Explicit-State Abstraction Explicit-State Abstraction: A New Method Abstractions for Generating Heuristic Functions Projections Explicit-State Abstractions Evaluation Malte Helmert 1 Patrik Haslum 2 org Hoffmann 3 J¨ Conclusion 1 Albert-Ludwigs-Universit¨ at Freiburg, Germany 2 NICTA & Australian National University, Australia 3 University of Innsbruck, Austria AAAI 2008, Nectar track
One-Slide Summary Explicit-State Abstraction Abstractions Abstraction heuristics Projections Heuristic estimate is goal distance in abstracted state space S ′ Explicit-State Abstractions obtained as homomorphism of original state space S . Evaluation Conclusion
One-Slide Summary Explicit-State Abstraction Abstractions Abstraction heuristics Projections Heuristic estimate is goal distance in abstracted state space S ′ Explicit-State Abstractions obtained as homomorphism of original state space S . Evaluation Conclusion Explicit-state abstraction heuristics You have seen other abstraction heuristics before; they are called pattern database heuristics. Ours can do the same and then some.
Outline Explicit-State Abstraction Abstractions 1 Abstractions Projections Explicit-State Projections 2 Abstractions Evaluation Conclusion Explicit-State Abstractions 3 Evaluation 4 Conclusion 5
Transition Graphs Explicit-State Abstraction Abstractions Definition (transition graph) Projections A transition graph is a 5-tuple � S, L, A, s 0 , S ⋆ � : Explicit-State Abstractions S : finite set of states Evaluation L : finite set of transition labels Conclusion A ⊆ S × L × S : labelled transitions s 0 ∈ S : initial state S ⋆ ⊆ S : goal states Assumption: States are assignments to a set of state variables.
Running Example Explicit-State Abstraction ALR ARL LLR RRL Abstractions Projections ALL ARR Explicit-State Abstractions LRR LLL RRR RLL Evaluation BLL BRR Conclusion LRL RLR BRL BLR Logistics problem with one package, two trucks, two locations: state variable package: { L, R, A, B } state variable truck A: { L, R } state variable truck B: { L, R }
Abstractions Explicit-State Abstraction Definition (abstraction, homomorphism) Abstractions Abstraction of transition graph T : pair �T ′ , α � where Projections T ′ is a transition graph with the same labels Explicit-State Abstractions α maps states of T to states of T ′ such that Evaluation initial state maps to initial state Conclusion goal states map to goal states transitions � s, l, s ′ � map to transitions � α ( s ) , l, α ( s ′ ) � Abstraction (and α ) is a homomorphism if T ′ only contains necessary goal states and transitions. Abstraction heuristic: h ( s ) = d ⋆ ( α ( s )) admissible, consistent
Example: Perfect Abstraction Explicit-State Abstraction Abstractions ALR ALR ARL ARL Projections LLR LLR RRL RRL Explicit-State Abstractions ALL ARR ALL ARR Evaluation Conclusion LLL RRR LRR LLL RRR RLL BLL BRR BLL BRR LRL LRL RLR RLR BRL BRL BLR BLR � perfect heuristic h ∗
Generating Abstractions Explicit-State Abstraction Abstractions Conflicting goals in generating abstractions: Projections Explicit-State obtain informative heuristic Abstractions keep representation small Evaluation Conclusion Abstractions have small representations if they have few abstract states succinct encoding for α
Outline Explicit-State Abstraction Abstractions 1 Abstractions Projections Explicit-State Projections 2 Abstractions Evaluation Conclusion Explicit-State Abstractions 3 Evaluation 4 Conclusion 5
Projections Explicit-State Abstraction Abstractions One idea to get succinct encodings: projections Projections � map states to abstract states with perfect hash function Explicit-State Abstractions Evaluation Definition (projection) Conclusion Projection π V ′ to variables V ′ ⊆ V : homomorphism α where α ( s ) = α ( s ′ ) iff s and s ′ agree on V ′ shorthand for atomic projections: π v := π { v } ( v ∈ V )
Example: Projection (1) Explicit-State Abstraction Project to { package } : Abstractions Projections ALR ARL ALR ARL Explicit-State Abstractions LLR RRL LLR RRL Evaluation ALL ALL ARR ARR Conclusion LRR LRR LLL LLL RRR RRR RLL RLL BLL BLL BRR BRR LRL LRL RLR RLR BRL BLR BRL BLR
Example: Projection (2) Explicit-State Abstraction Project to { package , truck A } : Abstractions Projections ALR ALR ARL ARL Explicit-State Abstractions LLR LLR RRL RRL Evaluation ALL ARR ALL ARR Conclusion LRR LLL RRR RLL LRR LLL RRR RLL BLL BLL BRR BRR LRL LRL RLR RLR BRL BRL BLR BLR
