Simulation-Based Admissible Dominance Pruning Alvaro Torralba, J - - PowerPoint PPT Presentation

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Simulation-Based Admissible Dominance Pruning Alvaro Torralba, J - - PowerPoint PPT Presentation

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Simulation-Based Admissible Dominance Pruning Alvaro Torralba, J org Hoffmann HSDIP Workshop at ICAPS June 8th, 2015 Alvaro Torralba, J org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 1 / 19 Motivation Cost-optimal

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SLIDE 1

Simulation-Based Admissible Dominance Pruning

´ Alvaro Torralba, J¨

  • rg Hoffmann

HSDIP Workshop at ICAPS June 8th, 2015

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 1 / 19

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SLIDE 2

Motivation

Cost-optimal planning: (V, O, I, G) A∗+ admissible heuristic h(s): estimates distance to goal Pruning methods:

1

Partial-order pruning

2

Symmetries

3

Dominance pruning

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 2 / 19

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SLIDE 3

Dominance Pruning

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 3 / 19

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SLIDE 4

Dominance Pruning

Detect “better than” states A B V = { at-T= {A, B}, at-P= {A, B, T} } I = {at-T A, at-P A} G = {at-P B} O = {move-T (A, B), move-T (B, A), load-P(A), . . . }

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 4 / 19

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SLIDE 5

Dominance Pruning

Detect “better than” states A B V = { at-T= {A, B}, at-P= {A, B, T} } I = {at-T A, at-P A} G = {at-P B} O = {move-T (A, B), move-T (B, A), load-P(A), . . . } What do you prefer? A B A B

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 4 / 19

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SLIDE 6

Dominance Pruning

Detect “better than” states A B V = { at-T= {A, B}, at-P= {A, B, T} } I = {at-T A, at-P A} G = {at-P B} O = {move-T (A, B), move-T (B, A), load-P(A), . . . } What do you prefer? A B A B Formally: relation of pair of states s t

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 4 / 19

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SLIDE 7

Admissible Pruning

t simulates s (s t) = ⇒ t is at least as good as s: h∗(s) ≥ h∗(t) If g(t) ≤ g(s) and s t then s can be discarded

I

s1 s2 s3 s4

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 5 / 19

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SLIDE 8

Admissible Pruning

t simulates s (s t) = ⇒ t is at least as good as s: h∗(s) ≥ h∗(t) If g(t) ≤ g(s) and s t then s can be discarded

I

s1 s2 s3 s4

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 5 / 19

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SLIDE 9

Admissible Pruning

t simulates s (s t) = ⇒ t is at least as good as s: h∗(s) ≥ h∗(t) If g(t) ≤ g(s) and s t then s can be discarded

I

s1 s2 s3 s4 s4 I

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 5 / 19

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SLIDE 10

Admissible Pruning

t simulates s (s t) = ⇒ t is at least as good as s: h∗(s) ≥ h∗(t) If g(t) ≤ g(s) and s t then s can be discarded

I

s1 s2 s3 s4 s4 I s1 s3

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 5 / 19

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SLIDE 11

Admissible Pruning

t simulates s (s t) = ⇒ t is at least as good as s: h∗(s) ≥ h∗(t) If g(t) ≤ g(s) and s t then s can be discarded

I

s1 s2 s3 s4 s4 I s1 s3 s5 s6 s7 Challenges:

1

How to find good dominance relations?

2

How to efficiently check dominance?

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 5 / 19

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SLIDE 12

Simulation Relation

Definition (Simulation)

A binary relation ⊆ S × S is a simulation for Θ if, whenever s t, for every transition s

l

− → s′ there exists t

l

− → t′ s.t. s′ t′. We call goal-respecting for Θ if, whenever s t, s ∈ SG implies that t ∈ SG. G A C B D l1 l2 l1 l2 l2 D B Thm: A unique coarsest goal-respecting simulation always exists and can be computed in time polynomial in the size of Θ

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 6 / 19

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SLIDE 13

Simulation Relation

Definition (Simulation)

A binary relation ⊆ S × S is a simulation for Θ if, whenever s t, for every transition s

l

− → s′ there exists t

l

− → t′ s.t. s′ t′. We call goal-respecting for Θ if, whenever s t, s ∈ SG implies that t ∈ SG. G A C B D 1 1 1 1 1 D B, C A Cost-simulation: replace labels by their cost → A cost-simulation on the state space of the planning task is a dominance relation Thm: A unique coarsest goal-respecting simulation always exists and can be computed in time polynomial in the size of Θ

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 6 / 19

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SLIDE 14

Compositional Approach

1

Consider a partition of the problem: Θ1, . . . , Θk

2

Compute a simulation for each part: 1, . . . , k

3

: s t iff ∀i ∈ [1, k] si i ti is a cost-simulation

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 7 / 19

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SLIDE 15

Compositional Approach

1

Consider a partition of the problem: Θ1, . . . , Θk

2

Compute a simulation for each part: 1, . . . , k

3

: s t iff ∀i ∈ [1, k] si i ti is a cost-simulation In our example:

