Graph-based Planning

Brian C. Williams Sept. 25th & 30th, 2002 16.412J/6.834J

Outline Introduction The Graph Plan Planning Problem Graph Construction Solution Extraction Properties Termination with Failure

Graph Plan Graphplan was developed in 1995 by

Avrim Blum and Merrick Furst, at CMU. Recent implementations by other

researchers have extended the capability of Graphplan to reason with temporally extended actions, metrics and non-atomic preconditions and effects.

Approach: Graph Plan1. Constructs compact encoding of state space from

operators and initial state, which prunes many invalid plans – Plan Graph.

2. Generates plan by searching for a consistent subgraph that achieves the goals.

PropositionInit State

ActionTime 1

PropositionTime 1

ActionTime 2

Plan graphs focus towards valid plans Plan graphs exclude many plans that are

infeasible. Plans that do not satisfy the initial or goal state. Plans with operators that threaten each other.

Plan graphs are constructed in polynomial time and are of polynomial in size.

The plan graph does not eliminate all infeasible plans.

Plan generation still requires focused search.

Example: Graph and Solution noGarb

cleanH

dinner

present

cleanH

noGarb

cleanH

dinner

present

1 Prop 1 Action 2 Prop 2 Action 3 Prop

Outline Introduction The Graph Plan Planning Problem Graph Construction Solution Extraction Properties Termination with failure

GraphPlan Planning Problem• State

• A consistent conjunction of propositions (positive literals)• E.g., (and (cleanhands)(quiet) (dinner) (present)(noGarbage))• Doesn’t commit to the truth of all propositions

• Initial State• Problem state at time i = 0

• E.g., (and (cleanHands) (quiet))

• Goal State• A partial state, represented by a conjunction of literals.

• E.g., (and (noGarbage) (dinner) (present))• The planner must put the system in a final state that satisfies

the conjunction.

GraphPlan Planning ProblemActions• E.g., (:operator carry

:precondition :effect (:and (noGarbage) (not (cleanHands)))

• Preconditions: propositions that must be true to apply operator.• A conjunction of propositions (no negated propositions).

• Effects: Propositions that operator changes, given preconditions.• A conjunction of propositions (adds) and their negation (deletes).

carry noGarb cleanH

• Parameterized schemas and objects are used to compactly encode actions.

Add Edge

Delete Edge

GraphPlan Planning Problem

(:operator cook :precondition (cleanHands):effect (dinner))

(:operator carry :precondition :effect (:and (noGarbage) (not (cleanHands)))

Action Execution at time i: If all propositions of :precondition appear in the state at i,Then state at i+1 is created from state at i, by

adding to i, all “add” propositions in :effects, removing from i, all “delete” propositions in :effects.

carry noGarb cleanH

cook dinner

cleanHands

GraphPlan Planning Problem• Noops

• Every proposition has a no-op action, which maintains it from time i to i+1.

• E.g., (:operator noop-P :precondition (P) :effect (P))

Noop-P P P

Example: Dinner Date ProblemInitial Conditions: (and (cleanHands) (quiet))

Goal: (and (noGarbage) (dinner) (present))

Actions:(:operator carry :precondition

:effect (and (noGarbage) (not (cleanHands)))

(:operator dolly :precondition :effect (and (noGarbage) (not (quiet)))

(:operator cook :precondition (cleanHands) :effect (dinner))

(:operator wrap :precondition (quiet) :effect (present))

+ noops

dinner

present

cleanH

noGarb

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dinner

present

Prop at 1 Action at 1 Prop at 2 Action at 2 Prop at 2

noop-dinner

noop-present

• Sets of concurrent actions performed at each time [i]• Uses model of concurrency as interleaving.

• If actions a and b occur at time i, then it must be valid toperform either a followed by b, OR b followed by a.

A Plan in GraphPlan <Actions[i] >

A Complete Consistent PlanGiven that the initial state holds at time 0, a plan is a solution iff:

• Complete:• The preconditions of every operator at time i, is satisfied by a proposition at time i.

• The goal propositions all hold in the final state.

• Consistent:• The operators at any time i can be executed in any order, without one of these operators undoing the:

•preconditions of another operator at time i. •effects of another operator at time i.

dinner

present

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noGarb

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Prop at 1 Action at 1 Prop at 2 Action at 2 Prop at 3

(noop dinner)

(noop present)

Example of a Complete Consistent Plan

Initial Conditions: (and (cleanHands) (quiet))

Goal: (and (noGarbage) (dinner) (present))

GraphPlan Algorithm Phase 1 – Plan Graph Expansion

Creates graph encoding pairwise consistency and reachability of actions and propositions from initial state.

