For Friday Read chapter 12, sections 3-5 None Program 2 Any - - PowerPoint PPT Presentation

For Friday

• Read chapter 12, sections 3-5
• None

Program 2

• Any questions?

Plan-Space Planners

• Plan-space planners search through the

space of partial plans, which are sets of actions that may not be totally ordered.

• Partial-order planners are plan-based and
• nly introduce ordering constraints as

necessary (least commitment) in order to avoid unnecessarily searching through the space of possible orderings

Partial Order Plan

• Plan which does not specify unnecessary
• rdering.
• Consider the problem of putting on your

socks and shoes.

Plans

• A plan is a three tuple <A, O, L>

– A: A set of actions in the plan, {A1 ,A2 ,...An} – O: A set of ordering constraints on actions {Ai <Aj , Ak <Al ,...Am <An}. These must be consistent, i.e. there must be at least one total

• rdering of actions in A that satisfy all the

constraints. – L: a set of causal links showing how actions support each other

Causal Links and Threats

• A causal link, Ap  QAc, indicates that

action Ap has an effect Q that achieves precondition Q for action Ac.

• A threat, is an action A t that can render a

causal link Ap  QAc ineffective because:

– O  {AP < At < Ac} is consistent – At has ¬Q as an effect

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.

Example

Op( Action: Go(there); Precond: At(here); Effects: At(there), ¬At(here) ) Op( Action: Buy(x), Precond: At(store), Sells(store,x); Effects: Have(x) )

• A0:

– At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill)

• A

– Have(Drill) Have(Milk) Have(Banana) At(Home)

POP Algorithm

• Stated as a nondeterministic algorithm where

choices must be made. Various search methods can be used to explore the space of possible choices.

• Maintains an agenda of goals that need to be

supported by links, where an agenda element is a pair <Q,Ai> where Q is a precondition of Ai that needs supporting.

• Initialize plan to null plan and agenda to conjunction
• f goals (preconditions of Finish).
• Done when all preconditions of every action in plan

are supported by causal links which are not threatened.

POP(<A,O,L>, agenda)

1) Termination: If agenda is empty, return <A,O,L>.

Use topological sort to determine a totally ordered plan.

2) Goal Selection: Let <Q,Aneed> be a pair on the agenda 3) Action Selection: Let A add be a nondeterministically chosen action that adds Q. It can be an existing action in A or a new action. If there is no such action return failure.

L’ = L  {Aadd  QAneed} O’ = O  {Aadd < Aneed} if Aadd is new then A’ = A  {Aadd} and O’=O’ È {A0 < Aadd <A} else A’ = A

4) Update goal set:

Let agenda’= agenda - {<Q,Aneed>} If Aadd is new then for each conjunct Qi of its precondition, add <Qi , Aadd> to agenda’

5) Causal link protection: For every action At that threatens a causal link Ap  QAc add an ordering constraint by choosing nondeterministically either

(a) Demotion: Add At < Ap to O’ (b) Promotion: Add Ac < At to O’

If neither constraint is consistent then return failure. 6) Recurse: POP(<A’,O’,L’>, agenda’)

Example

Op( Action: Go(there); Precond: At(here); Effects: At(there), ¬At(here) ) Op( Action: Buy(x), Precond: At(store), Sells(store,x); Effects: Have(x) )

• A0:

– At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill)

• A

– Have(Drill) Have(Milk) Have(Banana) At(Home)

Example Steps

• Add three buy actions to achieve the goals
• Use initial state to achieve the Sells

preconditions

• Then add Go actions to achieve new pre-

conditions

Handling Threat

• Cannot resolve threat to At(Home) preconditions
• f both Go(HWS) and Go(SM).
• Must backtrack to supporting At(x) precondition
• f Go(SM) from initial state At(Home) and

support it instead from the At(HWS) effect of Go(HWS).

• Since Go(SM) still threatens At(HWS) of

Buy(Drill) must promote Go(SM) to come after Buy(Drill). Demotion is not possible due to causal link supporting At(HWS) precondition of Go(SM)

Example Continued

• Add Go(Home) action to achieve At(Home)
• Use At(SM) to achieve its precondition
• Order it after Buy(Milk) and Buy(Banana)

to resolve threats to At(SM)

GraphPlan

• Alternative approach to plan construction
• Uses STRIPS operators with some

limitations

– Conjunctive preconditions – No negated preconditions – No conditional effects – No universal effects

Planning Graph

• Creates a graph of constraints on the plan
• Then searches for the subgraph that

constitutes the plan itself

Graph Form

• Directed, leveled graph

– 2 types of nodes:

• Proposition: P
• Action: A

– 3 types of edges (between levels)

• Precondition: P -> A
• Add: A -> P
• Delete: A -> P
• Proposition and action levels alternate
• Action level includes actions whose preconditions

are satisfied in previous level plus no-op actions (to solve frame problem).

