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Chapter 16

Planning Methods

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Chapter 16 Contents (1)

STRIPS STRIPS Implementation Partial Order Planning The Principle of Least Commitment Propositional Planning SAT Planning

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Chapter 16 Contents (2)

Planning Graphs GraphPlan ADL and PDDL Probabilistic Planning Dynamic World Planning Case-Based Planning Scheduling

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STRIPS (1)

Stanford Research Institute Problem Solver. An operator based planning system. STRIPS uses wffs in FOPC to describe the

world. For example: STRIPS was designed to enable a planner to

devise plans for a robot to solve problems in the blocks world.

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STRIPS (2) STRIPS defines operators as in the following rule

schemata:Precondition: AT(r, x)

Λ AT(o, x)Delete: AT(r, x)

AT(o, x)Add: AT(r, y)

AT(o, y) The preconditions specify what must be true for the

operator to be applied. The delete and add lists specify the changes that will

take place after the operator is applied.

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STRIPS Implementation

STRIPS uses resolution and means-ends analysis to devise plans:

The goal is negated, and the rule schemata are instantiated with objects from the real world.

If the resolution fails, then the goal has been achieved.

Otherwise, a plan is devised.

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Example 1

Three blocks – a,b,c Block a on table, block b is on c Predicates

On(x,y) means x is on y Clear(x) means x has no block on top of it t is the table

Goal On(c,a) get c on a

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Example 2

Start state

On(a,t)

On(b,c)

On(c,t)

Clear(b)

Clear(a)

Clear(t)

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Example 3

Operators MoveOnto(x,y) move x onto top y

– Preconditions On(x,z) Λ Clear(x) Λ Clear(y)– Delete On(x,z), Clear(y)– Add On(x,y), Clear(z)

moveOntoTable(x)– Preconditions On(x,y) Λ Clear(x)– Delete On(x,y)– Add On(x,t) Clear(y)

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Example 4

Approaches Forward chaining – search forward through

the space of possible plans until we find one

Build tree – first move choices– MoveOnto(a,b)– MoveOnto(b,a)– MoveOntoTable(b)

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Example 5

Choose MoveOntoTable(b) Preconditions On(b,y) Λ Clear(b) matched

by instantiating y with c. Apply

– Delete: On(b,c)– Add: On(b,t), Clear(c)

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Example 6

Current state description

On(a,t)

On(b,t)

On(c,t)

Clear(b)

Clear(a)

Clear(c)

Clear(t)

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Example 7

Applicable moves

MoveOnto(a,b)

MoveOnto(a,c)

MoveOnto(b,a)

MoveOnto(b,c)

MoveOnto(c,a)

MoveOnto(c,b)

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Example comments

Continue in this manner until a suitable plan is found

Does not use means-ends analysis Works only for small problems Could enhance by using backward

chaining – MoveOntoTable(b) has the effect of clearing c, which helps

Used unification and resolution

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Partial Order Planning (1)

A total order plan specifies the order in which all actions must be carried out.

A partial order plan can specify some actions in parallel – these actions can be carried out in any order relative to each other.

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Partial Order Planning (2)

A partial order plan can be implemented in one of several ways.

The partial order plan on the left is implemented in one of two ways, shown in the total order plans, center and right:

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The Principle of Least Commitment

In building a plan there will be some variables and objects that can be ignored, as they are superfluous to the goal of the plan.

Some variables do not need to be instantiated – for example it is preferable, where possible, to use MoveOnto (a, y) than to use MoveOnto (a, b).

This is the Principle of Least Commitment.

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Propositional Planning (1)

Any STRIPS plan can be expressed in propositional logic.

This will often involve increasing the number of variables.

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Propositional Planning (2) Any STRIPS plan can be expressed in propositional

logic. This will often involve increasing the number of

variables. For example:Clear (x)On (x, y)

These predicates can be represented as propositions:

X1 is equivalent to Clear (A)

X2 is equivalent to Clear (B)

X3 is equivalent to On (A, B)

X4 is equivalent to On (B, A)

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Propositional Planning (3) States can be represented as an assignment of

truth values to the propositions:

X1 Λ ¬X2 Λ X3 Λ ¬ X4

This state can be represented in STRIPS notation as:

Clear (A) Λ ¬Clear (B) Λ On (A, B) Λ ¬On (B, A) The following sentence represents all

states in which A is clear and B is not clear:

X1 Λ ¬X2

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Propositional Planning (4) Actions can also be represented as the preconditions and

the results of the action. We use the notation ¬X1’ to indicate that X1’ is no longer

true after the action. Hence, an action might be:

X1 Λ X2 Λ ¬X3 Λ ¬X4 Λ X1’ Λ ¬X2’ Λ X3’ Λ ¬X4’ This action is MoveOnto (A, B). This is a simple example, but propositional planning can

lead to very complex expressions being used. The advantage of using propositional planning is that

automated systems can be built to manipulate the plans. Requires n2 variables for n blocks

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Satisfiability (SAT) Planning

The satisfiability problem, of determining whether a given propositional logic sentence is satisfiable or not, is NP-Complete.

A number of efficient methods have been developed for devising plans by determining the satisfiability of propositional logic expressions.

Methods are either systematic, which involve checking all possible assignments of truth values, or stochastic.

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Planning Graphs (1)

Even-numbered levels represent states. Odd-numbered levels actions.

Level 0 contains the propositions that represent the start state. The arrows from level 0 to level 1 show how those propositions match the preconditions of the actions in level 1.

All possible actions are shown in the graph.

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Planning Graphs (2)

The partial planning graph shown includes persistence actions (things which do not change) as lines with squares on.

The heavy black lines show mutexes: Two propositions joined by such a line are mutually

exclusive, and cannot both be used in the same plan.

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Planning Graphs (3)

The planning graph for even a simple problem can be extremely complex.

By producing a complete planning graph for a problem, it can be determined whether a plan is possible, and the plan itself can also be derived.

Algorithms such as GraphPlan can be used to extract the plan.

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GraphPlan

Problems are expressed in STRIPS notation. GraphPlan iteratively builds a planning

graph, starting from the initial state and working towards the goal state.

All applicable operators are applied at each level to produce the next level .

When the propositions necessary for the goal are included in the current level, and they are not mutex, a possible solution may have been reached.

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ADL

Another method for representing planning problems.

ADL – Action Description Language: More expressive than STRIPS. Allows quantified expressions such as:

x.P(x) Λ ¬ Q(x) Preconditions can include disjunctions. Allows conditional effects – effects of actions

that are dependent on other factors.

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Probabilistic Planning

Thus far we have assumed that all actions are deterministic.

In fact, some actions are non-deterministic – their effects can vary.

It is possible to extend situation calculus to deal with non-deterministic actions.

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Dynamic World Planning

Our discussion so far has assumed the world is static.

In fact, the world is dynamic – things outside of the control of the planner change.

Execution monitoring is used to monitor the execution of plans: If something changes during execution, replanning

may be necessary. Another approach is conditional planning – this

includes every possible outcome in the plan.

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Case-Based Planning

Case-based planning involves storing each plan that is devised.

Plans (and partial plans) can be re-used later to solve other, similar problems.

Example: CHEF. A system that is used to devise recipes for Chinese food given a set of ingredients.

If presented with a set of ingredients it has not seen before, it is able to use similar sets of ingredients it has seen before to devise a new recipe.

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Scheduling

Scheduling is like planning, but also takes into account the length of time each action takes to execute.

Job-shop scheduling involves allocating machinery to a set of tasks.

The scheduler plans when each task will start, and how long it will take.

Scheduling can be treated as planning with constraints, where the constraints specify how long tasks will take.