Automatically Deriving Abstraction Heuristics PDB Abstractions - PowerPoint PPT Presentation

Transition Systems and Abstractions Automatically Deriving Abstraction Heuristics PDB Abstractions Explicit-State Abstractions Conclusion Malte Helmert Albert-Ludwigs-Universit¨ at Freiburg, Germany STAIR 2008

About This Talk Abstraction heuristics Transition Heuristic estimate is goal distance in abstracted state space S ′ Systems and Abstractions obtained as homomorphism of original state space S . PDB Abstractions Canonical example: pattern databases Explicit-State Abstractions Conclusion Abstraction heuristics in the search community A lot of thought has gone into developing (and analyzing) effective abstraction heuristics for particular search problems ( n 2 − 1 -puzzle, Rubik’s Cube, Top Spin, . . . ). This talk is about applying abstraction heuristics to problems where the search space is unknown to the algorithm designer.

Outline Transition Systems and Transition Systems and Abstractions 1 Abstractions PDB Abstractions Explicit-State Automatically Derived PDB Abstractions 2 Abstractions Conclusion Automatically Derived Explicit-State Abstractions 3 Conclusion 4

Transition Systems Transition Definition (transition system) Systems and Abstractions A transition system is a 5-tuple � S, L, A, s 0 , S ⋆ � : PDB Abstractions S : finite set of states Explicit-State Abstractions L : finite set of transition labels Conclusion A ⊆ S × L × S : labelled transitions s 0 ∈ S : initial state S ⋆ ⊆ S : goal states Objective: Find a shortest path from s 0 to some s ⋆ ∈ S ⋆ .

Factored Transition Systems Transition Systems and Abstractions We assume a factored representation of transition systems: PDB Abstractions states: assignments to set V of state variables Explicit-State Abstractions transitions and labels: given by set of operators Conclusion defined in terms of a condition and effect on subsets of V goal states: given by assignment to V ′ ⊆ V

Example: Blocksworld Transition Systems and Abstractions PDB Abstractions C A Explicit-State Abstractions B E A C E Conclusion D F B D F

Example: Pipesworld Transition Systems and Abstractions PDB Abstractions Explicit-State Abstractions Conclusion

Example: FreeCell Transition Systems and Abstractions PDB Abstractions Explicit-State Abstractions Conclusion

Abstractions Definition (abstraction, homomorphism) Transition Systems and Abstraction of transition system T : pair �T ′ , α � where Abstractions T ′ is a transition system with the same labels PDB Abstractions α maps states of T to states of T ′ such that Explicit-State Abstractions 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

Generating Abstractions Transition Systems and Abstractions Conflicting goals in generating abstractions: PDB obtain informative heuristic Abstractions Explicit-State keep representation small Abstractions Conclusion Abstractions have small representations if they have few abstract states succinct encoding for α

Pattern Databases Transition Systems and One idea to get succinct encodings: projections Abstractions � map states to abstract states with perfect hash function PDB Abstractions Definition (projection) Explicit-State Abstractions Projection π V ′ to variables V ′ ⊆ V : Conclusion homomorphism α where α ( s ) = α ( s ′ ) iff s and s ′ agree on V ′ Abstraction heuristics for projections are called pattern database (PDB) heuristics.

Example: Transition System ALR ARL Transition LLR RRL Systems and Abstractions ALL ARR PDB Abstractions LRR LLL RRR RLL Explicit-State Abstractions 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 }

Example: Projection Project to { package } : Transition Systems and Abstractions ALR ARL ALR ARL PDB Abstractions LLR RRL LLR RRL Explicit-State Abstractions 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

Automatically Derived Abstraction Heuristics Our research problem Automatically derive an effective abstraction heuristic Transition Systems and for a given transition system in factored representation. Abstractions PDB Abstractions Explicit-State Some important papers: Abstractions Edelkamp (ECP-01): Planning with PDBs Conclusion Edelkamp (AIPS-02): Symbolic PDBs Haslum et al. (AAAI-05): Constrained PDBs Haslum et al. (AAAI-07): Pattern selection Helmert et al. (ICAPS-07): Explicit-state abstractions Katz & Domshlak (ICAPS-08): Optimal cost partitioning Katz & Domshlak (ICAPS-08): Structural patterns