Θ1: (truck)

A B

dr dr lA lB

Θ2: (package)

A T B

lA lA lB lB dr dr dr

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 7 / 19

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SLIDE 16

Compositional Approach

1

Consider a partition of the problem: Θ1, . . . , Θk

2

Compute a simulation for each part: 1, . . . , k

3

: s t iff ∀i ∈ [1, k] si i ti is a cost-simulation In our example:

Θ1: (truck)

A B

dr dr lA lB

Θ2: (package)

A T B

lA lA lB lB dr dr dr

A 2 T 2 B

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 7 / 19

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SLIDE 17

Not so fast

Definition of s t: For every s

l

− → s′ there exists t

l

− → t′ s.t. s′ t′

Θ1: (truck)

A B

dr dr lA lB

Θ2: (package)

A T B

lA lA lB lB dr dr dr

A 2 T 2 B

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 8 / 19

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SLIDE 18

Not so fast

Definition of s t: For every s

l

− → s′ there exists t

l

− → t′ s.t. s′ t′

Θ1: (truck)

A B

dr dr lA lB

Θ2: (package)

A T B

lA lA lB lB dr dr dr

A 2 T 2 B T 2 B: T

lB

− → B and there is no B lB − →

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 8 / 19

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SLIDE 19

Not so fast

Definition of s t: For every s

l

− → s′ there exists t

l

− → t′ s.t. s′ t′

Θ1: (truck)

A B

dr dr lA lB

Θ2: (package)

A T B

lA lA lB lB dr dr dr

A 2 T 2 B T 2 B: T

lB

− → B and there is no B lB − → T

lB

− → B and T

lA

− → A are simulated by B →

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 8 / 19

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SLIDE 20

Not so fast

Definition of s t: For every s

l

− → s′ there exists t

l

− → t′ s.t. s′ t′

Θ1: (truck)

A B

dr dr lA lB

Θ2: (package)

A T B

lA lA lB lB dr dr dr

A 2 T 2 B T 2 B: T

lB

− → B and there is no B lB − → T

lB

− → B and T

lA

− → A are simulated by B

noop

− − − → B lA, lB do not have useful effects in the rest of the problem (Θ1)!

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 8 / 19

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SLIDE 21

Label-Dominance Simulation

Definition (Label Dominance)

l′ dominates l in Θ given if for every s

l

− → s′ ∈ Θ there exists s l′ − → t′ s.t. s′ t′

Definition (Label-Dominance Simulation)

A set R = {1, . . . , k} of binary relations i⊆ Si × Si is a label-dominance simulation for {Θ1, . . . , Θk} if, whenever s i t: s ∈ SG

i implies that t ∈ SG i

For every s

l

− → s′ in Θi, there exists t

l′

− → t′ in Θi s.t.:

1

s′ i t′,

2

c(l′) ≤ c(l), and

3

for all j = i, l′ dominates l in Θj given j

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 9 / 19

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SLIDE 22

Label-Dominance Simulation: Theoretical Results

Theorem

A coarsest label-dominance simulation always exists and can be computed in polynomial time For all i, set i:= {(s, t) | s, t ∈ Si, s ∈ Si

G or t ∈ Si G}

while ex. (i, s, t) s.t. not Ok(i, s, t) do Select one such triple (i, s, t) Set i:=i \{(s, t)} return R := {1, . . . , k}

Theorem

Combination of {1, . . . , k} is a cost-simulation for the planning task Θ1 ⊗ · · · ⊗ Θk

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 10 / 19

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SLIDE 23

Computation of Label-Dominance Simulation

Θ1: (truck) A B

dr dr lA lB

Θ2: (package) A T B

lA lA lB lB dr dr dr

Truck A1{ B } B1{ A } Package A2{ T, B } T2{ A, B } B2{ } noop simulates {lA, lB, dr} dr simulates {lA, lB} (noop ≡ dr) simulate {lA}

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 11 / 19

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SLIDE 24

Computation of Label-Dominance Simulation

Θ1: (truck) A B

dr dr lA lB

Θ2: (package) A T B

lA lA lB lB dr dr dr

Truck A1{ B } B1{ A } Package A2{ T, B } T2{ A, B } B2{ } noop simulates {lA, lB, dr} dr simulates {lA, lB} (noop ≡ dr) simulate {lA}

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 11 / 19

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SLIDE 25

Computation of Label-Dominance Simulation

Θ1: (truck) A B

dr dr lA lB

Θ2: (package) A T B

lA lA lB lB dr dr dr

Truck A1{ B } B1{ A } Package A2{ T, B } T2{ A, B } B2{ } noop simulates {lA, lB, dr} dr simulates {lA, lB} (noop ≡ dr) simulate {lA}

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 11 / 19

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SLIDE 26

Computation of Label-Dominance Simulation

Θ1: (truck) A B

dr dr lA lB

Θ2: (package) A T B

lA lA lB lB dr dr dr

Truck A1{ B } B1{ A } Package A2{ T, B } T2{ A, B } B2{ } noop simulates {lA, lB, dr} dr simulates {lA, lB} (noop ≡ dr) simulate {lA}

