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 planning: ( V , O , I , G ) A ∗ + admissible heuristic h(s): estimates distance to goal Pruning methods: Partial-order pruning 1 Symmetries 2 Dominance pruning 3 ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 2 / 19
Dominance Pruning ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 3 / 19
Dominance Pruning Detect “better than” states V = { at-T= { A, B } , at-P= { A, B, T } } I = { at-T A, at-P A } G = { at-P B } A B O = { move-T (A, B), move-T (B, A), load-P(A), . . . } ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 4 / 19
Dominance Pruning Detect “better than” states V = { at-T= { A, B } , at-P= { A, B, T } } I = { at-T A, at-P A } G = { at-P B } A B O = { move-T (A, B), move-T (B, A), load-P(A), . . . } What do you prefer? A B A B ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 4 / 19
Dominance Pruning Detect “better than” states V = { at-T= { A, B } , at-P= { A, B, T } } I = { at-T A, at-P A } G = { at-P B } A 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¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 4 / 19
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 s 1 s 2 s 3 I s 4 ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 5 / 19
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 s 1 s 2 s 3 I s 4 ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 5 / 19
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 s 1 s 2 s 3 I s 4 s 4 � I ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 5 / 19
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 s 1 � s 3 s 1 s 2 s 3 I s 4 s 4 � I ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 5 / 19
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 s 1 � s 3 s 1 s 5 Challenges: How to find good dominance 1 s 2 s 6 relations? How to efficiently check 2 s 3 s 7 I dominance? s 4 s 4 � I ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 5 / 19
Simulation Relation Definition (Simulation) A binary relation �⊆ S × S is a simulation for Θ if, whenever s � t , l → t ′ s.t. s ′ � t ′ . We call � l → s ′ there exists t for every transition s − − goal-respecting for Θ if, whenever s � t , s ∈ S G implies that t ∈ S G . l 1 A B l 2 l 1 G l 2 l 2 C D 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¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 6 / 19
Simulation Relation Definition (Simulation) A binary relation �⊆ S × S is a simulation for Θ if, whenever s � t , → s ′ there exists t l → t ′ s.t. s ′ � t ′ . We call � l for every transition s − − goal-respecting for Θ if, whenever s � t , s ∈ S G implies that t ∈ S G . 1 A B 1 Cost-simulation: replace labels by their cost 1 G 1 → A cost-simulation on the state space of 1 the planning task is a dominance relation C D D � B , C � A Thm: A unique coarsest goal-respecting simulation always exists and can be computed in time polynomial in the size of Θ ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 6 / 19
Compositional Approach Consider a partition of the problem: Θ 1 , . . . , Θ k 1 Compute a simulation for each part: � 1 , . . . , � k 2 � : s � t iff ∀ i ∈ [ 1 , k ] s i � i t i 3 � is a cost-simulation ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 7 / 19
Compositional Approach Consider a partition of the problem: Θ 1 , . . . , Θ k 1 Compute a simulation for each part: � 1 , . . . , � k 2 � : s � t iff ∀ i ∈ [ 1 , k ] s i � i t i 3 � is a cost-simulation In our example: Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 7 / 19
Compositional Approach Consider a partition of the problem: Θ 1 , . . . , Θ k 1 Compute a simulation for each part: � 1 , . . . , � k 2 � : s � t iff ∀ i ∈ [ 1 , k ] s i � i t i 3 � is a cost-simulation In our example: Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB A � 2 T � 2 B ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 7 / 19
Not so fast → t ′ s.t. s ′ � t ′ → s ′ there exists t l l Definition of s � t : For every s − − Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB A � 2 T � 2 B ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 8 / 19
Not so fast → t ′ s.t. s ′ � t ′ → s ′ there exists t l l Definition of s � t : For every s − − Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB A � 2 T � 2 B lB → B and there is no B lB T �� 2 B : T − − → ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 8 / 19
Not so fast → t ′ s.t. s ′ � t ′ → s ′ there exists t l l Definition of s � t : For every s − − Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB A � 2 T � 2 B lB → B and there is no B lB T �� 2 B : T − − → lB lA T − → B and T − → A are simulated by B → ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 8 / 19
Not so fast → t ′ s.t. s ′ � t ′ → s ′ there exists t l l Definition of s � t : For every s − − Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB A � 2 T � 2 B lB → B and there is no B lB T �� 2 B : T − − → lB lA noop T − → B and T − → A are simulated by B − − − → B lA , lB do not have useful effects in the rest of the problem ( Θ 1 )! ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 8 / 19
Label-Dominance Simulation Definition (Label Dominance) → s ′ ∈ Θ there exists s l ′ l l ′ dominates l in Θ given � if for every s − − → t ′ s.t. s ′ � t ′ Definition (Label-Dominance Simulation) A set R = {� 1 , . . . , � k } of binary relations � i ⊆ S i × S i is a label-dominance simulation for { Θ 1 , . . . , Θ k } if, whenever s � i t : s ∈ S G i implies that t ∈ S G i l l ′ → s ′ in Θ i , there exists t → t ′ in Θ i s.t.: For every s − − s ′ � i t ′ , 1 c ( l ′ ) ≤ c ( l ) , and 2 for all j � = i , l ′ dominates l in Θ j given � j 3 ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 9 / 19
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 ∈ S i , s �∈ S i G or t ∈ S i 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 Θ 1 ⊗ · · · ⊗ Θ k ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 10 / 19
Computation of Label-Dominance Simulation Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB Truck Package A � 1 { B } A � 2 { T, B } B � 1 { A } T � 2 { A, B } B � 2 { } noop simulates { lA , lB , dr } ( noop ≡ dr ) simulate { lA } dr simulates { lA , lB } ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 11 / 19
Computation of Label-Dominance Simulation Θ 1 : lA lB dr A B (truck) dr Θ 2 : dr dr dr lA lB A T B (package) lA lB Truck Package A � 1 { B } A � 2 { T, B } B � 1 { A } T � 2 { A, B } B � 2 { } noop simulates { lA , lB , dr } ( noop ≡ dr ) simulate { lA } dr simulates { lA , lB } ´ Alvaro Torralba, J¨ org Hoffmann Simulation Dominance Pruning HSDIP , June 2015 11 / 19
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