Automatically Derived Abstraction Heuristics Our research problem Automatically derive an effective abstraction heuristic Transition Systems and for a given transition system in factored representation. Abstractions PDB Abstractions Explicit-State Some important papers: Abstractions Edelkamp (ECP-01): Planning with PDBs Conclusion Edelkamp (AIPS-02): Symbolic PDBs Haslum et al. (AAAI-05): Constrained PDBs Haslum et al. (AAAI-07): Pattern selection Helmert et al. (ICAPS-07): Explicit-state abstractions Katz & Domshlak (ICAPS-08): Optimal cost partitioning Katz & Domshlak (ICAPS-08): Structural patterns

Outline Transition Systems and Transition Systems and Abstractions 1 Abstractions PDB Abstractions Explicit-State Automatically Derived PDB Abstractions 2 Abstractions Conclusion Automatically Derived Explicit-State Abstractions 3 Conclusion 4

Reference Transition Systems and Abstractions This part based on: PDB Abstractions Patrik Haslum, Adi Botea, Malte Helmert, Blai Bonet, Explicit-State Abstractions Sven Koenig. Conclusion Domain-Independent Construction of Pattern Database Heuristics for Cost-Optimal Planning. Proc. AAAI 2007 , pp. 1007–1012, 2007.

PDB Abstractions for Factored Transition Systems Transition Systems and Objective Abstractions Automatically derive an effective pattern database heuristic PDB Abstractions for a given transition system in factored representation. Explicit-State Abstractions Conclusion Guiding questions: 1 What is a pattern for a factored transition system? 2 How can we identify and exploit disjunctive patterns? 3 Which patterns do we choose?

Patterns for Factored Transition Systems 1 What is a pattern for a factored transition system? Transition Systems and Abstractions Most natural definition: PDB identify patterns with sets of state variables to project to Abstractions Explicit-State Abstractions Definition (abstracted transition system) Conclusion Let T be a factored transition system and V its variable set. Let P ⊆ V be a pattern. The abstracted transition system T ( P ) is obtained from T by restricting the initial state to P restricting operator conditions and effects to P removing goal conditions on variables not in P

Pattern Heuristics Transition Definition (pattern heuristic) Systems and Abstractions Let T be a factored transition system and V its variable set. PDB Abstractions Let P ⊆ V be a pattern. Explicit-State The pattern heuristic h P assigns to each state s of T Abstractions Conclusion the length of an optimal solution for T ( P ) , starting from the state obtained by restricting s to P . For all choices of P , heuristic h P is admissible and consistent. What can we do if we have multiple patterns P 1 , . . . , P k ?

Criterion for Disjunctive Patterns Transition 2 How can we identify and exploit disjunctive patterns? Systems and Abstractions PDB Abstractions Theorem (disjunctive patterns) Explicit-State Abstractions Let C be a pattern collection, i.e. a set of patterns of task T . Conclusion We say that an operator affects a pattern P if it can assign a new value to some variable v ∈ P . If no operator in T affects more than one pattern in C , P ∈C h P is admissible and consistent. then �

Finding Disjunctive Patterns Transition Systems and Abstractions Finding sets of disjunctive patterns in a pattern collection C : PDB Abstractions build compatibility graph for C Explicit-State vertices correspond to patterns P ∈ C Abstractions edge between two vertices iff no operator affects both Conclusion compute all maximal cliques of the graph using the algorithm of Tomita, Tanaka & Takahashi

The Canonical Heuristic Function Definition (canonical heuristic function) Let T be a factored transition system and V its variable set. Transition Let C be a pattern collection. Systems and Abstractions The canonical heuristic h C for pattern collection C is defined as PDB Abstractions Explicit-State h P ( s ) , h C ( s ) = � max Abstractions D∈ cliques ( C ) Conclusion P ∈D where cliques ( C ) is the set of all maximal cliques in the compatibility graph for C . For all choices of C , heuristic h C is admissible and consistent. It is the best possible admissible heuristic that can be derived from the information in the pattern databases in C . The full story includes “dominance pruning” to optimize speed.