Graph includes, as a subset, all plans that are complete and consistent.

Phase 2 - Solution Extraction Graph treated as a kind of constraint satisfaction problem

(CSP). Selects whether or not to perform each action at each time

point, by assigning CSP variables and testing consistency.

Outline Introduction The Planning Problem Graph Construction Solution Extraction Properties Termination with failure

Graph Properties

A Plan graph compactly encodes the space of consistent

plans, while pruning . . .

1. partial states and actions at each time i that are not reachable from the initial state.

2. pairs of actions and propositions that are mutually inconsistent at time i.

3. plans that cannot reach the goals.

Constructing the planning graph…(Reachability) Initial proposition layer

Contains propositions in initial state.

Example: Initial State, Layer 0

cleanH

Contains propositions in initial state Action layer i

If all action’s preconditions are consistent in i-1 Then add action to layer i

Proposition layer i+1 For each action at layer i Add all its effects at layer i+1

Example: Add Actions and Effects noGarb

cleanH

dinner

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Write down just the initial conditions Action layer i

If all action’s preconditions appear consistent in i-1 Then add action to layer i

Repeat until all goal propositions appear

Can a solution exist? noGarb

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Do all goalpropositionsappear?

Constructing the planning graph…(Consistency) Initial proposition layer

Write down just the initial conditions Action layer i

If action’s preconditions appear consistent in i-1 (non-mutex) Then add action to layer i

Identify mutual exclusions Actions in layer i Propositions in layer i + 1

Repeat until all goal propositions appear non-mutex

Mutual Exclusion: Actions Actions A,B are mutually exclusive at level i

if no valid plan could possibly contain both at i:

They Interfere A deletes B’s preconditions, or A deletes B’s effects, or Vice versa or

OR . . . .

What causes the exclusion? noGarb

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present

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noGarb

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cleanH

What causes the exclusion?

noGarb

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noGarb

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cleanH

Mutual Exclusion: Actions Actions A,B are mutually exclusive at level i

if no valid plan could possibly contain both at i:

They Interfere A deletes B’s preconditions, or A deletes B’s effects, or Vice versa or

They have competing needs: A & B have inconsistent preconditions

Layer 0: complete action mutexs noGarb

cleanH

dinner

present

cleanH

Mutual Exclusion: Proposition Layer

Propositions P,Q are inconsistent at i if no valid plan could possibly contain both at i If at i, all ways to achieve P exclude all ways to

achieve Q

Layer 1: add proposition mutexs noGarb

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Can a solution exist? noGarb

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Do all goalpropositionsappear non-mutex?

Graphplan Create plan graph level 1 from initial state Loop

If goal propositions of the highest level (nonmutex)

Then search graph for solution If solution found, then return and terminate

Else extend graph one more level

A kind of double search: forward direction checks necessary

(but insufficient) conditions for a solution, ...

Backward search verifies...

Searching for a SolutionRecursively find consistent actions achieving all goals at

t, t-1, . . . : For each goal proposition G at time t

For each action A making G true at t If A isn’t mutex with previously chosen action at t, Then select it

Finally, If no action of G works, Then backtrack on previous G.

Finally If action found for each goal at time t, Then recurse on preconditions of actions selected, t-1, Else backtrack.

Searching for a solution noGarb

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dinner

present

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Select action achieving Goal noGarb

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cleanH

Select action achieving Goal dinner,

Consistent with carry

Backtrack on noGarb

cleanH

dinner

present

cleanH

Select action dolly

Backtrack on noGarb

cleanH

dinner

present

cleanH

Select action achieving Goal dinner,

Consistent with dolly

cleanH

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present

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Select action achieving Goal present,

Consistent w dolly, cook

cleanH

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present

cleanH

Recurse on preconditions of dolly, cook & wrap

All satisfied by initial state

Avoiding redundancy No-ops are always favoured.

guarantees that the plan will not contain redundant actions.

Suppose the Plan Graph was Extended

noGarb

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noGarb

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Can a valid plan use carry? noGarb

cleanH

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present

cleanH

noGarb

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present

Using carry at level 2 is valid noGarb

cleanH

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present

cleanH

noGarb

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present

Using carry at level 2 is valid noGarb

cleanH

dinner

present

cleanH

noGarb

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present

Plan Graph Properties Propositions monotonically increase

once they are added to a layer they are never removed in successive layers;

Mutexes monotonically decrease once a mutex has decayed it can never reappear;

Memoized sets (to be discussed) monotonically decrease if a goal set is achievable at a layer

Then it will be achievable at all successive layers.

