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Projection Global Consistency: Application in AI Planning Pavel Surynek Charles University, Prague Czech Republic

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Outline of the presentation Problem: select a set of non-mutex actions supporting a goal Problem: select a set of non-mutex actions supporting a goal Obstacle: NP-complete Obstacle: NP-complete (Partial) solution: global consistency - projection consistency (Partial) solution: global consistency - projection consistency Application: AI Planning using planning graphs Application: AI Planning using planning graphs Experiments: several planning domains Experiments: several planning domains CSCLP 2006 Pavel Surynek

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Problem - support problem Goal = finite set of atoms Goal = finite set of atoms Action = triple (preconditions, positive effects, negative effects) Action = triple (preconditions, positive effects, negative effects) CSCLP 2006 Pavel Surynek Goal: A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 x atom 1 atom 2 atom 3atom 4 supports for atom 1 supports for atom 2 supports for atom 3 supports for atom 4 x x x x x x x A1 A2 A3 A4 A6 A9 A8 A10 A11 A12 A5 A7 solution - mutex free

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Mutex actions CSCLP 2006 Pavel Surynek 3 1 2 1 (two) actions load(small_truck,box1) load(big_truck,box2) are independent ( two) actions load(small_truck,box1) load(big_truck,box1) are dependent example of action example of action –load(small_truck, box1) = ({empty(truck), on(bottom, box1)}; {loaded(box1, truck)}; {¬empty(truck), ¬on(bottom, box1)}) generalized dependency = mutex generalized dependency = mutex

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Obstacle - NP completeness Instance: Instance: –goal augmented with finite sets of supporting actions –finite set of mutexes between actions Answer: Answer: –set of non-mutex actions supporting the goal NP-complete NP-complete –SAT instance in CNF ►►► support problem CSCLP 2006 Pavel Surynek

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Projection consistency Interpret support problem as a graph Interpret support problem as a graph Greedily find (vertex disjoint) cliques ►►► Greedily find (vertex disjoint) cliques ►►► ►►► clique decomposition ►►► clique decomposition At most one action from each clique can be selected At most one action from each clique can be selected CSCLP 2006 Pavel Surynek a real support problem Trucks, Cranes, Locations

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Counting argument Clique decomposition C 1, C 2,..., C k Clique decomposition C 1, C 2,..., C k Contribution of an action a Contribution of an action a c(a) = number of supported atoms Contribution of a clique C Contribution of a clique C c(C) = max a C c(a) Counting argument (simplest form) Counting argument (simplest form) if ∑ i=1...k c(C i ) < size of the goal ►►► ►►► the goal is unsatisfiable Generalized form = projection consistency Generalized form = projection consistency – w.r.t. sub-goals and singleton approach CSCLP 2006 Pavel Surynek

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Application: AI Planning Planning problem Planning problem – Initial state: set of atoms – Set of allowed actions – Goal: set of atoms (literals) Task Task – determine a sequence of actions transforming initial state to the goal Solution: planning graphs and GraphPlan algorithm (Blum & Furst, 1997) - support problem arise as a frequent sub-problem Solution: planning graphs and GraphPlan algorithm (Blum & Furst, 1997) - support problem arise as a frequent sub-problem CSCLP 2006 Pavel Surynek location B location C location A 5 6 location D location E location F 3 1 4 2 location B location C location A 5 location D location E location F 3 1 2 6 4

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Experiments: towers of Hanoi CSCLP 2006 Pavel Surynek Original puzzle (3 pegs, 4 discs, and 1 hand) Our generalization (more pegs, discs, and hands)

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Experiments: DWR CSCLP 2006 Pavel Surynek 5 6 3 1 4 2 9 8 7 Locations with several places for stacks of boxes and with several cranes Each crane can reach some stacks within location (not all) Trucks of various capacities (small - 1 box, big - 2 boxes)

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Experiments: Refueling planes CSCLP 2006 Pavel Surynek distance X distance Y distance Z Several planes dislocated at several airports Transport a fleet of planes at destination airport Airport - unlimited source of fuel, planes can refuel in-flight

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Experiments: Results CSCLP 2006 Pavel Surynek Significant improvements on problems with high action parallelism (Dock Worker Robots, Refueling Planes, Hanoi Towers with more hands)

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Conclusions Improvement of the GraphPlan algorithm – –Better method for finding mutex free set of actions supporting a goal – –We use projection consistency Experimental evaluation – –Projection consistency especially successful on problems with high action parallelism CSCLP 2006 Pavel Surynek

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© The McGraw-Hill Companies, Inc., 2005 8 - 1 Chapter 8 The Theory of NP-Completeness.

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