roots of planning

Roots of planning

n problem solving through state-spacesearch

n software agents (e.g. robotics)

EXAMPLE: “Shakey” (pg. 360)n a robot that roamed the halls of SRI in

the early 1970’sn actions were based on plans

The Problem

Find a sequence of actions that achievesa given goal when performed starting in

a given state.

PLANPLANPLANPLAN

START

FINISH

Initial State

Goal State

OperatorInstances

A Simple Planning Agent

plans ahead before acting (ch3)

selects actions based on explicit,logical representations of thecurrent state and the effects ofactions (ch6)

agent uses beliefs about actionsand their consequences tosearch for a solution over aspace of plans rather than aspace of situations


Algorithm: (pg. 338)

1. Generate a goal to achieve

2. Construct a plan to achieve goal fromcurrent state

3. Execute plan until finished

4. Begin again with new goal


n Assumptions:– Atomic time

– no concurrent actions allowed

– deterministic actions

– agent is the sole cause of change in theworld

– agent is omniscient

– Closed World Assumption

“The Blocks World”

Domain:– set of cubic blocks sitting on

a table

Actions:– blocks can be stacked

– can pick up a block andmove it to another position

– can only pick up one blockat a time

Goal:– to build a specified stack of

blocks

AAAACCCCBBBB

INITIAL STATE

GOAL STATE

BBBB

AAAACCCC

So what’s the difference?

Problem-Solving:

(State-space search)n representation of actions,

states, goals, plansn actions are represented by a

program that generatescomplete state descriptions

n “black box”n unbroken sequences of

actions beginning from theinitial state to the goal state

Planning:

(Situation Calculus)n “open up” the representation

of states, goals, and actionsn planner is free to add actions

to the plan wherever theyare needed

n most parts of the world areindependent of most otherparts -> divide and conquer

Important Points

n Planning agents uselookahead

n Planning agents differ fromproblem-solving agents– use of more flexible

representations of states,actions, goals, and plans

Situation Calculus

n Augment FOL so that it can reasonabout actions in time– Add situation variables to specify time

– Add predicate holds(f,s)

– Add function result(a,s)

n Example: agent-walks-to-location-y(∀∀ x)(∀∀ y)(∀∀ s)(at(Agent,x,s)) =

at(Agent,y,result(walk(y),s))

Situation Calculus

n You must also specify what doesn’tchange!

n Example: agent-walks-to-location-y– (∀∀ x)(∀∀ y)(∀∀ s)(at(Agent,x,s)) =

at(Agent,y,result(walk(x,y),s))

– (∀∀ x)(∀∀ y)(∀∀ s)(at(Bananas,x,s)) =at(Bananas,x,result(walk(x,y),s))

Practical Planning

n Restrict the language with which wedefine problems

n Use a special-purpose algorithm calleda planner rather

Important Points

n A planning problem isrepresented in situationcalculus by logicalsentences

n Planning by unguidedlogical inference isinefficient and gives noguarantees about theresulting plan

Representing States & Goals

n STRIPS:– describes states & operators in a restricted

language

n States:– a conjunction of “facts” (ground literals that

do not contain variable symbols)

n Goals:– a conjunction of positive literals

Representing States & Goals

Initial State:ontable(A) Λontable(B) Λ

on(C, B) Λ

clear(A) Λclear(C) Λ

handempty

Goal:ontable(B) Λ

on(C,B) Λon(A,C) Λ

clear(A) Λ

handempty

AAAACCCCBBBB

INITIAL STATE

GOAL STATE

BBBB

AAAACCCC

Representing Actions

1. Action descriptionName: pickup(x)

2. PreconditionPreconditions: ontable(x), clear(x), handempty

3. EffectEffect: holding(x), ~ontable(x), ~clear(x),

~handempty

An operator is APPLICABLE if all preconditions areAn operator is APPLICABLE if all preconditions aresatisfied.satisfied.