Planning graph

… … …

Constructing the planning graph

• Level P1: all literals from the initial state
• Add an action in level Ai if all its

preconditions are present in level Pi

• Add a precondition in level Pi if it is the

effect of some action in level Ai-1 (including no-ops)

• Maintain a set of exclusion relations to

eliminate incompatible propositions and actions (thus reducing the graph size)

Mutual Exclusion relations

• Two actions (or literals) are mutually

exclusive (mutex) at some stage if no valid plan could contain both.

• Two actions are mutex if:

– Interference: one clobbers others’ effect or precondition – Competing needs: mutex preconditions

• Two propositions are mutex if:

– All ways of achieving them are mutex

Mutual Exclusion relations

Inconsistent Effects Inconsistent Support Competing Needs Interference (prec-effect)

Dinner Date example

• Initial Conditions: (and (garbage) (cleanHands) (quiet))
• Goal: (and (dinner) (present) (not (garbage))
• Actions:

– Cook :precondition (cleanHands) :effect (dinner) – Wrap :precondition (quiet) :effect (present) – Carry :precondition :effect (and (not (garbage)) (not (cleanHands)) – Dolly :precondition :effect (and (not (garbage)) (not (quiet)))

Dinner Date example

Dinner Date example

Observation 1

Propositions monotonically increase

(always carried forward by no-ops) p ¬q ¬r p q ¬q ¬r p q ¬q r ¬r p q ¬q r ¬r A A B A B

Observation 2

Actions monotonically increase

p ¬q ¬r p q ¬q ¬r p q ¬q r ¬r p q ¬q r ¬r A A B A B

Observation 3

Proposition mutex relationships monotonically decrease

p q r … A p q r … p q r …

Observation 4

Action mutex relationships monotonically decrease

p q … B p q r s … p q r s … A C B C A p q r s … B C A

Observation 5

Planning Graph ‘levels off’.

• After some time k all levels are identical
• Because it’s a finite space, the set of literals

never decreases and mutexes don’t reappear.

Valid plan

A valid plan is a planning graph where:

• Actions at the same level don’t interfere
• Each action’s preconditions are made true

by the plan

• Goals are satisfied

GraphPlan algorithm

• Grow the planning graph (PG) until all

goals are reachable and not mutex. (If PG levels off first, fail)

• Search the PG for a valid plan
• If none is found, add a level to the PG and

try again

Searching for a solution plan

• Backward chain on the planning graph
• Achieve goals level by level
• At level k, pick a subset of non-mutex actions to

achieve current goals. Their preconditions become the goals for k-1 level.

• Build goal subset by picking each goal and

choosing an action to add. Use one already selected if possible. Do forward checking on remaining goals (backtrack if can’t pick non- mutex action)

Plan Graph Search

If goals are present & non-mutex:

Choose action to achieve each goal Add preconditions to next goal set

Termination for unsolvable problems

• Graphplan records (memoizes) sets of unsolvable

goals:

– U(i,t) = unsolvable goals at level i after stage t.

• More efficient: early backtracking
• Also provides necessary and sufficient conditions

for termination:

– Assume plan graph levels off at level n, stage t > n – If U(n, t-1) = U(n, t) then we know we’re in a loop and can terminate safely.

Dinner Date example

• Initial Conditions: (and (garbage) (cleanHands) (quiet))
• Goal: (and (dinner) (present) (not (garbage))
• Actions:

– Cook :precondition (cleanHands) :effect (dinner) – Wrap :precondition (quiet) :effect (present) – Carry :precondition :effect (and (not (garbage)) (not (cleanHands)) – Dolly :precondition :effect (and (not (garbage)) (not (quiet)))

Dinner Date example

Dinner Date example

Dinner Date example

Knowledge Representation

• Issue of what to put in to the knowledge

base.

• What does an agent need to know?
• How should that content be stored?

Knowledge Representation

• NOT a solved problem
• We have partial answers

Question 1

• How do I organize the knowledge I have?

Ontology

• Basically a hierarchical organization of

concepts.

• Can be general or domain-specific.

Question 2

• How do I handle categories?

Do I need to?

• What makes categories important?

Defining a category

• Necessary and sufficient conditions

Think-Pair-Share

• What is a chair?

Prototypes

In Logic

• Are categories predicates or objects?

Important Terms

• Inheritance
• Taxonomy

What does ISA mean?

Categories

• Membership
• Subset or subclass
• Disjoint categories
• Exhaustive Decomposition
• Partitions of categories

Other Issues

• Physical composition
• Measurement
• Individuation

– Count nouns vs. mass nouns – Intrinsic properties vs. extrinsic properties