Heuristics in Planning - Automated Planning - Lecture Slides, Slides of Computer Science

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Chapter 9

Heuristics in Planning

Lecture slides for

Automated Planning: Theory and Practice

Digression: the A* algorithm (on trees)

• Suppose we’re searching a tree in which each edge ( s , s' ) has a cost c ( s , s' )
  • If p is a path, let c ( p ) = sum of the edge costs
  • For classical planning, this is the length of p
• For every state s , let
  • g ( s ) = cost of the path from s 0 to s
  • h* ( s ) = least cost of all paths from s to goal nodes
  • f* ( s ) = g ( s ) + h* ( s ) = least cost of all paths from s 0 to goal nodes that go through s
• Suppose h ( s ) is an estimate of h* ( s )
  • Let f ( s ) = g ( s ) + h ( s )
    • f ( s ) is an estimate of f* ( s )
  • h is admissible if for every state s , 0 ≤ h ( s ) ≤ h* ( s )
  • If h is admissible then f is a lower bound on f*

g(s)

h*(s)

Hill Climbing

• Use h as a node-selection heuristic
  • Select the node v in C for which h ( v ) is smallest
• Why not use f?
• Do we care whether h is admissible?

u

C

FastForward (FF)

• Depth-first search

• Selection heuristic: relaxed

Graphplan

– Let v be a node in C

– Let Pv be the planning problem of

getting from v to a goal

– use Graphplan to find a solution for

a relaxation of Pv

– The length of this solution is a lower

bound on the length of a solution to

Pv

u

C

FastForward

• FF evaluates all the nodes in the set C of u ’s successors
• If none of them has a better heuristic value than u , FF does a breadth-first search for a state with a strictly better evaluation
• The path to the new state is added to the current plan, and the search continues from this state
• Works well because plateaus and local minima tend to be small in many benchmark planning problems
• Can’t guarantee how fast FF will find a solution, or how good a solution it will find - However, it works pretty well on many problems

AIPS-2000 Planning Competition

• FastForward did quite well

• In the this competition, all of the planning

problems were classical problems

• Two tracks:

– “Fully automated” and “hand-tailored” planners

– FastForward participated in the fully automated track

• It got one of the two “outstanding performance” awards

– Large variance in how close its plans were to optimal

• However, it found them very fast compared with the other

fully-automated planners

2004 International Planning

Competition

• FastForward’s author was one of the

competition chairs

• Thus FastForward did not participate

Plan-Space Planning

• Refine = select next flaw to work on
• Branch = generate resolvers
• Prune = remove some of the resolvers
• nondeterministic choice = resolver selection

Serializing and AND/OR Tree

• The search space is

an AND/OR tree

• Deciding what flaw to work on next = serializing this tree (turning it into a

state-space tree)

• at each AND branch,

choose a child to

expand next, and

delay expanding

the other children

… …

…

Operator o 1 … Operator o n

Goal g 1 Goal g 2 Constrain variable v

Order tasks

Partial plan p

Partial plan p

Goal g 1 Operator o (^1) Operator o n

Partial plan p 1 Partial plan p n

Goal g (^2) … Constrainvariable v …^ Ordertasks Goal g (^2) … Constrainvariable v …^ Ordertasks

One Serialization

Why Does This Matter?

• Different refinement strategies produce different serializations
  • the search spaces have different numbers of nodes
• In the worst case, the planner will search the entire serialized search space
• The smaller the serialization, the more likely that the planner will be efficient
• One pretty good heuristic: fewest alternatives first

A Pretty Good Heuristic

• Fewest Alternatives First (FAF)
  • Choose the flaw that has the smallest number of alternatives
  • In this case, unestablished precondition g 1

Case Study, Continued

• The best serialization contains Θ( b^2 k ) nodes
• The worst serialization contains Θ(2 kb^2 k ) nodes
• The size differs by an exponential factor
• But both serializations are doubly exponentially large
• This limits how good any flaw-selection heuristic can do
  • To do better, need good ways to do node selection, branching, pruning

Resolver Selection

• This is an “or” branch