1. Planning: Part 2 Partial Order Planning COMP151 April 2, 2007

2. Planning: Chapter 11 11.1 The planning problem 11.2 Planning with state-space search 11.3 Partial-order planning 11.4 Planning graphs 11.5 Planning with propositional logic 11.6 Analysis of planning approaches

3. Partial-order Planning Progression and regression planning are totally ordered plan search forms. They cannot take advantage of problem decomposition Decisions must be made on how to sequence actions on all the subproblems POP: Work on subgoals independently Construct plan by working on obvious or more important decisions first Least commitment strategy: delay choice during search

4. Shoe Example Goal(RightShoeOn  LeftShoeOn) Init() Action(RightShoe,PRECOND: RightSockOn EFFECT: RightShoeOn) Action(RightSock,PRECOND: EFFECT: RightSockOn) Action(LeftShoe,PRECOND: LeftSockOn EFFECT: LeftShoeOn) Action(LeftSock,PRECOND: EFFECT: LeftSockOn) Planner: combine two action sequences (1)leftsock, leftshoe (2)rightsock, rightshoe

5. Partial-order Planning (POP) Any planning algorithm that can place two actions into a plan without which comes first is a PO plan. (linearizations)

6. POP as a Search Problem Q: Assuming we have partial plans for subgoals, how can we construct a complete plan? A: Construct a search problem States are (mostly unfinished) plans Start with an empty plan An empty plan contains only start and finish actions Actions are actions on plans (not the actions in the real world) Add a step to the plan Impose an ordering on steps

7. POP as a Search Problem Each plan has 4 components: (1) A set of actions (steps of the plan) Start action: no precondition, effect = initial state Finish action: no effect, precondition = goal state Other actions: from partial plans (2) A set of ordering constraints: A ≺ B (A before B) Cycles represent contradictions

8. POP as a Search Problem Each plan has 4 components: (3) A set of causal links means “A achieves p for B”, where p is a precondition for B Action C conflicts with A  p B if C has effect ¬p and C could come between A and B The causal link protects p from being negated over the interval A to B (4) A set of open preconditions A precondition is open if it is not achieved by some action currently in the plan

9. POP as a Search Problem A consistent plan is one in which: there are no cycles in the ordering constraints no conflicts with the causal links A solution is a consistent plan with no open preconditions

10. Example of Solution Actions = { Rightsock, Rightshoe, Leftsock, Leftshoe, Start, Finish } Orderings = { Rightsock ≺ Rightshoe, Leftsock ≺ Leftshoe } Causal Links = { LeftSock  LeftSockOn LeftShoe, RightSock  RightSockOn RightShoe, LeftShoe  LeftShoeOn Finish, RightShoe  RightShoeOn Finish } Open preconditions = { }

11. POP Algorithm The initial plan contains Start and Finish, the ordering constraint Start ≺ Finish, no causal links, all the preconditions in Finish are open. Successor function : pick one open precondition p on an action B and generate a successor plan for every possible consistent way of choosing action A that achieves p. (see next slide for enforcing consistency) Test goal (no open preconditions)

12. Enforcing consistency When generating successor plan: The causal link A  p B and the ordering constraint A ≺ B are added to the plan. If A is new also add Start ≺ A and A ≺ Finish to the plan Resolve conflicts between new causal link and all existing actions Resolve conflicts between action A (if new) and all existing causal links. A conflict between A  p B and C is resolved by adding C ≺ A or B ≺ C.

13. Algorithm Summary Operators on partial plans Add link from existing plan to open precondition. Add a step to fulfill an open condition. Order one step w.r.t another to remove possible conflicts Gradually move from incomplete/vague plans to complete/correct plans Backtrack if an open condition is unachievable or if a conflict is irresolvable.

14. Example: Spare Tire Problem Init(At(Flat, Axle)  At(Spare,trunk)) Goal(At(Spare,Axle)) Action(Remove(Spare,Trunk) PRECOND: At(Spare,Trunk) EFFECT: ¬At(Spare,Trunk)  At(Spare,Ground)) Action(Remove(Flat,Axle)PRECOND: At(Flat,Axle) EFFECT: ¬At(Flat,Axle)  At(Flat,Ground)) Action(PutOn(Spare,Axle) PRECOND: At(Spare,Ground)  ¬At(Flat,Axle) EFFECT: At(Spare,Axle)  ¬Ar(Spare,Ground)) Action(LeaveOvernightPRECOND: EFFECT: ¬ At(Spare,Ground)  ¬ At(Spare,Axle)  ¬ At(Spare,trunk)  ¬ At(Flat,Ground)  ¬ At(Flat,Axle) )

15. Example: Spare Tire Problem Initial plan: Start with EFFECTS and Finish with PRECOND

16. Example: Spare Tire Problem Pick an open precondition: At(Spare, Axle) Only PutOn(Spare, Axle) is applicable Add causal link: PutOn(Spare, Axle)  At(Spare, Axle) Finish Add constraint : PutOn(Spare, Axle) ≺ Finish

17. Example: Spare Tire Problem Pick an open precondition: At(Spare, Ground) Only Remove(Spare, Trunk) is applicable Add causal link: Remove(Spare, Trunk)  At(Spare, Ground) PutOn(Spare,Axle) Add constraint : Remove(Spare, Trunk) ≺ PutOn(Spare,Axle)

18. Example: Spare Tire Problem Pick an open precondition: ¬At(Flat, Axle) LeaveOverNight is applicable conflict: LeaveOverNight also has the effect ¬ At(Spare,Ground), which conflicts with Remove(Spare, Trunk)  At(Spare, Ground) PutOn(Spare,Axle) To resolve, add constraint : LeaveOverNight ≺ Remove(Spare, Trunk)

19. Example: Spare Tire Problem Add causal link: LeaveOverNight  ¬At(Flat,Axle) PutOn(Spare,Axle)

20. Example: Spare Tire Problem Pick an open precondition: At(Spare, Trunk) Only Start is applicable Add causal link: Start  At(Spare, Trunk) Remove(Spare, Trunk) Conflict: causal link with effect ¬At(Spare,Trunk) in LeaveOverNight No re-ordering solution possible. backtrack

21. Example: Spare Tire Problem Remove LeaveOverNight, Remove(Spare, Trunk) and causal links Repeat step with Remove(Spare,Trunk) Add also RemoveFlatAxle and finish

22. Some details … What happens when a first-order representation that includes variables is used? Complicates the process of detecting and resolving conflicts. Can be resolved by introducing inequality constraint. CSP’s most-constrained-variable constraint can be used for planning algorithms to select a PRECOND.