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AI Planning

AI Planning. The planning problem. Inputs: 1. A description of the world state 2. The goal state description 3. A set of actions Output: A sequence of actions that if applied to the initial state, transfers the world to the goal state. An example – Blocks world. Blocks on a table

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AI Planning

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  1. AI Planning

  2. The planning problem • Inputs: 1. A description of the world state 2. The goal state description 3. A set of actions • Output: A sequence of actions that if applied to the initial state, transfers the world to the goal state

  3. An example – Blocks world • Blocks on a table • Can be stacked, but only one block on top of another • A robot arm can pick up a block and move to another position • On the table • On another block • Arm can pick up only one block at a time • Cannot pick up a block that has another one on it

  4. STRIPS Representation • State is a conjunction of positive ground literals On(B, Table) Λ Clear (A) • Goal is a conjunction of positive ground literals Clear(A) Λ On(A,B) Λ On(B, Table) • STRIPS Operators • Conjunction of positive literals as preconditions • Conjunction of positive and negative literals as effects

  5. More on action schema • Example: Move (b, x, y) • Precondition: Block(b)Λ Clear(b) Λ Clear(y) Λ On(b,x) Λ(b ≠ x) Λ (b ≠ y) Λ (y ≠ x) • Effect:¬Clear(y) Λ ¬On(b,x)Λ Clear(x) Λ On(b,y) • An action is applicable in any state that satisfies its precondition Delete list Add list

  6. STRIPS assumptions • Closed World assumption • Unmentioned literals are false (no need to explicitly list out) • STRIPS assumption • Every literal not mentioned in the “effect” of an action remains unchanged • Atomic Time (actions are instantaneous)

  7. STRIPS expressiveness • Literals are function free: Move (Block(x), y, z) • operators can be propositionalized (= actions) Move(b,x,y) and 3 blocks and table can be expressed as 48 purely propositional actions • No disjunctive goals: On(B, Table) V On(B, C) • No conditional effects: On(B, Table) if ¬On(A, Table)

  8. Planning algorithms • Planning algorithms are search procedures • Which state to search? • State-space search • Each node is a state of the world • Plan = path through the states • Plan-space search • Each node is a set of partially-instantiated operators and set of constraints • Plan = node

  9. State search • Search the space of situations, which is connected by operator instances (= actions) • The sequence of actions = plan • We have both preconditions and effects available for each operator, so we can try different searches: Forward vs. Backward

  10. Planning: Search Space C A B C A B C B A B A C B A C B A B C C B A B A C A C A A B C B C C B A A B C

  11. Forward state-space search (1) • Progression • Initial state: initial state of the problem • Actions: • Applied to a state if all the preconditions are satisfied • Succesor state is built by updating current state with add and delete lists • Goal test: state satisfies the goal of the problem

  12. Progression (forward search) ProgWS(world-state, goal-list, PossibleActions, path) If world-state satisfies all goals in goal-list, • Then return path. • Else Act = choose an action whose precondition is true in world-state • If no such action exists • Then fail • Else return ProgWS( result(Act, world-state), goal-list, PossibleActions, concatenate(path, Act) )

  13. Forward search in the Blocks world … …

  14. Forward state-space search (2) • Advantages • No functions in the declarations goals  search state is finite • Sound • Complete (if algorithm used to do the search is complete) • Limitations • Irrelevant actions  not efficient • Need heuristic or pruning procedure

  15. Backward state-space search (1) • Regression • Initial state: goal state of the problem • Actions: • Choose an action A that • Is relevant; has one of the goal literals in its effect set • Is consistent; does not negate another literal • Construct new search state • Remove all positive effects of A that appear in goal • Add all preconditions of A, unless already appears • Goal test: initial world state contains remaining goals

  16. Regression (backward search) RegWS(initial-state, current-goals, PossibleActions, path) • If initial-state satisfies all of current-goals • Then return path • Else Act = choose an action whose effect matches one of current-goals • If no such action exists, or the effects of Act contradict some of current-goals, then fail • G= (current-goals – goals-added-by(Act)) + preconds(Act) • If G contains all of current-goals, then fail • Return RegWS(initial-state, G, PossibleActions, concatenate(Act, path))

  17. Backward state-space search (2) • Advantages • Consider only relevant actions  much smaller branching factor • Limitations • Still need heuristic to be more efficient

  18. Comparing ProgWS and RegWS • Both algorithms are • sound (they always return a valid plan) • complete (if a valid plan exists they will find one) • Running time is O(bn) where b = branching factor, n = number of “choose” operators