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 11 / 19

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SLIDE 27

Pruning: Implementation Details

Insert in closed every state dominated by any expanded state

→ BDD Bg represents any state expanded/dominated with g

When s is generated or expanded:

1

Prune s if it is in Bg′ for some g′ ≤ g(s)

When s is expanded:

1

Insert all states dominated by s in Bg(s)

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 12 / 19

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SLIDE 28

Pruning: Implementation Details

Insert in closed every state dominated by any expanded state

→ BDD Bg represents any state expanded/dominated with g

When s is generated or expanded:

1

Prune s if it is in Bg′ for some g′ ≤ g(s)

When s is expanded:

1

Insert all states dominated by s in Bg(s)

Safety Belt: Stop if no state is pruned after 1000 expansions

◮ Don’t waste time if no useful dominance relation has been found ´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 12 / 19

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SLIDE 29

Experimental Results

M&S: Merge-DFP + bisimulation up to 100 000 transitions Pruning types:

◮ A: Baseline without pruning ◮ L: Label-dominance simulation ◮ S: Simulation ◮ B: Bisimulation ◮ P: Partial-order reduction

Heuristic: Blind or LM-cut

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 13 / 19

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SLIDE 30

Experimental Results: Blind Search

Coverage Evaluations Domain # A L S B P L S B P Airport 50 22

  • 7
  • 7
  • 1

1.2 1.2 1 4.4 Driverlog 20 7 +2 15.8 2 2 1 Floortile11 20 2 +4 +4 177 177 1.8 1.3 Gripper 20 8 +6 +6 +6 0 53968 53968 28353 1 Logistics00 28 10 +6 32.7 3.1 1.2 1 Miconic 150 55 +6

  • 1
  • 5

58.3 8.7 3.4 1 NoMystery 20 8 +10 +1 +1 2497 128 29.1 1.1 OpenStack11 20 17 +2 +2 +1 2.1 2 1.8 2 ParcPrint11 20 6 +5 +3 +1 +14 869 10 1.5 21826 Rovers 40 6 +2 +1 +1 33.4 9.6 1.7 2 Satellite 36 6 72.9 35.3 9.9 10.7 TPP 30 6 6.5 3.4 1 1 Trucks 30 6 +2 24.8 21.9 2.8 1 VisitAll11 20 9 30 25.5 1 1 Woodwork11 20 3 +9 +5 +4 +6 1059 116 92.2 514 Zenotravel 20 8 +1 41.6 1.5 1.1 1

  • 1271

605 +57 +16 +16 +8

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 14 / 19

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SLIDE 31

Experimental Results: LM-cut

Coverage Evaluations Domain # A L S B P L S B P Airport 50 28

  • 1
  • 1
  • 1

+1 1 1 1 4.7 Driverlog 20 13 1.9 1.2 1.2 1 Floortile11 20 7 +1 +1 6.4 6.4 1 1 Gripper 20 7 +7 +7 +7 0 14662 14662 10049 1 Logistics00 28 20 1.9 1.1 1.1 2.9 Miconic 150 141 2.1 1.5 1.1 1 NoMystery 20 14 +6 +3 6.5 3.1 1 1 OpenStack11 20 16 2.5 2.4 2.1 1.8 ParcPrint11 20 13 +7 5 1.2 1.1 1246 Rovers 40 7 +2 +1 +1 +2 6.1 3.8 1.2 4.4 Satellite 36 7 +3 +3 +3 +4 4.8 1.8 1.7 21.5 TPP 30 6 +1 +1 +1 1.2 1.1 1 1 Trucks 30 10 2.7 2.3 1 1 VisitAll11 20 10 +1 +1 7 6.8 1 1 Woodwork11 20 12 +5 +4 +4 +7 91.6 23.8 17 772 Zenotravel 20 13 3.6 1.6 1 1

  • 1271

833 +20 +14 +17 +38

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 15 / 19

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SLIDE 32

Experimental Results: Search Time

0.1 1 10 100 1000 0.1 1 10 100 1000 Search time LDsim (s) Search time baseline (s)

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 16 / 19

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SLIDE 33

Experimental Results: Total Time

0.1 1 10 100 1000 0.1 1 10 100 1000 Total time LDsim (s) Total time baseline (s)

Max partition size: 100 000 transitions

0.1 1 10 100 1000 0.1 1 10 100 1000 Total time LDsim-10k (s) Total time baseline (s)

Max partition size: 10 000 transitions

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 17 / 19

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SLIDE 34

Conclusions

Novel method of dominance pruning useful for many domains Overhead in computing the relation and comparing states during the search Future work:

◮ Find coarser relations ◮ Reduce overhead ◮ Irrelevance pruning

→ SoCS talk on Thursday (joint session with ICAPS)!

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 18 / 19

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SLIDE 35

Thank you for your attention!

Questions?

´ Alvaro Torralba, J¨

  • rg Hoffmann

Simulation Dominance Pruning HSDIP , June 2015 19 / 19