Background Experiments An Empirical Case Study on Symmetry Handling in Cost-Optimal Planning as Heuristic Search Silvan Sievers 1 Martin Wehrle 1 Malte Helmert 1 Michael Katz 2 1 University of Basel Basel, Switzerland 2 IBM Research Haifa, Israel September 23, 2015
Background Experiments Motivation Successful usage of symmetries: Planning: duplicate pruning in A ⋆ , improved merge-and-shrink heuristics Heuristic search: symmetrical/dual lookups
Background Experiments Motivation Successful usage of symmetries: Planning: duplicate pruning in A ⋆ , improved merge-and-shrink heuristics Heuristic search: symmetrical/dual lookups Contribution of this work: Quantitative analysis of symmetries in planning benchmarks Empirical comparison of different symmetry-based techniques (adapted to planning)
Background Experiments Outline Background 1 Experiments 2 Symmetries in Planning Benchmarks Symmetrical Lookups for Planning Comparison of Symmetry-based Techniques
Background Experiments Classical Planning SAS + planning task Π : Finite-domain state variables Initial state: complete variable assignment Goal description: partial variable assignment Operators: preconditions, effects, cost
Background Experiments Classical Planning SAS + planning task Π : State transition graph T Π : Finite-domain state variables 00 Initial state: complete variable assignment o a o b Goal description: partial 01 10 variable assignment Operators: preconditions, o b o a effects, cost 11
Background Experiments Structural Symmetries (Shleyfman et al. 2015) Structural symmetry of a planning task Π : Maps facts (variable/value pairs) to facts and operators to operators Induced symmetry σ on the state transition graph T Π = ( V , E ) is a goal-stable automorphism: ( s , o , s ′ ) ∈ E iff ( σ ( s ) , σ ( o ) , σ ( s ′ ) ∈ E s goal state iff σ ( s ) goal state
Background Experiments Structural Symmetries (Shleyfman et al. 2015) Structural symmetry of a planning task Π : Maps facts (variable/value pairs) to facts and operators to operators Induced symmetry σ on the state transition graph T Π = ( V , E ) is a goal-stable automorphism: ( s , o , s ′ ) ∈ E iff ( σ ( s ) , σ ( o ) , σ ( s ′ ) ∈ E s goal state iff σ ( s ) goal state Example symmetry: σ ( o a ) = o b 00 σ ( o b ) = o a o a o b 01 10 o b o a 11
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 s 0 Before search: find (some) generators of the automorphism group During search: Run A ⋆ as usual When expanding state s , replace successors by orbit representatives, but save regular operators → symmetrical duplicate pruning s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group X X During search: Run A ⋆ as usual When expanding state s , replace successors by orbit representatives, but save regular operators → symmetrical duplicate pruning s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group X X During search: Run A ⋆ as usual When expanding state s , replace successors by orbit representatives, but save regular operators → symmetrical duplicate pruning s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group X X During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators → symmetrical duplicate pruning s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group X X During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators → symmetrical duplicate pruning s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group X X During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators X → symmetrical duplicate pruning s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group X X During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators X → symmetrical duplicate pruning s ∗ s ∗ Credits to A. Shleyfman
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators X → symmetrical duplicate pruning Non-standard plan extraction: s ∗ s ∗ Compute the “real” state sequence Find operators connecting the Credits to A. Shleyfman sequence
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators X → symmetrical duplicate pruning Non-standard plan extraction: s ∗ s ∗ Compute the “real” state sequence Find operators connecting the Credits to A. Shleyfman sequence
Background Experiments Orbit Space Search (Domshlak et al. 2015) Orbit: equivalence class of symmetrical states s 0 Before search: find (some) generators of the automorphism group During search: Run A ⋆ as usual X When expanding state s , replace successors by orbit representatives, but save regular operators X → symmetrical duplicate pruning Non-standard plan extraction: s ∗ s ∗ Compute the “real” state sequence Find operators connecting the Credits to A. Shleyfman sequence
Background Experiments Symmetrical Lookups for Planning (For heuristic search: Felner et al. 2005, Zahavi et al. 2008) Before search: find (some) generators of the automorphism group During search, for a given state s and heuristic h : Compute (a subset of) the orbit containing s : S := { s , s 1 , . . . s m } h ( s ) := max { h ( s ′ ) | s ′ ∈ S } Compute heuristic as ¯ Properties: S can be chosen arbitrarily ¯ h ( s ) is still admissible (if h is)
Background Experiments Bidirectional Pathmax for Planning (For heuristic search: Felner et al. 2011) Symmetrical lookups usually render heuristics inconsistent Consistency: h ( s ) ≤ cost ( o ) + h ( s ′ ) for a transition from s to s ′ with operator o Bidirectional pathmax (BPMX) rule: h ( s ′ ) = max ( h ( s ′ ) , h ( s ) − cost ( o ))
Background Experiments Merge-and-Shrink Heuristic (Helmert et al. 2014) Represent state space as set T of small finite transition systems, with a shared label set L State space corresponds to product of transition systems Transform transition systems to obtain distance heuristic for state space
Background Experiments Factored Symmetries (Sievers et al. 2015) Work on a set T of transition systems as encountered during the merge-and-shrink computation Locally map abstract states to abstract states within elemets of T and globally map transition labels to transition labels in L Goal states must be preserved
Background Experiments Factored Symmetries (Sievers et al. 2015) Work on a set T of transition systems as encountered during the merge-and-shrink computation Locally map abstract states to abstract states within elemets of T and globally map transition labels to transition labels in L Goal states must be preserved Example: σ ( o 1 ) = o 1 a 1 b 1 c 1 c 2 d 0 o 1 o 2 o 3 o 1 σ ( o 2 ) = o 2 o 2 o 1 o 1 o 2 o 2 o 1 o 1 o 3 σ ( o 3 ) = o 3 o 3 a 0 b 0 o 3 c 0 d 1 d 2
Background Experiments Factored Symmetries (Sievers et al. 2015) Work on a set T of transition systems as encountered during the merge-and-shrink computation Locally map abstract states to abstract states within elemets of T and globally map transition labels to transition labels in L Goal states must be preserved Example: σ ( o 1 ) = o 1 a 1 b 1 c 1 c 2 d 0 o 1 o 2 o 3 o 1 σ ( o 2 ) = o 2 o 2 o 1 o 1 o 2 o 2 o 1 o 1 o 3 σ ( o 3 ) = o 3 o 3 a 0 b 0 o 3 c 0 d 1 d 2 Usage: improve merging strategies
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