The graph will eventually reach a fix point, that is, a level where facts and mutexes no longer change.

Fix point Example: Door Domain

Fix point Example: Door DomainMove from room A to room B pre: robot is in A and the door is open add: robot is in B del: robot is in A

Open door pre: door is closed add: door is open del: door is closed

Close door pre: door is open add: door is closed del: door is open

Closed

Opened

Layer 4

Closed

Layer 1

Closed

Opened

Layer 2

Closed

Opened

Layer 3

Layer 4 is the fixed point (called level out) of the graph

Graph Search Properties Graphplan may need to expand well beyond the

fix point to find a solution. Why?

Gripper ExampleMove from one room to another pre: robot in the first room add: robot in the second room del: robot in the first room

Pick a ball up in a gripper pre: robot has a free gripper and is in same room as ball add: robot holding the ball del: gripper free, ball in room.

Drop a ball in a room pre: robot holding the ball, robot in destination room add: ball in destination room, gripper free. del: ball being held.

Gripper Example In Gripper, the fix point occurs at layer 4,

all facts concerning the locations of the balls and the robot are pairwise non-mutex after 4 steps.

The distance to the solution layer depends on the number of balls to be moved. For large numbers of balls Graphplan searches

copies of the same layers repeatedly. For example, for 30 balls

the solution layer is at layer 59, 54 layers contain identical facts, actions and mutexes.

Search Properties (cont) Plans do not contain redundant steps.

By preferring No-ops

Graphplan guarantees parallel optimality. The plan will take as short a time as possible.

Graphplan doesn’t guarantee sequential optimality Might be possible to achieve all goals at some layer with

fewer actions at that layer.

Graphplan returns failure if and only if no plan exists.

Simple Termination If the fix point is reached and either:

One of the goals is not asserted or Two of the goals are mutex

Then Graphplan can return "No solution" without any search at all.

Else there may be higher order exclusions which Graphplan cannot detect

Requires more sophisticated additional test for termination.

Why Continue After the Fix Point? facts, actions and mutexes no longer change after the fix

point, N-ary exclusions DO change.

New layers add time to the graph. Time allows actions to be spaced so that N-ary mutexes have

time to decay.

While new things can happen between two successive layers progress is being made.

Track n-ary mutexes. Terminate on their fix point.

Memoization and Termination A goal set at a layer may be unsolvable

Example, in Gripper. If a goal set at layer k cannot be achieved,

Then that set is memoized at layer k. To prevent wasted search effort

Checks each new goal set at k against memos of k for infeasibility.

If an additional layer is built and searching it creates no new memos, then the problem is unsolvable.

Recap: Graph Plan Graphplan was developed in 1995 by Avrim Blum

and Merrick Furst, at CMU. Graphplan searches a compact encoding of the

state space from the operators and initial state, which prunes many invalid plans, violating reachability and mutual exclusion.

Recent implementations by other researchers have extended the capability of Graphplan to reason with temporally extended actions, metrics and non-atomic preconditions and effects.

Appendix

Termination Test If the Graph leveled off at layer n, and a search

stage, t >n, has passed in which the memoized sets at n+1 are the same as at layer n, Then Graphplan can output "No Solution".

Layer n

Layer n + 1

Layer t

Stn = St

n+1 Where Smk = the sets of goals found

unsolvable at layer k during search from m

Termination Property Claim: Graphplan returns with failure iff the problem

is unsolvable.

Proof: If the problem is unsolvable Graphplan returns with failure. The number of sets found unsolvable at layer n from layer t will never be smaller than the number at n from layer t+1, and there is a finite maximum number of goal sets. This means that, if the problem is unsolvable, eventually two successive layers will contain the same memoized sets.

If Graphplan outputs "No Solution," then the problem is unsolvable. Suppose the fix point is at layer n and Graphplan has completed an

unsuccessful search from layer t > n. A plan to achieve any set unsolvable at layer n+1 must, one step

earlier, achieve some set unsolvable at layer n. Suppose Graphplan returns "No Solution" but the problem is

solvable:

If Stn = St

n+1 then S' and S'' must both be in Stn+1. This means that some set

in Stn+1 will need to be achieved in n+1. this situation is contradictory.

Layer n

Layer N + 1

Layer q

S’ in Stn S’’ in St

Layer q + 1

Solution Layer q

S’’S’

Graph-based Planning

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