Planning: Part 1Representation andState-space Search

COMP151March 30, 2007

Example Planning Problem

Table

A

C

E

D

J

F

LG

K

I

H

B

Goal: Stack of blocks: FCGHJB (bottom to top)Possible Actions: move any single block which has nothing on top of it.

Example Planning ProblemGoal: Stack of blocks: FCGHJB (bottom to top)Possible Actions: move any single block which has nothing on top of it.

Considerations:Can this be decomposed into subproblems?Can we identify irrelevant/useless actions?Can we select better actions?

Analysis:12 blocks, 13 destinations (including table)156 potential movesn156 nodes in a search tree (n = # moves)

Planning: Chapter 11

• 11.1 The planning problem• 11.2 Planning with state-space search• 11.3 Partial-order planning• 11.4 Planning graphs• 11.5 Planning with propositional logic• 11.6 Analysis of planning approaches

Planning?

• Planning is the task of determining a sequence of actions that will achieve a goal.

• Domain independent heuristics and strategies must be based on a domain independent representation– General planning algorithms require a way to

represent states, actions and goals– STRIPS, ADL, PDDL are languages based on

propositional or first-order logic• Classical planning environment:

fully observable, deterministic, finite, static and discrete.

Difficulty of real world problems

• Assume a problem-solving agentusing some search method …– Which actions are relevant?

• Exhaustive search vs. backward search

– What is a good heuristic functions?• Good estimate of the cost of the state?• Problem-dependent vs, -independent

– How to decompose the problem?• Most real-world problems are nearly decomposable.• Decomposability means that working on one subgoal does

not undo other subgoals

Problem Representation

• State– What is true about the (hypothesized) world?

• Goal– What must be true in the final state of the world?

• Actions– What can be done to change the world?– Preconditions and effects

• We’ll represent all these as logic predicates

Representing States and Goals

• States and goals are represented as aconjunction of positive, ground, function-free literals

• Propositional literals: Poor Unknown• FO-literals (grounded and function-free):

At(Plane1, Melbourne) At(Plane2, Sydney)• States:

– Closed world assumption: anything not explicitly true is false

• Goals:– A partially specified state– A goal is satisfied if state contains all literals in goal.

Representing Actions

• An action schema represents a parameterized actions– Action name and parameter list– Precondition

• conj. of function-free literals– Effect

• conj of function-free literals• add-list: predicates that will be true• delete-list: predicates that will be false

• Example:– Action(Fly(p,from, to),

PRECOND: At(p,from) Plane(p) Airport(from) Airport(to)EFFECT: ¬AT(p,from) At(p,to))

Applicability of Actions

• An action is applicable in any state that satisfies the precondition.

• For FO action schema applicability involves a substitution for the variables in the PRECOND.

At(P1,JFK) At(P2,SFO) Plane(P1) Plane(P2) Airport(JFK) Airport(SFO)

Satisfies : At(p,from) Plane(p) Airport(from) Airport(to)

With ={p/P1,from/JFK,to/SFO}

Thus the action is applicable.

Effect of Actions

• Executing action a in state s results in state s’,where s’ is same as s except– Any positive literal P in the effect of a is added to s’– Any negative literal ¬P is removed from s’

s: At(P1,JFK) At(P2,SFO) Plane(P1) Plane(P2) Airport(JFK) Airport(SFO)

EFFECT: ¬AT(p,from) At(p,to):

s’: At(P1,SFO) At(P2,SFO) Plane(P1) Plane(P2) Airport(JFK) Airport(SFO)

• STRIPS assumption: every literal NOT in the effect remains unchanged

avoids representational frame problem

Planning Languages

• STRIPS: Stanford Research Institute Problem Solver– important limit: function-free literals

• allows for propositional representation• function symbols lead to infinitely many states and actions

• ADL: Action Description Language– A number of extension to STRIP (next slide)

Action(Fly(p:Plane, from: Airport, to: Airport),PRECOND: At(p,from) (from to)EFFECT: ¬At(p,from) At(p,to))

• PDDL: Planning Domain Definition Language:– standardization effort – Includes sublanguages for STRIPS and ADL.

STRIPS ADL

Only positive literals in states Positive and negative literals in states

Closed world assumption Open world assumption

Effect P ^ ¬Q means add P and delete Q

Effect P ^ ¬Q means add P, ¬Q and delete ¬P , Q

Only ground literals in goals Quantified variables in goals

Goals are conjunctions Goals contain conjunction and disjunction

Effects are conjunctions Conditional effects are allowed (when P: E)

Equality predicate (x = y) built in

Variables can have types

air cargo transport: STRIPSInit(At(C1, SFO) At(C2,JFK) At(P1,SFO) At(P2,JFK)

Cargo(C1) Cargo(C2) Plane(P1) Plane(P2) Airport(JFK) Airport(SFO))

Goal(At(C1,JFK) At(C2,SFO))Action(Load(c,p,a)

PRECOND: At(c,a) At(p,a) Cargo(c) Plane(p) Airport(a)EFFECT: ¬At(c,a) In(c,p))

Action(Unload(c,p,a)PRECOND: In(c,p) At(p,a) Cargo(c) Plane(p) Airport(a)EFFECT: At(c,a) ¬In(c,p))

