1. Performance of Coordinating Concurrent Hierarchical Planning Agents Using Summary Information Brad Clement and Ed Durfee University of Michigan Artificial Intelligence Laboratory Recent research has provided methods for coordinating the individually formed concurrent hierarchical plans (CHiPs) of a group of agents in a shared environment. A reasonable criticism of this technique is that the summary information can grow exponentially as it is propagated up a plan hierarchy. This paper analyzes the complexity of the coordination problem to show that in spite of this exponential growth, coordinating CHiPs at higher levels is still exponentially cheaper than at lower levels. In addition, this paper offers heuristics, including “fewest threats first” (FTF) and “expand most threats first” (EMTF), that take advantage of summary information to smartly direct the search for a global plan. Experiments show that for a particular domain these heuristics greatly improve the search for the optimal global plan compared to a “fewest alternatives first” (FAF) heuristic that has been successful in Hierarchical Task Network (HTN) Planning.

2. Multi-level Coordination A B DADA DBDB A B DADA DBDB A B DADA DBDB A B DADA DBDB A B DADA DBDB temporal constraints blocked

3. Coordinating at Abstract Levels Resolve conflicts at high level to minimize search time Better solutions may exist at lower levels coordination levels crispersolutions lower coordination cost flexibility

4. Concurrent Hierarchical Plans (CHiPs) and Summary Information pre, in, & postconditions - sets of literals over a set of propositions summary information –external preconditions at(A, 0, 0) –external postconditionsat(A, 0, 4) –internal conditionsat(A, 1, 1) –must, may, always, sometimes at(A, 1, 2) must sometimes hold at(A, 0, 1) may sometimes hold havePower(A) must always hold B - before B B BB BBB A B DADA DBDB 0 1 2 01234

5. Deriving Summary Information Recursive procedure bottoming out at primitives Derived from those of immediate subplans O(n 2 c 2 ) for n non-primitive plans in hierarchy and c conditions in each set of pre, in, and postconditions Proven procedures for determining must/may - achieve/undo/clobber Properties of summary conditions are proven based on procedure

6. Summary Information Summarize conditions of potential refinements at abstract levels Reason about abstract plan interactions among agents –resolve all conflicts at abstract level –prune inconsistent refinement choices at abstract levels –make refinement choices based on task interactions

7. Concurrent Hierarchical Plan Coordination Agents individually derive summary information for their plan hierarchies Coordinator requests summary information for expansions of agents’ hierarchies from the top down After each expansion, try to resolve threats by adding ordering constraints Algorithm shown to be sound and complete

8. Search for Coordinated Plan search state –set of expanded plans –set of blocked subplans –set of temporal constraints search operators –expand –block –constrain blocked temporal constraints

9. Reasoning at Abstract Levels Can Improve Performance Total Cost mid-level best top-level best primitive-level best A B DADA DBDB Computation Cost Execution Cost

10. Easier to Coordinate at Higher Levels Complexity of identifying threats among plans is O(n 2 c´ 2 ) for n plan steps and c´ summary conditions per step or O(b 2d c 2 ) b - branching factor i - level d - depth c - conditions per plan c´=O(b d-i c) n=O(bi)n=O(bi)

11. Easier to Coordinate at Higher Levels The number of orderings to test grows doubly exponentially down the hierarchy O(b i !) Resolving threats for a partial order plan is NP- complete (reduced from Hamiltonian Path) b - branching factor i - level d - depth c - conditions per plan

12. Search Techniques Prune inconsistent global plans Branch & bound - abstract solutions help prune space where cost is higher “Expand most threats first” (EMTF) –expand subplan involved in most threats –focuses search on driving down to source of conflict “Fewest threats first” (FTF) –search plan states with fewest threats first –or subplans involved in most threats are blocked first

13. evacuate no switch one switch two switches no switch one switch two switches cwccw go to farthest switch & go to farthest go to safe loc move NEO Domain Experiments 4 - 8 locations 2 & 3 transports no, partial, & complete overlap in locations visited

14. Summary Information vs. FAF CPU Time in units of 1/100 CPU sec. FAF only found solutions for 6 problems

15. Contributions Sound and complete concurrent hierarchical plan coordination algorithm Complexity analysis showing that resolving conflicts at higher levels is much easier than at lower levels Search techniques including FTF and EMTF heuristics that take advantage of summary information Preliminary experiments showing that these techniques can greatly improve the search for optimal plans