1. Cost-Optimal Symbolic Pattern Database Planning with State Trajectory and Preference Constraints Stefan Edelkamp University of Dortmund

2. 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?

3. 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

4. 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

5. 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

6. Symbolic Bidirectional Search Direction: Time BDD Size State Set Size Intersection found

7. 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

8. Genetic Pattern Databases  Not used in Competition (MoChArt)  Problem of Greedy Bin Packing: Selection Strategy influences Efficiency Many Patterns to Choose From Proposal: 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

9. PDDL3 Sequential Plan Semantics

10. 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 ?)

11. 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

12. Cost-Optimal Search

13. 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.

14. 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

15. Results GA-Optimization

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

17. 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)