STRIPS Planning Set of operators , where each operator has Set of - PDF document

“STRIPS” Planning • Set of operators , where each operator has – Set of parameters – Set of preconditions – Set of effects , consisting of add effects and delete effects. • Set of objects to instantiate operator’s parameters fully instantiated operator ≡ action • Set of propositions representing initial state • Set of propositions representing goals Planning problem : Find sequence of actions that, starting in initial state, achieve all the goals Graph-Plan 1

Approaches to STRIPS planning • Search through space of world states – forward search, – regression search – bi-directional search – means-ends analysis – . . . • Search through space of plans – total order planning – partial order planning • Search through planning graph Graph-Plan 2

GraphPlan Approach 1. Construct a “PlanGraph” that contains all valid plans + other stuff (invalid plans) up to a maximum depth 2. Search PlanGraph for valid plan . . . then return that plan Graph-Plan 3

Simple Cake-Eating Domain • Initial: HaveCake ∧ ¬ EatenCake • Goal: HaveCake ∧ EatenCake • Actions:   Eat Op PreC: HaveCake   ¬ HaveCake ∧ EatenCake Eff:   Bake ¬ HaveCake Op PreC:   Eff: HaveCake • PlanGraph A S 0 S 1 A 1 S 2 0 Bake(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Eat(Cake) Eat(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Graph-Plan 4

Parts of a PlanGraph A S 0 S 1 A 1 S 2 0 Bake(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Eat(Cake) Eat(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) “2-leveled” Graph �S 0 , A 0 , S 1 , A 1 , . . . � • S 0 : propositions in initial state • A i : each action whose preconditions all occur in level S i − 1 • S i : each prop’n that is ADDed/DELETEd by ⋆ an action in level A i ⋆ a “No-Op” (persistance) • Mutex links ⋆ between actions in level A i ⋆ between propositions in level S i “ mut ually ex clusive” “cannot occur in same plan” Graph-Plan 5

Mutex Conditions#1: Actions A S 0 S 1 A 1 S 2 0 Bake(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Eat(Cake) Eat(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Between 2 actions O 1 and O 2 , same level A i : • Inconsistent effects O 1 : Eff negates O 2 : Eff EatenCake , NoOp(HaveCake) disagree wrt “ HaveCake ” = ¬ HaveCake EatenCake:Eff = HaveCake NoOp(HaveCake):Eff • Interference O 1 : Eff negates O 2 : PreC EatenCake interfers with NoOp(HaveCake) : = EatenCake:Eff ¬ HaveCake = NoOp(HaveCake):PreC HaveCake • Competing Needs O 1 : PreC negates O 2 : PreC = Bake:PreC ¬ HaveCake = Eat:PreC HaveCake Graph-Plan 6

Mutex Conditions#2: Propositions A S 0 S 1 A 1 S 2 0 Bake(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Have(Cake) Eat(Cake) Eat(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Eaten(Cake) Between 2 propositions ρ 1 and ρ 2 , same level S i : • Negation ρ 1 = ¬ ρ 2 • Inconsistent Support Every action achieving ρ 1 (from S i − 1 ) is mutex with every action achieving ρ 2 In S 1 : HaveCake mutex EatenCake as only way to achieve HaveCake : NoOp(HaveCake) is mutex with only way to achieve EatenCake : Eat N.b. : Not mutex at S 2 ! Graph-Plan 7

Planning Graphs • A valid plan is “2-leveled” graph – two kinds of nodes (propositions, actions) alternates: proposition level, action level – 5 kinds of edges ⋆ precondition ( S i → A i ) ⋆ add effect ( A i → S i +1 ) ⋆ delete effect ( A i → S i +1 ) ⋆ mutex-action ( A i ↔ A i ) ⋆ mutex-prop ( S i ↔ S i ) – Include action O at action-level A i if all preconditions at proposition-level S i – Include proposition ρ at proposition-level S i if it is add/delete effect of action O ∈ A i − 1 (including no-op actions) Restriction: Allow actions O 1 , O 2 at same time t ONLY if don’t interfere with each other • PlanningGraph ≈ valid plan but without no-interfere restriction Graph-Plan 8

GraphPlan Algorithm function Graphplan( problem ) returns solution or failure graph ← Initial-Planning-Graph( problem ) goals ← Goals[ problem ] loop do if goals all non-mutex in last level of graph then do solution ← Extract-Solution( graph , goals , Length( graph )) if solution � = failure then return solution else if No-Solution-Possible( graph ) then return failure graph ← Expand-Graph( graph , problem ) end Graph-Plan 9

Flat-Tire Domain Fl= Flat; Sp= Spare; Ax= Axel; Tr= Trunk; Gr= Ground • Initial: At(Fl, Ax) ∧ At(Sp, Tr) • Goal: At(Sp, Ax) • Actions:   TakeOutSpare PreC: At(Sp, Tr) Op   Eff: ¬ At(Sp, Tr) ∧ At(Sp, Gr)   RemoveFlat Op PreC: At(Fl, Ax)   Eff: ¬ At(Fl, Ax) ∧ At(Fl, Gr)   PutOnSpare Op PreC: At(Sp, Gr) ∧ ¬ At(Fl, Ax)   Eff: ¬ At(Sp, Gr) ∧ At(Sp, Ax)   LeaveOverNight {} PreC: Op   ¬ At(Sp, Gr) ∧ ¬ At(Sp, Ax) ∧ ¬ At(Sp, Tr) Eff:   ∧ ¬ At(Fl, Gr) ∧ ¬ At(Fl, Ax) Graph-Plan 10

