For Wednesday• Read chapter 12, sections 3-5• Program 2 progress due

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 only 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 ordering.

• 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 ordering 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 È {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 of 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 È {Aadd ® QAneed} O’ = O È {Aadd < Aneed} if Aadd is new then

A’ = A È {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 of both Go(HWS) and Go(SM).

• Must backtrack to supporting At(x) precondition of 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

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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

InconsistentEffects

InconsistentSupport

CompetingNeeds

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)

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Observation 2

Actions monotonically increase

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Observation 3

Proposition mutex relationships monotonically decrease

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Observation 4

Action mutex relationships monotonically decrease

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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 SearchIf goals are present & non-mutex:

Choose action to achieve each goalAdd 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