Precondition-Choice Functions Landmarks: {carAB, carAC}

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

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SLIDE 80

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Precondition-Choice Functions Landmarks: {carAB, carAC}, {carAB, carBC}

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-81
SLIDE 81

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Precondition-Choice Functions Landmarks: {carAB, carAC}, {carAB, carBC}, {carAC, carBC}

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-82
SLIDE 82

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Landmarks: {carAB, carAC}, {carAB, carBC}, {carAC, carBC}. (Action costs: Uniform 1.)

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-83
SLIDE 83

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Landmarks: {carAB, carAC}, {carAB, carBC}, {carAC, carBC}. (Action costs: Uniform 1.) Optimal cost partitioning: h(I) = 1.5 < h∗(I): Set hLA = hLB = hLC = 0.5.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-84
SLIDE 84

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Landmarks: {carAB, carAC}, {carAB, carBC}, {carAC, carBC}. (Action costs: Uniform 1.) Optimal cost partitioning: h(I) = 1.5 < h∗(I): Set hLA = hLB = hLC = 0.5. Minimum cost hitting set: h(I) = 2 = h∗(I): E.g., H := {carAB, carAC}.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-85
SLIDE 85

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Hitting Sets Over Landmarks!

I A B C

{carAB, carAC} {carAB, carBC} {carAC, carBC}

Landmarks: {carAB, carAC}, {carAB, carBC}, {carAC, carBC}. (Action costs: Uniform 1.) Optimal cost partitioning: h(I) = 1.5 < h∗(I): Set hLA = hLB = hLC = 0.5. Minimum cost hitting set: h(I) = 2 = h∗(I): E.g., H := {carAB, carAC}. Hitting sets are admissible: Let L1, . . . , Ln be disjunctive action landmarks for s. Let H be a minimum-cost hitting set. Then

a∈H cost(a) ≤ h∗(s). (Simply because by definition every plan

must hit every Li.)

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 7/13

slide-86
SLIDE 86

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

From Landmarks to h+! [Bonet and Helmert (2010)]

  • Theorem. Let s be a state, and let L1, . . . , Ln be the collection of disjunctive action landmarks for s

resulting from all precondition-choice functions and cuts. Let H be a minimum-cost hitting set. Then

  • a∈H cost(a) = h+(s).
  • Proof. Any relaxed plan must hit L1, . . . , Ln so

a∈H cost(a) ≤ h+(s).

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

slide-87
SLIDE 87

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

From Landmarks to h+! [Bonet and Helmert (2010)]

  • Theorem. Let s be a state, and let L1, . . . , Ln be the collection of disjunctive action landmarks for s

resulting from all precondition-choice functions and cuts. Let H be a minimum-cost hitting set. Then

  • a∈H cost(a) = h+(s).
  • Proof. Any relaxed plan must hit L1, . . . , Ln so

a∈H cost(a) ≤ h+(s). We now prove that any

hitting set H contains a relaxed plan.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

slide-88
SLIDE 88

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

From Landmarks to h+! [Bonet and Helmert (2010)]

  • Theorem. Let s be a state, and let L1, . . . , Ln be the collection of disjunctive action landmarks for s

resulting from all precondition-choice functions and cuts. Let H be a minimum-cost hitting set. Then

  • a∈H cost(a) = h+(s).
  • Proof. Any relaxed plan must hit L1, . . . , Ln so

a∈H cost(a) ≤ h+(s). We now prove that any

hitting set H contains a relaxed plan. With RH := {p | p can be reached in delete-relaxation using

  • nly H}, assume to the contrary that G ⊆ RH. Consider the cut L defined by RH, RH:

s G a (1) a (1) a (2) a (2) L L RH RH

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

slide-89
SLIDE 89

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

From Landmarks to h+! [Bonet and Helmert (2010)]

  • Theorem. Let s be a state, and let L1, . . . , Ln be the collection of disjunctive action landmarks for s

resulting from all precondition-choice functions and cuts. Let H be a minimum-cost hitting set. Then

  • a∈H cost(a) = h+(s).
  • Proof. Any relaxed plan must hit L1, . . . , Ln so

a∈H cost(a) ≤ h+(s). We now prove that any

hitting set H contains a relaxed plan. With RH := {p | p can be reached in delete-relaxation using

  • nly H}, assume to the contrary that G ⊆ RH. Consider the cut L defined by RH, RH:

s G a (1) a (1) a (2) a (2) L L RH RH

Case (1): If prea ⊆ RH then adda ⊆ RH so a ∈ L.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

slide-90
SLIDE 90

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

From Landmarks to h+! [Bonet and Helmert (2010)]

  • Theorem. Let s be a state, and let L1, . . . , Ln be the collection of disjunctive action landmarks for s

resulting from all precondition-choice functions and cuts. Let H be a minimum-cost hitting set. Then

  • a∈H cost(a) = h+(s).
  • Proof. Any relaxed plan must hit L1, . . . , Ln so

a∈H cost(a) ≤ h+(s). We now prove that any

hitting set H contains a relaxed plan. With RH := {p | p can be reached in delete-relaxation using

  • nly H}, assume to the contrary that G ⊆ RH. Consider the cut L defined by RH, RH:

s G a (1) a (1) a (2) a (2) L L RH RH

Case (1): If prea ⊆ RH then adda ⊆ RH so a ∈ L. Case (2): If prea ⊆ RH then our precondition-choice function can select p ∈ prea \ RH so, again, a ∈ L. So H does not hit L, in contradiction.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 8/13

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SLIDE 91

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 4

Well, isn’t it just beautiful?

