1. Maintaining Arc-consistency over Mutex Relations in Planning Graphs during Search Pavel Surynek Roman Barták Charles University, Prague Czech Republic

2. Maintaining Arc-consistency over Mutex Relations in Planning Graphs during Search FLAIRS 2007 Pavel Surynek and Roman Barták AI Planning Problem AI Planning Optimal Solution Planning Graphs Arc-consistency Our Contribution Justification Experiments What it is about

3. AI Planning Problems - example / motivation Dock Worker Robots (Ghallab, Nau, Traverso, 2003) location B location C location A 5 6 location D location E location F 3 1 4 2  Initial state of the planning world  Actions transforming states of the planning world location B location C location A 5 location D location E location F 3 1 2 6 4  take(box1,craneA) load(box1,small_truck)  Goal state of the planning world  Task: Transform using actions FLAIRS 2007 Pavel Surynek and Roman Barták

4. AI Planning Problems - optimal solution / existence  Initial state = set of atoms (empty(small_truck), on(box1, box3),...)  Perform actions A1=move(small_truck,locationA) A2=take(craneB,box2),... A1A2An...  Action = a triple (i) precondition = set of atoms, that must be contained in a state (ii) positive effect = set of atoms added to the state (iii) negative effect = set of atoms deleted from the state AxAyAz...  Reach the goal = set of literals (required atoms and forbidden atoms) FLAIRS 2007 Pavel Surynek and Roman Barták  Optimal solution = shortest possible sequence  parallel actions (must not conflict)

5. Conflicting Actions - mutual exclusion FLAIRS 2007 Pavel Surynek and Roman Barták  (two) actions load(small_truck,box1) load(big_truck,box1) are dependent  (two) actions load(small_truck,box1) load(big_truck,box2) are independent 3 1 2 1  Independent actions can be performed in parallel  Dependent cannot occur together at a single time step ►►► actions are mutually excluded = mutex...  A set of actions at a single time step is mutex free

6. Planning Graphs (Blum, Furst, 1997) FLAIRS 2007 Pavel Surynek and Roman Barták x x x x x x x x x x x x x x x x  Goal (mutex free) =  Initial state =  A layer represents set of states new atom new action

7. GraphPlan / Plan Extraction (Blum, Furst, 1997) FLAIRS 2007 Pavel Surynek and Roman Barták x x x x x x x x x x x x x x x x  each action supports its positive effect (set of atoms)  for each atom, we have several supporting actions ►►► choice points  mutex free set of actions supporting the goal induces another goal at lower layer

8. Goal/Sub-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 Our Contribution - Arc-consistency FLAIRS 2007 Pavel Surynek and Roman Barták  Determining a set of mutex free actions - very frequent and time consuming operation ►►►address the sub-problem cleverly  View the layer as a Constraint Satisfaction Problem (CSP) and maintain arc-consistency (MAC) during search for mutex free actions  Select action A1 A1 A2 A3  Propagate changes by AC A4 A6 A9 A8 A10 A11 A12 A5 A7  Solved using one (backtrack) decision (the rest is AC propagation)

9. Why Arc-consistency - several notes FLAIRS 2007 Pavel Surynek and Roman Barták  AC represents a good compromise between time consumption and strength of propagation  Stronger consistency - Singleton AC provides slightly better propagation, but very time consuming  Weaker consistency - Forward checking is fast, but propagates only in neighborhood of the change  Planning graphs are large - hundreds of atoms per layer, hundreds of supporting actions per atom ►►► large search space

10. Experimental evaluation - Towers of Hanoi FLAIRS 2007 Pavel Surynek and Roman Barták  Original puzzle (3 pegs, 4 discs, and 1 hand)  Our generalization (more pegs, discs, and hands)

11. Experimental evaluation - Dock Worker Robots FLAIRS 2007 Pavel Surynek and Roman Barták 5 6 3 1 4 2 9 8 7  Locations with several places for stacks for 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)

12. Experimental evaluation - Refueling planes FLAIRS 2007 Pavel Surynek and Roman Barták 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

13. Experimental evaluation - Results FLAIRS 2007 Pavel Surynek and Roman Barták  Significant improvements on problems with high action parallelism (Dock Worker Robots, Refueling Planes, Hanoi Towers with more hands)  Almost no improvement on problems with no action parallelism (Dock Worker Robots with one truck, Refueling lanes with one plane, Hanoi Towers with one hands)

14. Conclusions FLAIRS 2007 Pavel Surynek and Roman Barták  Improvement of the GraphPlan algorithm Better method for finding mutex free set of actions supporting a goal We use maintaining arc-consistency (AC)  Experimental evaluation Several real-life planning problems (DWR, Planes, Hanoi) Maintaining AC especially successful on problems with high action parallelism