Cost-Optimal Symbolic Pattern Database Planning with State

Trajectoryand Preference Constraints

Stefan Edelkamp

University of Dortmund

Motivation

Our BDD Planner MIPS compute Step-Optimal Propositional Plans

How can it be extended to compute Cost-Optimal Plans for PDDL3 to take part in the 2006 International Planning Competition?

Overview

BDD-based Planning Forward, Backward Partitioned Images Bidirectional Search

Symbolic Pattern Databases Abstraction Databases Genetic PDBs

Sequential PDDL3 Planning Encoding Cost-Functions Cost-Optimal Breadth-First Branch-And-Bound

Results, Conclusion, Future Work

BDD-based Planning

Symbolic Representation of Planning State Sets

1 Bit per Proposition Inefficient Use Multivariate (SAS+) Encoding

Variable Ordering is important Breadth-First Symbolic Search: Si(x)

represents all states reachable in i steps Symbolic Heuristic Search: Available through Boolean Representation of the Heuristic

Images

T(x,x‘) encodes Transition Relation for Image(x‘) = x States(x) T(x,x‘) Pre-Image(x) = x States(x‘) T(x,x‘)

Forward and Backward Search very much the same

Partition Computation: 1 Transition Relation for each Planning Operator ( and v commute) Disjunctive Partition

Symbolic Bidirectional Search

Direction:

Time BDD Size State Set Size

Intersection found

Pattern Databases

Not used in Competition Backward Search only executed in Abstract State

Space Abstraction (Set SAS+-variables to Don‘t

Care/Smaller Ranges) Precomputed Partition in BDDs H[0,…m], computed

with Backward BFS Guides Search in Concrete State Space Pattern Selection Strategy: Bin-Packing Faster and Smaller than Explicit-PDBs

Genetic Pattern Databases

Not used in Competition (MoChArt) Problem of Greedy Bin Packing:

Selection Strategy influences Efficiency

Many Patterns to Choose FromProposal: Automate Pattern Selection

Problem Genetic Algorithm with Variable-

Selection Vector Genes Selection based on mean heuristic

value as fitness (one PDB) During learning PDBs are constructed

but not used

PDDL3 Sequential Plan Semantics

Symbolic PDDL3 Planning

State Trajectory Constraints PDDL3-to-PDDL2 Approach Poster on Main Conference

Goal Constraints Soft Constraints evaluated at Intersection States BnB Pruning at Intersection States

Temporal Constraints e.g. hold-after (t p): i>t: Openi Openi p

Unidirectional Search (? Bidirectional ?)

BDDs for Linear Expressions

Preference (preference pi Pi) Introduce Boolean Variable bi for Pi, s.t. Indicator Function: bi Pi

Metric F(x) = a1*v1 + … an*vn Compute minF,maxF Encode Range [0,maxF-minF] Construct BDD for F

Bartzis & Bultan (2006): Space & Time: O(n *(a1+…+an)) Encoding crucial, if well-chosen better than ADDs

Cost-Optimal Search

Correctness

Theorem: The latest plan stored by the symbolic search planner Cost-Optimal-Symbolic-BFS has minimal cost.

Proof: The algorithm applies full duplicate detection and traverses entire planning state space. It generates each planning state exactly once. Only clearly inferior states are pruned in the intersection (when evaluated in Eval and taken into conjunct with Bound). Therefore, Metric empty only if there is no state in the intersection that has an improved bound.

Memory Savings

Locality k: Number of Previous Layers to be looked up for Duplicate Elimination Undirected Graphs: k=2 Layers, Planning Graphs: k<5

Store only Layers that correspond to Concrete State Space m-fold reduction, m=#Automata Automata Transition correspond to Axioms

Results GA-Optimization

Conclusion

1st Approach to PDB optimization Solves Pattern Selection Problem Some Memory Saving Strategies Results: See IPC-5

Future Work

Implement Bartzis & Bultan‘s Method: So far we are using Buddy‘s Functionality to come up with same result but with more work (fixpoint computation)

Bidirectional Constraint Search Natural Numbers, Real-Time Variables: Use Büchi-

Automata Representation for Presburger Arithmetic as e.g. suggested by Felix Klaedtke CAV-06

Combine Symbolic Search and Externalization ( ICAPS 05)