Actions in “Blocks World”

n pickup(x)– picks up block ‘x’

from table

n putdown(x)– if holding block ‘x’,

puts it down on table

n stack(x,y)– if holding block ‘x’,

puts it on top ofblock ‘y’

n unstack(x,y)– picks up block ‘x’ that

is currently onblock ‘y’

Important Points

n The STRIPS languagedescribes actions in termsof their preconditions andeffects. It captures much ofthe expressive power ofsituation calculus, but notall domains and problemscan be described in theSTRIPS language

Planning as Search

n Situation Space– start at initial

state, applyoperators untilreached goal(e.g. searching)

n Plan Space– search through a

space of plans

See Figure 11.2 (pg. 340)

See Figure 11.5 (pg. 348)

FinishStartpickup(A)

unstack(C,B)

putdown(A)

stack(A,C)

...putdown(C)

stack(C,A)

stack(C,B)

Start

Finishon(A,C)

holding(A)pickup(A)

handempty

stack(A,C)

Situation-Space Planners

n Progression– Forward chaining from initial state to goal

state

n Regression– Backward chaining from goal state to initial

state

Progression Situation Space

n Like a state-space search exceptSTRIPS operators specified instead aset of next-move functions

n Use any search method you like (e.g.BFS, DFS, A*)

n PROBLEM:– huge search space to explore, so usually

very inefficient

AAAACCCCBBBB

INITIAL STATE

GOAL STATE

BBBB

AAAACCCC

Progression Situation-Space

Algorithm:1. Start from initial state2. Find all operators whose

preconditions are true inthe initial state

3. Compute effects ofoperators to generatesuccessor states

4. Repeat steps 2-3, until anew state satisfies thegoal conditions

Important Points

n There are 2 approaches tosolving planning problems:– situation-space planners

– plan-space planners

n Progression is one type ofsituation-space planners

Regression Situation-Space

n Usually more efficient than progression– progression: many operators are

applicable at each state

– regression: only a small number ofoperators are applicable for achieving agiven goal

n PROBLEM:– cannot always find a plan even if one

exists!

AAAACCCCBBBB

INITIAL STATE

GOAL STATE

BBBB

AAAACCCC

Regression Situation-Space

Algorithm:1. Start with goal node

corresponding to goal tobe achieved

2. Choose an operator thatwill add one of the goals

3. Replace that goal withthe operator’spreconditions

4. Repeat steps 2-3 untilyou have reached theinitial state

Important Points

n Regression is the secondtype of situation-spaceplanner

n It is more efficient thanprogression

Where Situation-Space fails…

n If the order of solving a set of goals failsbecause solving a later goal undoes anearlier goal

n Does not allow for interleaving of stepsin any solution it finds

“Sussman Anomaly”

n Solving on(A,B) first will be undonewhen solving the second goal on(B,C)and vice versa.

AAAACCCC

BBBBINITIAL STATE GOAL STATE

BBBBAAAA

CCCC

Plan-Space Planners

n Use when ordering of sub-goals matters

n Make commitments only as necessaryn Least Commitment planning:

– never making a choice unless required todo so

• Advantage: don’t have to backtrack later!

Important Points

n Use situation spaceplanners when goals arenot interleaved

n Use plan space plannerswhen goals are interleaved– the ordering of goals affects

the outcome

Definition of a plan

n A plan is formally defined as a data structureconsisting of the following 4 components:– A set of plan steps– A set of step ordering constraints

– A set of variable binding constraints– A set of causal links

Plan(STEPS:{S1:Op(ACTION: Start),

S2:Op(ACTION: Finish,

PRECOND: On(c, table) ∧ On(b,c) ∧ On(a,b) } ,

ORDERINGS: {S1 < S2},

BINDINGS: {},

LINKS: {})

Representation for Plans

Partial Order Plans:n can represent plans

in which some stepsare ordered withrespect to eachother and othersteps are unordered

Total Order Plans:n plans consist of a

simple list of steps

Partial vs. Total Order Plans

Start

LeftSock

RightSock

LeftShoe

RightShoe

Finish

Start

Finish

RightSock

LeftSock

LeftShoe

RightShoe

Partial Order Plan: Total Order Plan:

Start

Finish

LeftSock

RightSock

RightShoe

LeftShoe

Start

Finish

LeftSock

RightSock

LeftShoe

RightShoe

Start

Finish

RightSock

RightShoe

LeftSock

LeftShoe

Start

Finish

RightSock

LeftSock

RightShoe

LeftShoe

Start

Finish

LeftSock

LeftShoe

RightSock

RightShoe

Important Points

n Partial-ordering constraintsand uninstantiatedvariables allow us to followa least-commitmentapproach

Chapter 11 - Planning

Section 11.5

Partial-Order PlanningExample

Partial-Order Planning Example

n Return to shopping problem:– Buy milk, bananas, a drill, and then return

back home

n Start with the definitions of the initialand final states

n Also give the actions involved in theexample

State Definitions

n Initial StateOp(ACTION: Start,

EFFECT: At(Home) ∧ Sells(HWS,Drill) ∧

Sells(SM, Milk) ∧ Sells(SM,Bananas))

n Goal StateOp(ACTION: Finish,

PRECOND: Have(Drill) ∧ Have(Milk) ∧

Have(Bananas) ∧ At(Home))

