Performance of Coordinating Concurrent Hierarchical Planning Agents Using Summary Information Brad Clement and Ed Durfee University of Michigan Artificial Intelligence Laboratory

Overview Background Claims Other results What is concurrent hierarchical plan coordination? What is summary information? Claims Coordinating at abstract levels is much easier than coordinating at detailed levels in finding some solution. (complexity analysis) Coordinating at abstract levels is better at finding optimal solutions. search techniques and heuristics that leverage summary information preliminary experimental results Other results CHiP coordination algorithm is sound and complete. Resolving threats in a partial order plan is NP-complete.

Multi-level Coordination DA B DB

Multi-level Coordination DA DA A A B DB B DB

Multi-level Coordination DA DA A B DB B DB DA A B DB

Multi-level Coordination DA DA A B DB B DB DA DA A A DB B DB B

Multi-level Coordination DA DA A A B DB B DB DA DA DA A A A DB B DB B B DB

Multi-level Coordination DA DA A A B DB B DB DA DA DA A A A DB B DB B B DB blocked temporal constraints

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

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 postconditions at(A, 0, 4) internal conditions at(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 A B DA DB 1 2 3 4 B B B - before B B B B B

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

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

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 blocked

Reasoning at Abstract Levels Can Improve Performance DA DB Total Cost top-level best mid-level best primitive-level best Computation Cost Execution Cost

Easier to Coordinate at Higher Levels b - branching factor i - level d - depth c - conditions per plan Number of summary conditions per plan grows exponentially up the hierarchy O(bd-ic)

Easier to Coordinate at Higher Levels b - branching factor i - level d - depth c - conditions per plan Number of summary conditions per plan grows exponentially up the hierarchy O(bd-ic) Number of plans per level grows exponentially down the hierarchy O(bi)

Easier to Coordinate at Higher Levels b - branching factor i - level d - depth c - conditions per plan Complexity of identifying threats among plans is O(n2c´2) for n plan steps and c´ summary conditions per step or O(b2dc2)

Easier to Coordinate at Higher Levels b - branching factor i - level d - depth c - conditions per plan The number of orderings to test grows doubly exponentially down the hierarchy O(bi!)

Easier to Coordinate at Higher Levels b - branching factor i - level d - depth c - conditions per plan Resolving threats for a partial order plan is NP-complete (reduced from Hamiltonian Path)

Reasoning at Abstract Levels Can Improve Performance DA DB Total Cost top-level best mid-level best primitive-level best Computation Cost Execution Cost

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

NEO Domain Experiments Compare FAF’s and our strategies for ordering search states and ordering expansions 4 - 8 locations 2 & 3 transports no, partial, & complete overlap in locations visited

NEO Domain Experiments evacuate evacuate no switch one switch two switches no switch one switch two switches cw ccw go to farthest switch & go to farthest go to safe loc move move move move move move move move

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

FTF-EMTF found solutions for 23 problems, 14 optimal FTF-ExCon found solutions for 19 problems, 12 optimal FAF-FAF found solutions for 22 problems, 14 optimal DFS-ExCon found solutions for 6 problems, 3 optimal (not shown)

Future Work What properties of plan hierarchies benefit which heuristics? For different domains, how can the hierarchies be restructured to take advantage of different heuristics? How can greater numbers of agents be continually coordinated as they accomplish, change, or add plans/goals?

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