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CostOptimal Symbolic Pattern Database Planning with State Trajectory and Preference Constraints

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Title: CostOptimal Symbolic Pattern Database Planning with State Trajectory and Preference Constraints


1
Cost-Optimal Symbolic Pattern Database Planning
with State Trajectoryand 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
Intersection found
  • Direction
  • Time
  • BDD Size
  • State Set Size

7
Pattern Databases
  • ? Not used in Competition ?
  • Backward Search only executed in Abstract State
    Space
  • Abstraction (Set SAS-variables to Dont
    Care/Smaller Ranges)
  • Precomputed Partition in BDDs H0,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) ?? igtt 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) a1v1 anvn
  • Compute minF,maxF
  • Encode Range 0,maxF-minF
  • Construct BDD for F
  • Bartzis Bultan (2006)
  • Space Time O(n (a1an))
  • 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 k2 Layers,
  • Planning Graphs klt5
  • Store only Layers that correspond to Concrete
    State Space
  • m-fold reduction, mAutomata
  • 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 Bultans Method So far we
    are using Buddys 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)
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