  19. Efficiency of Backward Search a1 a2 • Backward search can also have a very large branching factor • E.g., many relevant actions that don’t regress toward the initial state • As before, deterministic implementations can waste lots of time trying all of them a3 a3 … a50 a1 a2 goal initial state

  20. Lifting p(a,a) foo(a,a) • Can reduce the branching factor of backward search if we partially instantiate the operators • this is called lifting p(a,b) foo(x,y) precond: p(x,y) effects: q(x) foo(a,b) p(a,c) q(a) foo(a,c) … foo(a,y) q(a) p(a,y)

  21. The Search Space is Still Too Large • Backward-search generates a smaller search space thanForward-search, but it still can be quite large • Suppose a, b, and c are independent, d must precede all of them, and d cannot be executed • We’ll try all possible orderings of a, b, and c before realizing there is no solution a d b c b d a b d a goal b a c d b d c a c d b

  22. A ground version of the STRIPS algorithm.

  23. Blocks world: STRIPS operators • UnStack(x,y) • Pre: on(x, y), ae • Del: on(x, y), ae • Add: holding(x), clear(y) • Stack(x, y) • Pre: holding(x), clear(y) • Del: holding(x), clear(y) • Add: on(x, y), ae • Pickup(x) Pre: on(x, Table), clear(x), ae Del: on(x, Table), ae Add: holding(x) • Putdown(x) Pre: holding(x) Del: holding(x) Add: on(x, Table), ae

  24. STRIPS Planning • Current state: • on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. • Goal • on(A,C), on(D, A) C D A B D A C

  25. STRIPS Planning D Plan: on(A,C), on(D,A) Goalstack: A C on(A,C) Stack(A, C) holding(A), clear(C) holding(A) Pickup(A) on(A,Table), clear(A), ae C A B D on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. Current State

  26. STRIPS Planning D Plan: on(A,C), on(D,A) Goalstack: A C on(A,C) Stack(A, C) holding(A), clear(C) • Pickup(A) • Pre: on(A,Table), clear(A), ae • Del: on(A, Table), ae, • Add: holding(A) holding(A) Pickup(A) C A B D holding(A), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D). on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. Current State

  27. STRIPS Planning D Plan: on(A,C), on(D,A) Goalstack: A C Pickup(A) on(A,C) Stack(A, C) • Stack(A, C) • Pre: holding(A), clear(C) • Del: holding(A), clear(C) • Add: on(A, C), ae A C B D holding(A), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D). on(A,C), on(C, B), on(B,table), on(D,table), clear(A), clear(D), ae. Current State

  28. STRIPS Planning D Plan: on(A,C), on(D,A) Goalstack: A C Pickup(A) on(D, A) Stack(A, C) Stack(D,A) holding(D), clear(A) holding(D) Pickup(D) on(D,Table), clear(D), ae A C B D on(A,C), on(C, B), on(B,table), on(D,table), clear(A), clear(D), ae. on(A,C), on(C, B), on(B,table), holding(D), clear(A), clear(D) on(A,C), on(C, B), on(B,table), on(D,A), clear(A), ae Current State

  29. STRIPS Planning D Plan: on(A,C), on(D,A) Goalstack: A C Pickup(A) on(D, A) Stack(A, C) Stack(D,A) holding(D), clear(A) Pickup(D) holding(D) D A C B on(A,C), on(C, B), on(B,table), holding(D), clear(A), clear(D) on(A,C), on(C, B), on(B,table), on(D,A), clear(A), ae Current State

  30. STRIPS Planning: Getting it Wrong! D Plan: on(A,C), on(D,A) Goalstack: A C on(D,A) Stack(D, A) holding(D), clear(A) holding(D) Pickup(D) on(D,Table), clear(D), ae C A B D on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. on(A,table), on(C, B), on(B,table), holding(D), clear(A), clear(C), clear(D) Current State

  31. STRIPS Planning: Getting it Wrong! D Plan: on(A,C), on(D,A) Goalstack: A C Pickup(D) on(D,A) Stack(D, A) D C A B on(A,table), on(C, B), on(B,table), on(D,A), clear(C), clear(D), ae. on(A,table), on(C, B), on(B,table), holding(D), clear(A), clear(C), clear(D) Current State