Action Definitions

n ActionsOp(ACTION: Go(there),

PRECOND: At(here),

EFFECT: At(there) ∧ ¬ At(here))

Op(ACTION: Buy(x),

PRECOND: At(store) ∧ Sells(store, x),

EFFECT: Have(x))

Initial Plan

n Bold arrows are causal links

n Thin arrows are ordering constraints

Figure 11.6 From Artificial Intelligence: A Modern Approach

Extending the Plan

n Since there is no other way to achieve the Have(x)preconditions of the goal state, planner mustextend plan by adding actions that will achievethem

n Extend plan by adding Buy actions

Figure 11.7a From Artificial Intelligence: A Modern Approach

Causal Links Protect Conditions

n Causal link between Buy(Drill) andFinish means it was added to achievethe Finish precondition Have(Drill)

n If a new step comes that deletesHave(Drill) condition the planner willmake sure it won’t go betweenBuy(Drill) and Finish.

Achieving Preconditions

n Now planner achieves the Sells preconditions bylinking them to the initial state by causal links

n Since this is the only operator that achieves the Sellsprecondition, the planner must choose this one

Figure 11.17b From Artificial Intelligence: A Modern Approach

How Well Are We Doing?

n Large improvement over the problem-solving approach– Planner was able to choose the right

actions without considering all of the otheralternatives

– Don’t need to worry about ordering ofactions because the POP will do that later

Adding the Go Actions

n Planner must now extend the plan by adding Goactions to achieve At condition of Buy action

n This will get us to the HWS and the SM


Problem with the Go Actions I

n If planner links both Go actions to the initial state,both actions will interact with each other

n Problem: agent can’t be at two places at once


Problem with Go Actions II

n If the agent takes one link, then theprecondition At(Home) for the other linkwill be negated

n This is a threat, and must be resolvedbefore continuing

Problem with Go Actions III

n We realize that neither promotion nordemotion can solve this threat

n No matter which of the two is chosenfirst, At(Home) precondition of the otheris deleted

n When this occurs, the planner must giveup on partial plan and back up to aprevious choice

Solution to Go Actions

n Instead of linking both Go actions to initial state, theplanner decides to go from Home to the HWS, thenfrom the HWS to the SM

n This is done by linking Go(HWS) to Go(SM)

Modified Figure 11.11 From Artificial Intelligence: A Modern Approach

Another Threat

n Go(SM) step threatens the At(HWS) precondition ofthe Buy(Drill) step which is protected

n Might go from HWS to the SM and forget to buy thedrill!

n Solution: promotion– constrain Go(SM) to come after the Buy(Drill) step

– add an ordering constraint from Buy(Drill) to Go(SM)

Modified Figure 11.11 From Artificial Intelligence: A Modern Approach

Final Step I

n Only At(Home) precondition of goal stateleft to achieve

n Now planner must extend plan by addingGo(Home) step to achieve this

n Try to achieve At(x) by linking At(Home) tothe initial state, this will threaten At(Home)precondition of At(HWS) step

n There is no way to resolve this threat, soplanner must back up and try again

Final Step II

n Now planner tries to link At(x) to theGo(HWS) step, then this will also be athreat

n Threatens At(HWS) step of the Go(SM)step

n Again, can’t resolve this, so plannermust backtrack and try another path

Final Step III

n It now tries to link to Go(SM) step

n Another threat:– Threatens At(SM) preconditions of the

Buy(Milk) and Buy(Bananas) steps

n But this can be solved with promotion– Constrain the Go(Home) step to come after

the Buy(Milk) and Buy(Bananas) steps

n Now we’re done!

Final Plan


Advantages of POP

n POP can take a problem that would takemany search states for problem-solvingapproach and solve it in only a few steps

n Least commitment nature of planner means itonly needs to search where subplans interact

n Causal links allow planner to know when tostop a bad plan without wasting timeexpanding irrelevant parts of the plan

Summary

n POP starts with a minimal plan and adds to it until allpreconditions have been satisfied

n POP is a regression planner

n POP can take a problem that would take manysearch states for problem-solving approach and solveit in only a few steps

n Causal links protect conditions

n If threat occurs, promote/demote threatening steps

n POP can be extended to allow partially-instantiatedoperators– need to worry about possible threats

roots of planning

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