Last time: Problem-Solving Problem solving: Goal formulation - - PDF document

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Last time: Problem-Solving

Problem solving:

Goal formulation Problem formulation (states, operators) Search for solution

Problem formulation:

Initial state ?

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? ? ?

Last time: Problem-Solving

Problem types: Problem types:

single state: accessible and deterministic environment multiple state:

?

contingency:

?

exploration:

?

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exploration:

?

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Last time: Finding a solution

Solution: is ??? Function General-Search(problem, strategy) returns a solution, or

failure initialize the search tree using the initial state problem

loop do if there are no candidates for expansion then return failure Basic idea: offline, systematic exploration of simulated state-space by generating successors of explored states (expanding)

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if there are no candidates for expansion then return failure

choose a leaf node for expansion according to strategy

if the node contains a goal state then return the

corresponding solution

else expand the node and add resulting nodes to the search

tree

end

Last time: Finding a solution

Function General-Search(problem, strategy) returns a solution, or Function General Search(problem, strategy) returns a solution, or

failure initialize the search tree using the initial state problem

loop do if there are no candidates for expansion then return failure

choose a leaf node for expansion according to strategy

if the node contains a goal state then return the

di l ti

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corresponding solution

else expand the node and add resulting nodes to the search

tree

end Strategy: The search strategy is determined by ???

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Last time: search strategies

Uninformed: Use only information available

in the problem formulation

B dth fi t

Breadth-first Uniform-cost Depth-first Depth-limited Iterative deepening

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p g

Informed: Use heuristics to guide the search

Best first A*

Evaluation of search strategies

Search algorithms’ four criteria: Search algorithms four criteria:

Completeness: Time complexity: Space complexity: Optimality:

Complexity are measured in terms of:

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Complexity are measured in terms of:

b – max branching factor of the search tree d – depth of the least-cost solution m – max depth of the search tree (may be infinity)

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Last time: uninformed search strategies

Uninformed search: Uninformed search:

Use only information available in the problem formulation

Breadth-first Uniform-cost

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Uniform cost

Depth-first Depth-limited Iterative deepening

Comparing uninformed search strategies

Criterion Breadth Uniform DepthFirst DepthLim Iterative Bidirectional Time b^ d b^ d b^ m b^ l b^ d b^ (d/2) Space b^ d b^ d bm bl bd b^ (d/2) Optimal? Yes Yes No No Yes Yes

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Complete? Yes Yes No Yes if l≥d Yes Yes

b – max branching factor of the search tree d – depth of the least-cost solution m – max depth of the state-space (may be infinity) l – depth cutoff

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This time: informed search

I nformed search: I nformed search:

Use heuristics to guide the search

Best first A* Heuristics

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Heuristics Hill-climbing Simulated annealing

Best-first search

Idea: use an evaluation function for each

Idea: use an evaluation function for each node; estimate of “desirability”

Implementation:

QueueingFn = insert successors in decreasing

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Q g

g

• rder of desirability

Special cases:

greedy search A* search

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Romania with step costs in km

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329 253

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Greedy search

Estimation function:

Estimation function:

h(n) = estimate of cost from n to goal (heuristic)

For example:

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Properties of Greedy Search

Complete? Complete? Time?

S ?

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Space? Optimal?

Properties of Greedy Search

Complete?

99 105

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120

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Properties of Greedy Search

Complete? Complete? Time?

Space?

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Space? Optimal?

A* search

Idea combine the ad antages of nifo m

Idea: combine the advantages of uniform

cost and greedy approach

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A* search

A* sea ch ses an admissible he istic

A* search uses an admissible heuristic,

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Theorem: A* search is optimal

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Optimality of A* (standard proof)

G2 : suboptimal goal has been generated and is generated and is in the queue. n : an unexpanded node on a shortest path to an optimal l G

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goal G1.

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Optimality of A* (more useful proof)

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f-contours

How do the contours look like when h(n) =0?

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Properties of A*

Complete?

Complete? Time?

S ?

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Space? Optimal?

Proof of lemma: pathmax

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Admissible heuristics

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Relaxed Problem

How to determine an admissible heuristics? How to determine an admissible heuristics?

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Relaxed Problem

Example: Example:

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Next time

Iterative improvement

Iterative improvement Hill climbing Simulated annealing

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