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Planning with Resources at Multiple Levels of Abstraction Brad Clement, Tony Barrett, Gregg Rabideau Artificial Intelligence Group Jet Propulsion Laboratory California Institute of Technology {bclement, Ed Durfee Artificial Intelligence Lab University of Michigan Paper to appear in proceedings of ECP-01

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Motivation Hierarchical refinement search can reduce the search space exponentially when there is no need to backtrack up the hierarchy. (Korf ’87, Knoblock ’91) Often planning problems have interacting goals such that backtracking is necessary. How can this performance advantage be exploited for such problems? By summarizing the constraints of an abstract task’s refinement, the planner can reduce the complexity exponentially. (Clement & Durfee 2000) How can this work be applied to the use of metric resources?

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Contributions Algorithm summarizing metric resource usage for abstract activities based on their decompositions Complexity analyses showing that scheduling is exponentially cheaper at higher levels of abstraction using summary information Experiments in a multi-rover domain that support the analyses Search techniques for directing the refinement of activities in an iterative repair planner that further improve performance

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Resource Usage Depletable resource –usage carries over after end of task –gas = gas - 5 Non-depletable –usage is only local –zero after end of task –machines = machines - 2 Replenishing a resource –negative usage –gas = gas + 10 –can be depletable or non-depletable interval of task

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Mars Rover morning activities move(A, B) sunbathe soak rays use –4W 20 min soak rays use –5W 20 min soak rays use –6W 20 min low path go(A,1) use 6W 10 min go(1,2) use 6W 10 min go(2,B) use 12W 20 min go(A,B) use 8W 50 min go(A,3) use 8W 15 min go(3,B) use 12W 25 min middle path high path A B C D E F 12 3

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Summarizing Resource Usage soak rays low Battery power usage for three possible decompositions morning activities move(A, B) sunbathe soak rays use –4W 20 min soak rays use –5W 20 min soak rays use –6W 20 min low path go(A,1) use 6W 10 min go(1,2) use 6W 10 min go(2,B) use 12W 20 min go(A,B) use 8W 50 min go(A,3) use 8W 15 min go(3,B) use 12W 25 min middle path high path soak rays medium soak rays high

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summarized resource usage Captures uncertainty of decomposition choices and temporal uncertainty of partially ordered actions Summarizing Resource Usage

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Resource Summarization Algorithm Can be run offline for a domain model Run separately for each resource Recursive from leaves up hierarchy Summarizes parent from summarizations of immediate children Considers all legal orderings of children Considers all subintervals where upper and lower bounds of children’s resource usage may be reached Exponential with number of immediate children, so summarization is really constant for one resource and O(r) for r resources

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Resource Summarization Algorithm OR

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Resource Summarization Algorithm Serial AND

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Resource Summarization Algorithm Parallel AND

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Resource Summarization Algorithm for each consistent ordering of endpoints –for each subtask/subinterval summary usage assignment use Parallel-AND to combine subtask/subinterval usages by subinterval use Serial-AND over the chain of subintervals use OR computation to combine profiles Each iteration generates a profile

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Complexity Analysis Iterative repair planners (such as ASPEN) heuristically pick conflicts and resolve them by moving activities and choosing alternative decompositions of abstract activities. Moving an activity hierarchy to resolve a conflict is O(vnc 2 ) for v state or resource variables, n hierarchies in the schedule, and c constraints in hierarchy per variable. Summarization can collapse the constraints per variable making c smaller. In the worst case, where no constraints are collapsed because they are over different variables, the complexity of moving and activity hierarchies at different levels of expansion is the same.... level 0 1 d 12n branching factor b c constraints per hierarchy v variables

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Complexity Analysis In the other extreme, where constraints are always collapsed when made for the same variable, the number of constraints c is the same as the number of activities and grows b i for b children per activity and depth level i. Thus, the complexity of scheduling operations grows O(vnb 2i ). Along another dimension, the number of temporal constraints that can cause conflicts during scheduling grows exponentially (O(b i )) with the number of activities as hierarchies are expanded. In addition, by using summary information to prune decomposition choices with greater numbers of conflicts, exponential computation is avoided. Thus, reasoning at abstract levels can resolve conflicts exponentially faster.... level 0 1 d 12n branching factor b c constraints per hierarchy v variables

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Decomposition Strategies Expand most threats first (EMTF) –instead of moving activity to resolve conflict, decompose with some probability (decomposition rate) –expands activities involved in greater numbers of conflicts (threats) Level expansion –repair conflicts at current level of abstraction until conflicts cannot be further resolved –then decompose all activities to next level and begin repairing again Relative performance of two techniques depends decomposition rate selected for EMTF

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Decomposition Strategies FTF (fewest-threats-first) heuristic tests each decomposition choice and picks those with fewer conflicts with greater probability. rover_move path1path2path3 10 conflicts 20 conflicts15 conflicts

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Multi-Rover Domain 2 to 5 rovers Triangulated field of 9 to 105 waypoints 6 to 30 science locations assigned according to a multiple travelling salesman algorithm Rovers’ plans contain 3 shortest path choices to reach next science location Paths between waypoints have capacities for a certain number of rovers Rovers cannot be at same location at the same time Rovers cannot cannot cross a path in opposite directions at the same time Rovers communicate with the lander over a shared channel for telemetry--different paths require more bandwidth than others

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Experiments using ASPEN for a Multi-Rover Domain Performance improves greatly when activities share a common resource. Rarely shared resources (only path variables)Mix of rarely shared (paths) and often shared (channel) resources Often shared (channel) resource only

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Experiments using ASPEN for a Multi-Rover Domain CPU time required increases dramatically for solutions found at increasing depth levels.

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Picking branches that result in fewer conflicts (FTF) greatly improves performance. Expanding activities involved in greater numbers of conflicts is better than level-by-level expansion when choosing a proper rate of decomposition Experiments in ASPEN for a Multi-Rover Domain

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Summary Algorithm summarizing metric resource usage for abstract activities based on their decompositions Complexity analyses showing that scheduling operations are exponentially cheaper at higher levels of abstraction when summarizing activities collapses state/resource constraints and temporal constraints Experiments in multi-rover domain support analyses Reasoning at abstract levels is wasteful when hierarchies must be fully expanded and OR branch selection is not important Search techniques for directing the refinement of activities further improve performance

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Future Work How does abstract reasoning affect plan quality using iterative repair? How should complex resources be abstracted? –resource usage functions –window variables –geometrical constraints –volatile objects (file system) What is the effect of abstracting over resource classes? What are efficient protocols for coordinating a group of agents’ plans continually?

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