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

slide-92
SLIDE 92

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 4

Well, isn’t it just beautiful? More concretely: Improved LM-cut, runtime-effective in cases with large search space reduction [Bonet and Helmert (2010); Bonet and Castillo (2011)].

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

slide-93
SLIDE 93

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 4

Well, isn’t it just beautiful? More concretely: Improved LM-cut, runtime-effective in cases with large search space reduction [Bonet and Helmert (2010); Bonet and Castillo (2011)]. State of the art method for computing h+ [Haslum et al. (2012)].

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

slide-94
SLIDE 94

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

The Impact of Stage 4

Well, isn’t it just beautiful? More concretely: Improved LM-cut, runtime-effective in cases with large search space reduction [Bonet and Helmert (2010); Bonet and Castillo (2011)]. State of the art method for computing h+ [Haslum et al. (2012)]. State of the art method for computing h++, i. e., h+ computed in compilation ΠC, which converges to h∗ [Haslum et al. (2012)].

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 9/13

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SLIDE 95

Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

Last Slide

And now: No questions. Off to dinner! p.s.: Apologies and thanks to everybody who worked on landmarks but is not mentioned here!

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 10/13

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Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 References

References I

Blai Bonet and Julio Castillo. A complete algorithm for generating landmarks. In Fahiem Bacchus, Carmel Domshlak, Stefan Edelkamp, and Malte Helmert, editors, Proceedings of the 21st International Conference

  • n Automated Planning and Scheduling (ICAPS 2011). AAAI Press, 2011.

Blai Bonet and Malte Helmert. Strengthening landmark heuristics via hitting sets. In Helder Coelho, Rudi Studer, and Michael Wooldridge, editors, Proceedings of the 19th European Conference on Artificial Intelligence (ECAI’10), pages 329–334, Lisbon, Portugal, August 2010. IOS Press. Patrik Haslum, John Slaney, and Sylvie Thi´

  • ebaux. Minimal landmarks for optimal delete-free planning. In Blai

Bonet, Lee McCluskey, Jos´ e Reinaldo Silva, and Brian Williams, editors, Proceedings of the 22nd International Conference on Automated Planning and Scheduling (ICAPS 2012), pages 353–357. AAAI Press, 2012. Malte Helmert and Carmel Domshlak. Landmarks, critical paths and abstractions: What’s the difference anyway? In Alfonso Gerevini, Adele Howe, Amedeo Cesta, and Ioannis Refanidis, editors, Proceedings of the 19th International Conference on Automated Planning and Scheduling (ICAPS 2009), pages 162–169. AAAI Press, 2009.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 11/13

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SLIDE 97

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References II

Erez Karpas and Carmel Domshlak. Cost-optimal planning with landmarks. In C. Boutilier, editor, Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI 2009), pages 1728–1733, Pasadena, California, USA, July 2009. Morgan Kaufmann. Michael Katz and Carmel Domshlak. Optimal additive composition of abstraction-based admissible heuristics. In Jussi Rintanen, Bernhard Nebel, J. Christopher Beck, and Eric Hansen, editors, Proceedings of the 18th International Conference on Automated Planning and Scheduling (ICAPS 2008), pages 174–181. AAAI Press, 2008. Julie Porteous, Laura Sebastia, and J¨

  • rg Hoffmann. On the extraction, ordering, and usage of landmarks in
  • planning. In A. Cesta and D. Borrajo, editors, Recent Advances in AI Planning. 6th European Conference
  • n Planning (ECP-01), Lecture Notes in Artificial Intelligence, pages 37–48, Toledo, Spain, September
  • 2001. Springer-Verlag.

Silvia Richter and Matthias Westphal. The LAMA planner: Guiding cost-based anytime planning with

  • landmarks. Journal of Artificial Intelligence Research, 39:127–177, 2010.

Silvia Richter, Malte Helmert, and Matthias Westphal. Landmarks revisited. In Dieter Fox and Carla Gomes, editors, Proceedings of the 23rd National Conference of the American Association for Artificial Intelligence (AAAI-08), pages 975–982, Chicago, Illinois, USA, July 2008. AAAI Press.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 12/13

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References III

Simon Vernhes, Guillaume Infantes, and Vincent Vidal. Problem splitting using heuristic search in landmark

  • rderings. In Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI-2013),

Beijing, China, August 2013. AAAI Press. Lin Zhu and Robert Givan. Landmark extraction via planning graph propagation. In ICAPS 2003 Doctoral Consortium, pages 156–160, 2003.

Julie Porteous Laura Sebastia J¨

  • rg Hoffmann

Landmarks, the Universe, and Everything 13/13