Flat-Tire in GraphPlan S 0 A S 1 A 1 S 2 0 At(Spare,Trunk) At(Spare,Trunk) At(Spare,Trunk) TakeOutSpare At(Spare,Trunk) At(Spare,Trunk) TakeOutSpare RemoveFlat RemoveFlat At(Flat,Axle) At(Flat,Axle) At(Flat,Axle) At(Flat,Axle) At(Flat,Axle) LeaveOvernight LeaveOvernight At(Spare,Axle) At(Spare,Axle) At(Spare,Axle) At(Spare,Axle) PutOnSpare At(Flat,Ground) At(Flat,Ground) At(Flat,Ground) At(Flat,Ground) At(Flat,Ground) At(Spare,Ground) At(Spare,Ground) At(Spare,Ground) At(Spare,Ground) At(Spare,Ground) Graph-Plan 11

Trace of GraphPlan Algorithm #1 • S 0 : initial facts (include ¬ facts) • As At(Sp,Ax) �∈ S 0 do not call Extract-Solution • Expand-Graph forms A 0 with ⋆ 3 “real” actions ⋆ 5 no-op actions; S 1 is effects Expand-Graph then finds ⋆ 4 action-mutex within A 0 ⋆ 4 prop-mutex within S 1 • As At(Sp,Ax) �∈ S 1 do not call Extract-Solution • Expand-Graph forms A 1 with ⋆ 4 “real” actions ⋆ 7 no-op actions S 2 is effects Graph-Plan 12

Mutex wrt FlatTire • Inconsistent Effects RemoveSpare + LeaveOvernight = RemoveSpare:Eff At(Sp,Gr) = LeaveOvernight:Eff ¬ At(Sp,Gr) • Inteference RemoveFlat + LeaveOvernight = RemoveFlat:PreC At(Sp,Ax) = ¬ At(Sp,Ax) LeaveOvernight:Eff • Competing Needs RemoveFlat + PutOnSpare = RemoveFlat:PreC At(Fl,Ax) = PutOnSpare:Eff ¬ At(Fl,Ax) • Inconsistent Support At(Sp,Ax) + At(Fl,Ax) in S 2 At(Sp,Ax) by PutOnSpare At(Fl,Ax) by NoOp[At(Fl,Ax)] and PutOnSpare mutex NoOp[At(Fl,Ax)] (Can’t put 2 objects in same place at same time) Graph-Plan 13

Trace of GraphPlan Algorithm #2 • “All” goal literals, At(Sp, Ax) , in S 2 none are mutex . . . • So there MAY be solution . . . call Extract-Solution Extract-Solution(. . . ) Let G n be the GOAL at last level, S n For each i = n.. 1 ⋆ Let H i be a conflict-free subset of A i − 1 , that covers G i (in S i ) ⋆ Let G i − 1 be preconditions of H i . . . until reach state in S 0 satisfying all goals Action-set H is “conflict-free” ≡ no pair of H are mutex, and no pair of preconditions (in G ) are mutex Graph-Plan 14

Trace of Extract-Solution = • G 2 { At(Sp,Ax) } = H 2 { PutOnSpare } • G 1 = { At(Sp,Gr) , ¬ At(Fl,Ax) } What is H 1 ? – Achieve At(Sp,Gr) by TakeOutSpare – Achieve ¬ At(Fl,Ax) by #1. LeaveOvernight #2. RemoveFlat But not #1, as LeaveOvernight is mutex with TakeOutSpare = { TakeOutSpare , RemoveFlat } ⇒ H 1 • G 0 = { At(Sp,Tr) , At(Fl,Ax) } As in G 0 ⊂ S 0 , DONE! Graph-Plan 15

Extending PlanGraph Add action level A i : ForEach action ( ∗ ) O If O ’s preconditions all true in prop-level S i , and NOT mut-ex, Then add O to level A i include precondition-links create mutex ( O :actions-I-am-exclusive-of) Add prop-level S i +1 : ForEach effect ρ of each action in action-level A i Add ρ to prop-level S i +1 Add S ← ρ add- or delete- links Mark ρ 1 , ρ 2 as mutex if each way of generating ρ 1 is mutex to each way of generating ρ 2 ( ∗ ) each instantiation of each operator; including “no-op”s Graph-Plan 16

Correctness Graphplan is sound and complete: ∗ any plan Graphplan finds is a legal plan ∗ if ∃ legal plan then Graphplan will find one. If ∃ valid plan using ≤ t time steps, Theorem: then plan is subgraph of (depth- t ) Planning Graph. + If Goals not satisfiable by any valid plan, then GraphPlan will halt, w/failure, in finite time. (extends most partial-order planners) Graph-Plan 17

Leveling Off • GraphPlan ≈ Iterative deepening When to stop?? • Lemma: If no valid plan exists, then ∃ a prop-level S n s.t. all future proposition levels are identical to S n – Identical ≡ same propositions, mutual exclusions – graph has “ leveled off after S n ” • Corollary: No solution exists if – a goal does not appear in S n or – S n has mutually exclusive goals • Subtlety: { on(A,B) , on(B,C) , on(C,A) } Graph-Plan 18

Termination Condition • Let S t i denote set of memoized goal sets at level i after an unsuccessful stage t • Theorem: If the graph has leveled off at level n and stage t has passed in which | S t − 1 | = | S t n | , then no valid plan exists n Graph-Plan 19