Action(Fly(p,from,to)PRECOND: At(p,from) Plane(p) Airport(from) Airport(to)EFFECT: ¬ At(p,from) At(p,to))

[Load(C1,P1,SFO), Fly(P1,SFO,JFK), Load(C2,P2,JFK), Fly(P2,JFK,SFO), Unload(C1, P1, SFO), Unload(C2, P2, SFO)]

air cargo transport: ADLInit(At(C1, SFO) At(C2,JFK) At(P1,SFO) At(P2,JFK)

Cargo(C1) Cargo(C2) Plane(P1) Plane(P2) Airport(JFK) Airport(SFO))

Goal(At(C1,JFK) At(C2,SFO))Action(Load(c: Cargo, p: Plane, a: Airport)

PRECOND: At(c,a) At(p,a)EFFECT: ¬At(c,a) In(c,p))

Action(Unload(c : Cargo, p: Plane, a: Airport)PRECOND: In(c,p) At(p,a) EFFECT: At(c,a) ¬In(c,p))

Action(Fly(p: Plane, from: Airport, to: Airport)PRECOND: At(p,from) (from ≠ to) EFFECT: ¬ At(p,from) At(p,to))

[Load(C1,P1,SFO), Fly(P1,SFO,JFK), Load(C2,P2,JFK), Fly(P2,JFK,SFO), Unload(C1, P1, SFO), Unload(C2, P2, SFO)]

spare tire problem: ADLInit(At(Flat, Axle) At(Spare,trunk))Goal(At(Spare,Axle))Action(Remove(Spare,Trunk)

PRECOND: At(Spare,Trunk)EFFECT: ¬At(Spare,Trunk) At(Spare,Ground))

Action(Remove(Flat,Axle)PRECOND: At(Flat,Axle)EFFECT: ¬At(Flat,Axle) At(Flat,Ground))

Action(PutOn(Spare,Axle)PRECOND: At(Spare,Groundp) ¬At(Flat,Axle)EFFECT: At(Spare,Axle) ¬At(Spare,Ground))

Action(LeaveOvernightPRECOND:EFFECT: ¬ At(Spare,Ground) ¬ At(Spare,Axle) ¬ At(Spare,trunk) ¬ At(Flat,Ground) ¬ At(Flat,Axle) )

Do we need to add negative facts in initial state?

blocks world: ADL

Init(On(A, Table) On(B,Table) On(C,Table) Block(A) Block(B) Block(C) Clear(A) Clear(B) Clear(C))

Goal(On(A,B) On(B,C))Action(Move(b,x,y)

PRECOND: On(b,x) Clear(b) Clear(y) Block(b) (b x) (b y) (x y) EFFECT: On(b,y) Clear(x) ¬ On(b,x) ¬ Clear(y))

Action(MoveToTable(b,x)PRECOND: On(b,x) Clear(b) Block(b) (b x) EFFECT: On(b,Table) Clear(x) ¬ On(b,x))

Planning with state-space search

• Both forward and backward search possible• Progression planners

– forward state-space search– consider the effect of all possible actions in a given

state• Regression planners

– backward state-space search– Determine what must have been true in the previous

state in order to achieve the current state

Progression and regression

initial state

goal

Progression algorithm

• Formulation as state-space search problem:– Initial state and goal test: obvious– Successor function: generate from applicable actions– Step cost = each action costs 1

• Any complete graph search algorithm is a complete planning algorithm.– E.g. A*

• Inherently inefficient: – (1) irrelevant actions lead to very broad search tree– (2) good heuristic required for efficient search

Regression algorithm

• How to determine predecessors?– What are the states from which applying a given action leads to the

goal?Goal state = At(C1, B) At(C2, B) … At(C20, B)Relevant action for first conjunct: Unload(C1,p,B)Works only if pre-conditions are satisfied.Previous state= In(C1, p) At(p, B) At(C2, B) … At(C20, B)Subgoal At(C1,B) should not be present in this state.

• Actions must not undo desired literals (consistent)• Main advantage: only relevant actions are considered.

– Often much lower branching factor than forward search.

Regression algorithm

• General process for predecessor construction– Give a goal description G– Let A be an action that is relevant and consistent– The predecessors are as follows:

• Any positive effects of A that appear in G are deleted.• Each precondition literal of A is added , unless it already

appears.

• Any standard search algorithm can be added to perform the search.

• Termination when predecessor satisfied by initial state.– In FO case, satisfaction might require a substitution.

Heuristics for state-space search

• Neither progression or regression are very efficient without a good heuristic.– How many actions are needed to achieve the goal?– Exact solution is NP hard, find a good estimate

• Two approaches to find admissible heuristic:– The optimal solution to the relaxed problem.

• Remove all preconditions from actions

– The subgoal independence assumption:The cost of solving a conjunction of subgoals is approximated by the sum of

the costs of solving the subproblems independently.

Exercises

Define state, goal and actions using STRIPS or ADL• Sorting a deck of cards by suit and value• Changing your socks• Cleaning the kitchen:

– Cleaning stove or refrigerator will get floor dirty

– To clean oven: first apply oven cleaner, then remove cleaner

– Floor must be swept before it is washed

– Garbage must be taken out before floor can be swept

– Cleaning refrigerator generates garbage and messes up counters

– Washing counters or floor gets sink dirty