AI Planning

The planning problem Inputs: Output: 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 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 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 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

STRIPS assumptions Closed World assumption STRIPS 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 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)

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

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

Planning: Search Space 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

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

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) )

Forward search in the Blocks world … …

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

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

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))

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

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

Efficiency of Backward Search … a50 a3 initial state goal 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

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

The Search Space is Still Too Large Backward-search generates a smaller search space than Forward-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 d a b c d b a d b a goal b d a c d b c a d c b

A ground version of the STRIPS algorithm.

Blocks world: STRIPS operators 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 C D A B D A C Current state: Goal 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)

STRIPS Planning Plan: Goalstack: on(A,C), on(D,A) 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

STRIPS Planning Plan: Goalstack: on(A,C), on(D,A) 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

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

STRIPS Planning Plan: Goalstack: on(A,C), on(D,A) 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

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

STRIPS Planning: Getting it Wrong! D Plan: Goalstack: on(A,C), on(D,A) 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), holding(D), 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

STRIPS Planning: Getting it Wrong! D Plan: Goalstack: on(A,C), on(D,A) 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

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

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

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.

Total Order Plans: Partial Order Plans: 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

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 Ap Ac Q (clear b)

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

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

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.

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 Ap Ac Q

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!

More analysis 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 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

POP in the Blocks world

POP in the Blocks world

POP in the Blocks world

Example 2 A0: Start A¥ : Finish 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 x1 = SM start x2 = SM At(x3) x3 = H Sells(SM, M) Sells(SM,B) At(H) GO (x3,SM) At(SM) At(x1) Sells(x1,M) At(x2) Sells(x2, B) Buy (M,x1) Buy (B,x2) Have(M) Have(B) Have(M) Have(B) finish