Graph-based Planning

1. Graph-based Planning Brian C. Williams Sept. 25th & 30th, 2002 16.412J/6.834J

2. Outline • Introduction • The Graph Plan Planning Problem • Graph Construction • Solution Extraction • Properties • Termination with Failure

3. Graph Plan • Graphplan was developed in 1995 by Avrim Blum and Merrick Furst, at CMU. • Recent implementations by other researchers have extended the capability of Graphplan to reason with temporally extended actions, metrics and non-atomic preconditions and effects.

4. Proposition Init State Action Time 1 Proposition Time 1 Action Time 2 Approach: Graph Plan • Constructs compact encoding of state space from operators and initial state, which prunes many invalid plans – Plan Graph. • Generates plan by searching for a consistent subgraph that achieves the goals.

5. Plan graphs focus towards valid plans • Plan graphs exclude many plans that are infeasible. • Plans that do not satisfy the initial or goal state. • Plans with operators that threaten each other. • Plan graphs are constructed in polynomial time and are of polynomial in size. • The plan graph does not eliminate all infeasible plans. • Plan generation still requires focused search.

6. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Example: Graph and Solution cleanH quiet noGarb cleanH quiet dinner present noGarb cleanH quiet dinner present carry dolly cook wrap carry dolly cook wrap

7. Outline • Introduction • The Graph Plan Planning Problem • Graph Construction • Solution Extraction • Properties • Termination with failure

8. GraphPlan Planning Problem • State • A consistent conjunction of propositions (positive literals) • E.g., (and (cleanhands)(quiet) (dinner) (present)(noGarbage)) • Doesn’t commit to the truth of all propositions • Initial State • Problem state at time i = 0 • E.g., (and (cleanHands) (quiet)) • Goal State • A partial state, represented by a conjunction of literals. • E.g., (and (noGarbage) (dinner) (present)) • The planner must put the system in a final state that satisfies the conjunction.

9. noGarb cleanH carry GraphPlan Planning Problem Actions • E.g., (:operator carry :precondition :effect (:and (noGarbage) (not (cleanHands))) • Preconditions: propositions that must be true to apply operator. • A conjunction of propositions (no negated propositions). • Effects: Propositions that operator changes, given preconditions. • A conjunction of propositions (adds) and their negation (deletes). Add Edge Delete Edge • Parameterized schemas and objects are used to compactly encode actions.

10. noGarb cleanH carry dinner cook cleanHands GraphPlan Planning Problem Action Execution at time i: If all propositions of :precondition appear in the state at i, Then state at i+1 is created from state at i, by adding to i, all “add” propositions in :effects, removing from i, all “delete” propositions in :effects. (:operator cook :precondition (cleanHands) :effect (dinner)) (:operator carry :precondition :effect (:and (noGarbage) (not (cleanHands)))

11. P P Noop-P GraphPlan Planning Problem • Noops • Every proposition has a no-op action, which maintains it from time i to i+1. • E.g., (:operator noop-P :precondition (P) :effect (P))

12. Example: Dinner Date Problem Initial Conditions: (and (cleanHands) (quiet)) Goal: (and (noGarbage) (dinner) (present)) Actions: (:operator carry :precondition :effect (and (noGarbage) (not (cleanHands))) (:operator dolly :precondition :effect (and (noGarbage) (not (quiet))) (:operator cook :precondition (cleanHands) :effect (dinner)) (:operator wrap :precondition (quiet) :effect (present)) + noops

13. Prop at 1 Action at 1 Prop at 2 Action at 2 Prop at 2 A Plan in GraphPlan <Actions[i] > • Sets of concurrent actions performed at each time [i] • Uses model of concurrency as interleaving. • If actions a and b occur at time i, then it must be valid toperform either a followed by b, OR b followed by a. dinner present cook wrap cleanH quiet noGarb cleanH dinner present carry noop-dinner noop-present

14. A Complete Consistent Plan Given that the initial state holds at time 0, a plan is a solution iff: • Complete: • The preconditions of every operator at time i, is satisfied by a proposition at time i. • The goal propositions all hold in the final state. • Consistent: • The operators at any time i can be executed in any order, without one of these operators undoing the: • preconditions of another operator at time i. • effects of another operator at time i.

15. Prop at 1 Action at 1 Prop at 2 Action at 2 Prop at 3 Example of a Complete Consistent Plan dinner present Initial Conditions: (and (cleanHands) (quiet)) Goal: (and (noGarbage) (dinner) (present)) cook wrap cleanH quiet noGarb cleanH dinner present carry (noop dinner) (noop present)

16. GraphPlan Algorithm • Phase 1 – Plan Graph Expansion • Creates graph encoding pairwise consistency and reachability of actions and propositions from initial state. • Graph includes, as a subset, all plans that are complete and consistent. • Phase 2 - Solution Extraction • Graph treated as a kind of constraint satisfaction problem (CSP). • Selects whether or not to perform each action at each time point, by assigning CSP variables and testing consistency.

17. Outline • Introduction • The Planning Problem • Graph Construction • Solution Extraction • Properties • Termination with failure

18. Graph Properties A Plan graph • compactly encodes the space of consistent plans, • while pruning . . . • partial states and actions at each time i that are not reachable from the initial state. • pairs of actions and propositions that are mutually inconsistent at time i. • plans that cannot reach the goals.

