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The planning problem 2  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 3  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

STRIPS Representation 4  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

More on action schema 5  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 listAdd list

STRIPS assumptions 6  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)

STRIPS expressiveness 7  Literals are function free: Move (Block(x), y, z)  operators can be propositionalized 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)

Planning algorithms 8  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

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

Planning: Search Space 10 A C B ABC AC B C B A B A C B A C BC A C A B A C B B C A AB C A B C A B C

Forward state-space search (1) 11  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

Progression (forward search) 12 ProgWS(world-state, goal-list, PossibleActions, path) If world-state satisfies all goals in goal-list, 1. Then return path. 2. Else Act = choose an action whose precondition is true in world-state a) If no such action exists b) Then fail c) Else return ProgWS( result(Act, world-state), goal-list, PossibleActions, concatenate(path, Act) )

Forward search in the Blocks world 13 … …

Forward state-space search (2) 14  Advantages  No functions in the declarations of 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

Backward state-space search (1) 15  Regression  Initial state: goal state of the problem  Actions:  Choose an action 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, unless already appears  Goal test: state is the initial world state

Regression (backward search) 16 RegWS(initial-state, current-goals, PossibleActions, path) 1. If initial-state satisfies all of current-goals 2. Then return path 3. Else Act = choose an action whose effect matches one of current-goals a. If no such action exists, or the effects of Act contradict some of current-goals, then fail b. G = (current-goals – goals-added-by(Act)) + preconds(Act) c. If G contains all of current-goals, then fail d. Return RegWS(initial-state, G, PossibleActions, concatenate(Act, path))

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

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

Blocks world: STRIPS operators 19  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  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

STRIPS Planning 20  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(C,D) A C BD C A D

STRIPS Planning 21 on(A,C), on(D,A) on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. Current State Plan: Goalstack: on(A,C) Stack(A, C) holding(A), clear(C) holding(A) Pickup(A) on(A,Table), clear(A), ae A C BD C A D

STRIPS Planning 22 on(A,C), on(D,A) on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. Current State Plan: Goalstack: on(A,C) Stack(A, C) holding(A), clear(C) holding(A) Pickup(A) A C BD C A D Pickup(x) Pre: on(x,Table), clear(x), ae Del: on(x, Table), ae, Add: holding(x) holding(A), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D).

STRIPS Planning 23 holding(A), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D). on(A,C), on(D,A) Current State Plan: Goalstack: on(A,C) Stack(A, C) A C BD C A D Stack(x, y) Pre: holding(x), clear(y) Del: holding(x), clear(y) Add: on(x, y), ae Pickup(A) on(A,C), on(C, B), on(B,table), on(D,table), clear(A), clear(D), ae.

STRIPS Planning 24 on(A,C), on(D,A) on(A,C), on(C, B), on(B,table), on(D,table), clear(A), clear(D), ae. Current State Plan: Goalstack: on(D, A) Stack(D,A) holding(D), clear(A) holding(D) Pickup(D) on(D,Table), clear(D), ae A C BD C A D 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 Stack(A, C) Pickup(A)

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

STRIPS Planning: Getting it Wrong! 26 on(A,C), on(D,A) on(A,table), on(C, B), on(B,table), on(D,table), clear(A), clear(C), clear(D), ae. Current State Plan: Goalstack: on(D,A) Stack(D, A) holding(D), clear(A) holding(D) Pickup(D) on(D,Table), clear(D), ae A C BD C A D on(A,table), on(C, B), on(B,table), holding(D), clear(A), clear(C), clear(D)

STRIPS Planning: Getting it Wrong! 27 on(A,table), on(C, B), on(B,table), holding(D), clear(A), clear(C), clear(D) on(A,C), on(D,A) on(A,table), on(C, B), on(B,table), on(D,A), clear(C), clear(D), ae. Current State Plan: Goalstack: on(D,A) Stack(D, A) Pickup(D) A C B D C A D

STRIPS Planning: Getting it Wrong! 28 on(A,C), on(D,A) on(A,table), on(C, B), on(B,table), on(D,A), clear(C), clear(D), ae. Current State Plan: Goalstack: Stack(D, A) Pickup(D) A C B D C A D 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

STRIPS planning (Goalstack planning) 29  Works on one subgoal at a time  Insists on completely achieving that subgoal before considering other subgoals  May have to backtrack:  If it chooses the wrong order to work on the subgoals  If it chooses the wrong action to achieve a subgoal  Searches backwards from goal – uses goal to guide choice of actions

Limitation of state-space search 30  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

Search through the space of plans 31  Nodes are partial plans, links are plan refinement operations and a solution is a node (not a path).  This can be powerful if the plan representation and refinements change the search space.  POP creates partial-order plans following a “least commitment” principle.