Example: Projection (2) Explicit-State Abstraction Project to { package , truck A } : Abstractions Projections ALR ARL ALR ARL Explicit-State Abstractions LLR LLR RRL RRL Evaluation ALL ALL ARR ARR Conclusion LRR LRR LLL LLL RRR RRR RLL RLL BLL BLL BLR BLR LRL RLR LRL RLR BRL BRR BRL BRR
Problems of Projections Explicit-State Abstraction abstraction heuristics for projections are Abstractions pattern database (PDB) heuristics Projections must keep number of reflected variables (pattern) small Explicit-State Abstractions Evaluation price in heuristic accuracy: Conclusion consider generalization of running example: N trucks, M locations (still one package) consider any pattern that is proper subset of V h ( s 0 ) ≤ 2 � no better than atomic projection to package (maximizing over patterns or additive patterns do not help either)
Outline Explicit-State Abstraction Abstractions 1 Abstractions Projections Explicit-State Projections 2 Abstractions Evaluation Conclusion Explicit-State Abstractions 3 Evaluation 4 Conclusion 5
Explicit-State Abstraction Heuristics: Main Idea Explicit-State Abstraction Abstractions Projections Explicit-State Main idea Abstractions (due to Dr¨ ager, Finkbeiner & Podelski, 2006): Evaluation Conclusion Instead of perfectly reflecting a few state variables, reflect all state variables, but in a potentially lossy way.
Explicit-State Abstraction Heuristics: Key Insights Explicit-State Abstraction Key insights: 1 Information of two abstractions A and A ′ of the same Abstractions transition system can be composed by a simple Projections graph-theoretic operation (synchronized product A ⊗ A ′ ). Explicit-State Abstractions 2 Under suitable conditions (factored transition systems), Evaluation the complete state space can be recovered Conclusion using only atomic projections: � π v is isomorphic to π V . v ∈V � build fine-grained abstractions from coarse ones 3 When intermediate results become too big, we can shrink them by aggregating some abstract states.
Computing Explicit-State Abstractions Explicit-State Abstraction Generic abstraction computation algorithm abs := all atomic projections π v ( v ∈ V ). Abstractions while abs contains more than one abstraction: Projections select A 1 , A 2 from abs Explicit-State Abstractions shrink A 1 and/or A 2 until size ( A 1 ) · size ( A 2 ) ≤ N Evaluation abs := abs \ {A 1 , A 2 } ∪ {A 1 ⊗ A 2 } Conclusion return the remaining abstraction N : parameter bounding number of abstract states Questions for practical implementation: Which abstractions to select? � composition strategy How to shrink an abstraction? � shrinking strategy How to choose N ?
Outline Explicit-State Abstraction Abstractions 1 Abstractions Projections Explicit-State Projections 2 Abstractions Evaluation Conclusion Explicit-State Abstractions 3 Evaluation 4 Conclusion 5
Guiding Questions for Evaluation Explicit-State Abstraction Comparison to state of the art Is this competitive with the state of the art? Abstractions Compare scaling behaviour to other heuristics: Projections blind, h max , PDB Explicit-State Abstractions � next slide Evaluation Conclusion Comparison to pattern databases How does this compare to well-chosen PDB heuristics? compare to approach of Haslum et al. (2007) compare scaling behaviour and runtime compare heuristic quality, preprocessing time, search time � details in the ICAPS 2007 paper
Comparison to State of the Art Explicit-State Abstraction Comparison to state of the art Is this competitive with the state of the art? Abstractions Projections Compare scaling behaviour to other heuristics: Explicit-State blind, h max , PDB Abstractions Evaluation h max Domain blind PDB abs Conclusion 14 15 15 Pipes-NoTankage 19 Pipes-Tankage 13 10 10 7 4 5 6 Satellite 6 Logistics 18 6 6 16 PSR 5 3 4 4 7 5 6 6 TPP total 68 42 46 54
Comparison to Pattern Databases: Theory Explicit-State As powerful as PDBs Abstraction PDB heuristics are a special case of our abstraction heuristics, Abstractions and arise naturally as a side product. Projections Explicit-State Get additivity for free Abstractions If P and P ′ are additive patterns, then Evaluation for all h -preserving abstractions A of π P and A ′ of π P ′ , Conclusion the abstraction heuristic for A ⊗ A ′ dominates h P + h P ′ . Greater representational power In some planning domains where PDBs have unbounded error ( Gripper , Schedule , two Promela variants), we can obtain perfect heuristics in polynomial time with suitable composition/shrinking strategies.
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