19. Constructing the planning graph…(Reachability) • Initial proposition layer • Contains propositions in initial state.

20. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Example: Initial State, Layer 0 cleanH quiet

21. Constructing the planning graph…(Reachability) • Initial proposition layer • Contains propositions in initial state • Action layer i • If all action’s preconditions are consistent in i-1 • Then add action to layer i • Proposition layer i+1 • For each action at layer i • Add all its effects at layer i+1

22. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Example: Add Actions and Effects cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

23. Constructing the planning graph…(Reachability) • Initial proposition layer • Write down just the initial conditions • Action layer i • If all action’s preconditions appear consistent in i-1 • Then add action to layer i • Proposition layer i+1 • For each action at layer i • Add all its effects at layer i+1 • Repeat until all goal propositions appear

24. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Can a solution exist? cleanH quiet noGarb cleanH quiet dinner present Do all goal propositions appear? carry dolly cook wrap

25. Constructing the planning graph…(Consistency) • Initial proposition layer • Write down just the initial conditions • Action layer i • If action’s preconditions appear consistent in i-1 (non-mutex) • Then add action to layer i • Proposition layer i+1 • For each action at layer i • Add all its effects at layer i+1 • Identify mutual exclusions • Actions in layer i • Propositions in layer i + 1 • Repeat until all goal propositions appear non-mutex

26. Mutual Exclusion: Actions • Actions A,B are mutuallyexclusive at level iif no valid plan could possibly contain both at i: • They Interfere • A deletes B’s preconditions, or • A deletes B’s effects, or • Vice versa or • OR . . . .

27. 1 Prop 1 Action 2 Prop 2 Action 3 Prop What causes the exclusion? cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

28. 1 Prop 1 Action 2 Prop 2 Action 3 Prop What causes the exclusion? cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

29. 1 Prop 1 Action 2 Prop 2 Action 3 Prop What causes the exclusion? cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

30. 1 Prop 1 Action 2 Prop 2 Action 3 Prop What causes the exclusion? cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

31. Mutual Exclusion: Actions • Actions A,B are mutuallyexclusive at level iif no valid plan could possibly contain both at i: • They Interfere • A deletes B’s preconditions, or • A deletes B’s effects, or • Vice versa or • They have competing needs: • A & B have inconsistent preconditions

32. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Layer 0: complete action mutexs cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

33. P Q A1 A2 M N Mutual Exclusion: Proposition Layer Propositions P,Q are inconsistent at i • if no valid plan could possibly contain both at i • If at i, all ways to achieve P exclude all ways to achieve Q

34. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Layer 1: add proposition mutexs cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap None!

35. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Can a solution exist? cleanH quiet noGarb cleanH quiet dinner present Do all goal propositions appear non-mutex? carry dolly cook wrap

36. Outline • Introduction • The Planning Problem • Graph Construction • Solution Extraction • Properties • Termination with failure

37. Graphplan • Create plan graph level 1 from initial state • Loop • If goal  propositions of the highest level (nonmutex) • Then search graph for solution • If solution found, then return and terminate • Else extend graph one more level A kind of double search: forward direction checks necessary (but insufficient) conditions for a solution, ... Backward search verifies...

38. Searching for a Solution Recursively find consistent actions achieving all goals at t, t-1, . . . : • For each goal proposition G at time t • For each action A making G true at t • If A isn’t mutex with previously chosen action at t, • Then select it • Finally, • If no action of G works, • Then backtrack on previous G. • Finally • If action found for each goal at time t, • Then recurse on preconditions of actions selected, t-1, • Else backtrack.

39. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Searching for a solution Select action achieving Goal noGarb cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

40. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Searching for a solution Select action achieving Goal dinner, Consistent with carry cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap Backtrack on noGarb

41. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Searching for a solution Backtrack on noGarb cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap Select action dolly

42. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Searching for a solution Select action achieving Goal dinner, Consistent with dolly cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

43. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Searching for a solution Select action achieving Goal present, Consistent w dolly, cook cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap

44. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Searching for a solution Recurse on preconditions of dolly, cook & wrap cleanH quiet noGarb cleanH quiet dinner present carry dolly cook wrap All satisfied by initial state

45. Avoiding redundancy • No-ops are always favoured. • guarantees that the plan will not contain redundant actions.

46. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Suppose the Plan Graph was Extended cleanH quiet noGarb cleanH quiet dinner present noGarb cleanH quiet dinner present carry dolly cook wrap carry dolly cook wrap

47. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Can a valid plan use carry? cleanH quiet noGarb cleanH quiet dinner present noGarb cleanH quiet dinner present carry dolly cook wrap carry dolly cook wrap

48. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Using carry at level 2 is valid cleanH quiet noGarb cleanH quiet dinner present noGarb cleanH quiet dinner present carry dolly cook wrap carry dolly cook wrap

49. 1 Prop 1 Action 2 Prop 2 Action 3 Prop Using carry at level 2 is valid cleanH quiet noGarb cleanH quiet dinner present noGarb cleanH quiet dinner present carry dolly cook wrap carry dolly cook wrap

50. Outline • Introduction • The Planning Problem • Graph Construction • Solution Extraction • Properties • Termination with failure