32 Left Sock Start Finish Right Shoe Left Shoe Right Sock Start Right Sock Finish Left Shoe Right Shoe Left Sock Start Right Sock Right Sock Right Sock Right Sock Right Sock Left Sock Left Sock Left Sock Left Sock Left Sock Left Sock Right Shoe Right Shoe Right Shoe Right Shoe Right Shoe Left Shoe Left Shoe Left Shoe Left Shoe Finish Left Shoe on Right Shoe on Left Sock onRight Sock on Partial Order Plans: Total Order Plans:

P.O. plans in POP 33  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 Ap Ac Q (clear b)

Threats to causal links 34 Step At threatens link if: 1. At has (~Q) as an effect 2. At could come between Ap and Ac, i.e., O is consistent with Ap < At < Ac Ap Ac Q

Threat Removal 35  Threats must be removed to prevent a plan from failing  Demotion adds the constraint A t < A p to prevent clobbering, i.e. push the clobberer before the producer  Promotion adds the constraint A c < A t to prevent clobbering, i.e. push the clobberer after the consumer

Initial (Null) Plan 36  Initial plan has  A = { A 0, A  }  O = {A 0 < A  }  L = {}  A 0 (Start) has no preconditions but all facts in the initial state as effects.  A  (Finish) has the goal conditions as preconditions and no effects.

POP algorithm 37 POP((A, O, L), agenda, PossibleActions): 1. If agenda is empty, return (A, O, L) 2. Pick (Q, An) from agenda 3. Ad = choose an action that adds Q. a. If no such action exists, fail. b. Add the link Ad Ac to L and the ordering Ad < Ac to O c. If Ad is new, add it to A. 4. Remove (Q, An) from agenda. If Ad is new, for each of its preconditions P add (P, Ad) to agenda. 5. For every action At that threatens any link 1. Choose to add At < Ap or Ac < At to O. 2. If neither choice is consistent, fail. 6. POP((A, O, L), agenda, PossibleActions) Q Ap Ac Q

Sussman Anomaly 38 0A00A0 (on C A)(on-table A)(on-table B)(clear C)(clear B)  A  (on A B)(on B C) (on-table C)

Work on open precondition (on B C) 39 0A00A0 (on C A)(on-table A)(on-table B)(clear C)(clear B)  A  (on A B)(on B C) A1: move B from Table to C (on B C)-(on-table B)-(clear C) (clear B)(clear C)(on-table B) (on-table C)

Work on open precondition (on A B) 40 0A00A0 (on C A)(on-table A)(on-table B)(clear C)(clear B)  A  (on A B)(on B C) A1: move B from Table to C (on B C)-(on-table B)-(clear C) (clear B)(clear C)(on-table B) A2: move A from Table to B (clear A)(clear B)(on-table A) (on A B)-(on-table A)-(clear B) (on-table C)

Work on open precondition (on-table C) 41 0A00A0 (on C A)(on-table A)(on-table B)(clear C)(clear B)  A  (on A B)(on B C) A1: move B from Table to C (on B C)-(on-table B)-(clear C) (clear B)(clear C)(on-table B) A2: move A from Table to B (clear A)(clear B)(on-table A) (on A B)-(on-table A)-(clear B) A3: move C from A to Table (clear C)(on C A) -(on C A)(on-table C)(clear A) (on-table C)

Analysis 42  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!

More analysis 43  Does POP make the least possible amount of commitment?  Lifted POP: Using Operators, instead of ground actions,  Unification is required

POP in the Blocks world 44 On(x,y), Cl(x), ~Cl(y), ~On(x,z) PutOn(x,y) Cl(x), Cl(y), On(x,z) PutOnTable(x) On(x, z) Cl(x) On(x,Table), Cl(x), ~On(x,z)

POP in the Blocks world 45

POP in the Blocks world 46

POP in the Blocks world 47

POP in the Blocks world 48

Example 2 49  A 0 : Start  At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill)  A  : Finish  Have(Drill) Have(Milk) Have(Banana) At(Home) Have(y) Buy (y,x) At(x), Sells(x,y) GO (x,y) At(x)At(y) ~At(x)

POP Example 50 finish start

POP Example 51 finish start Have(M) Have(B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 52 finish start Have(M) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Have(M) Have(B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 53 finish start Have(M) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Have(M) Have(B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 54 finish start Have(M) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Have(M) Have(B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 55 finish start Have(B) Have(M) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 56 finish start Have(B) Have(M) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 57 finish start Have(B) x 1 = SM Have(M) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 58 finish start Have(B) x 1 = SM Have(M) x 2 = SM Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 59 finish start Have(B) x 1 = SM Have(M) x 2 = SM GO (x 3,SM) At(x 3 ) At(SM) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 60 finishGO (x 3,SM) start Have(B) x 1 = SM Have(M) x 2 = SM At(x 3 ) At(SM) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 61 finishGO (x 3,SM) start Have(B) x 1 = SM Have(M) x 2 = SM At(x 3 ) At(SM) Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 62 finishGO (x 3,SM) start Have(B) x 1 = SM Have(M) x 2 = SM At(x 3 ) At(SM) x 3 = H Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)

POP Example 63 finishGO (x 3,SM) start Have(B) x 1 = SM Have(M) x 2 = SM At(x 3 ) At(SM) x 3 = H Buy (M,x 1 ) At(x 1 ) Sells(x 1,M) Buy (B,x 2 ) Have(M) Have(B) At(x 2 ) Sells(x 2, B) Sells(SM, M) Sells(SM,B) At(H)