  32. STRIPS Planning: Getting it Wrong! D Plan: on(A,C), on(D,A) Goalstack: A C Pickup(D) Stack(D, A) • Now What? • We chose the wrong goal first • A is no longer clear. • stacking D on A messes up the preconditions for actions to accomplish on(A, C) • either have to backtrack, or else we must undo the previous actions C D A B on(A,table), on(C, B), on(B,table), on(D,A), clear(C), clear(D), ae. Current State

  33. Limitation of state-space search • Linear planning or Total order planning • Example • Initial state: all the blocks are clear and on the table • Goal: On(A,B) Λ On(B,C) • If search achieves On(A,B) first, then needs to undo it in order to achieve On(B,C) • Have to go through all the possible permutations of the subgoals

  34. Search through the space of plans • Nodes are partial plans, links are plan refinement operations and a solution is a node (not a path). • POP creates partial-order plans following a “least commitment” principle.

  35. Total Order Plans: Partial Order Plans: Start Start Start Start Start Start Start Right Sock Right Sock Left Sock Left Sock Right Sock Left Sock Left Sock Right Sock Left Sock Left Sock Right Sock Right Sock Right Shoe Left Shoe Left Sock on Right Sock on Left Shoe Right Shoe Right Shoe Left Sock Right Shoe Left Shoe Left Sock Right Sock Right Shoe on Left Shoe on Right Shoe Left Shoe Right Shoe Left Shoe Right Shoe Left Shoe Finish Finish Finish Finish Finish Finish Finish

  36. Q Ac Ap P.O. plans in POP • Plan = (A, O, L), where • A is the set of actions in the plan • O is a set of temporal orderings between actions • L is a set of causal links linking actions via a literal Causal link means that Ac has precondition Q that is established in the plan by Ap. move-a-from-b-to-table move-c-from-d-to-b (clear b)

  37. Q Ac Ap Threats to causal links Step Atthreatens link if: • At has (~Q) as an effect • At could come between Ap and Ac, i.e., O is consistent with Ap < At < Ac

  38. Threat Removal • Threats must be removed to prevent a plan from failing • Demotion adds the constraint At < Ap to prevent clobbering, i.e. push the clobberer before the producer • Promotion adds the constraint Ac < At to prevent clobbering, i.e. push the clobberer after the consumer

  39. Initial (Null) Plan • Initial plan has • A = { A0, A¥} • O = {A0 < A¥} • L = {} • A0 (Start) has no preconditions but all facts in the initial state as effects. • A¥ (Finish) has the goal conditions as preconditions and no effects.

  40. Q Ac Ap POP algorithm POP((A, O, L), agenda, PossibleActions): • If agenda is empty, return (A, O, L) • Pick (Q, An) from agenda • Ad = choose an action that adds Q. • If no such action exists, fail. • Add the link Ad An to L and the ordering Ad < An to O • If Ad is new, add it to A. • Remove (Q, An) from agenda. If Ad is new, for each of its preconditions P add (P, Ad) to agenda. • For every action At that threatens any link • Choose to add At < Ap or Ac < At to O. • If neither choice is consistent, fail. • POP((A, O, L), agenda, PossibleActions) Q

  41. Analysis • POP can be much faster than the state-space planners because it doesn’t need to backtrack over goal orderings (so less branching is required). • Although it is more expensive per node, and makes more choices than RegWS, the reduction in branching factor makes it faster, i.e., n is larger but b is smaller!

  42. More analysis • Does POP make the least possible amount of commitment? • Lifted POP: Using Operators, instead of ground actions, • Unification is required

  43. PutOn(x,y) PutOnTable(x) POP in the Blocks world Cl(x), Cl(y), On(x,z) On(x,y), Cl(x), ~Cl(y), ~On(x,z) On(x, z) Cl(x) On(x,Table), Cl(x), ~On(x,z)

  44. POP in the Blocks world

  45. POP in the Blocks world

  46. POP in the Blocks world

  47. POP in the Blocks world

  48. Buy (y,x) GO (x,y) At(x), Sells(x,y) Have(y) At(x) At(y) ~At(x) Example 2 • A0: Start • At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill) • A¥ : Finish • Have(Drill) Have(Milk) Have(Banana) At(Home)

  49. start Buy (B,x2) Buy (M,x1) GO (x3,SM) finish POP Example x1 = SM x2 = SM At(x3) x3 = H Sells(SM, M) Sells(SM,B) At(H) At(SM) At(x1) Sells(x1,M) At(x2) Sells(x2, B) Have(M) Have(B) Have(M) Have(B)

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