Solving Problems by

Searching

CS 486/686: Introduction to Artificial Intelligence

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Announcements

• Enrollment: Wait list, Grads

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• My Office: DC 2306B (Accessed through AI Lab)

• Slide versions2

Previously on CS 486

• Fully Observable vs Partially

Observable

• Deterministic vs Stochastic

• Episodic vs Dynamic

• Discrete vs Continuous

• Single agent vs Multi agent

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Introduction

• Search was one of the first topics

studied in AI

- Newell and Simon (1961) General Problem

Solver

• Central component to many AI systems

- Automated reasoning, theorem proving,

robot navigation, scheduling, game

playing,...

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Search Problems

• Search problem consists of

• a state space

• a successor function (actions, cost)

• a start state

• Goal test

(N, 1.0)

(E, 1.0)

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Search Problems

• Path: a sequence of states and operators

• May be associated with a cost c

• Solution: a sequence of actions from the start

state to a goal state

• Optimal solution: a path with minimum cost

…

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Example: Traveling in Romania

Start

End

•States:

•Initial State:

•Successor Function:

•Goal test:

•Solution:

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Examples of Search Problems

•States:

•Initial State:

•Successor Function:

•Goal test:

•Solution:

•States:

•Initial State:

•Successor Function:

•Goal test:

•Solution:8

Examples of Search Problems

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Our Definition Excludes...

Chance

Continuous states

All of the above

Partial

Observability

Adversaries

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What is is a state space?

The world state includes every last detail of the environment

A search state keeps only the details needed for planning (abstraction)

• Problem: Pathing

• States: (x,y) location

• Actions: NSEW

• Successor: update location

only

• Goal test: is (x,y)=END

• Problem: Eat-All-Dots

• States: {(x,y), dot booleans}

• Actions: NSEW

• Successor: update location and

possibly a dot boolean

• Goal test: dots all false

Adapted from UC Berkeley’s CS188 Course

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Representing Search

• State space graph

- Vertices correspond to states

(one vertex for each state)

- Edges correspond to

successors

- Goal test is a set of goal nodes

• We search for a solution by

building a search tree and

traversing it to find a goal

state

S

G

d

b

pq

c

e

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a

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Search Tree

• A search tree:

• Start state is the root of

the tree

• Children are successors

• A plan is a path in the tree.

A solution is a path from

the root to a goal node.

• For most problems we do

not actually generate the

entire tree

S

B

A

G

S

A B

GB G

G13

Quiz

• Given this state graph, how large is the

search tree?

S

B

A

G

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Expanding Nodes

• Expanding a node

- Applying all legal operators to the state

contained in the node

- Generating nodes for all corresponding

successor states

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Example: Traveling in Romania

Start

End

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Generic Search Algorithm

• Initialize with initial state of the problem

• Repeat

- If no candidate nodes can be expanded return failure

- Choose leaf node for expansion, according to search

strategy

- If node contains goal state, return solution

- Otherwise, expand the node. Add resulting nodes to the

tree

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Generic Search Algorithm

Implementation Details

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• Need to keep track of nodes to be expanded (frontier)

• Implement using a queue:

0. Insert node for initial state

Implementation Details

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• Need to keep track of nodes to be expanded (frontier)

• Implement using a queue:

0. Insert node for initial state

• Repeat

1. If queue is empty, return failure

2. Dequeue a node

Implementation Details

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• Need to keep track of nodes to be expanded (frontier)

• Implement using a queue:

0. Insert node for initial state

• Repeat

1. If queue is empty, return failure

2. Dequeue a node

3. If node contains goal state, return solution

4. Expand node

Implementation Details

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• Need to keep track of nodes to be expanded (frontier)

• Implement using a queue:

0. Insert node for initial state

• Repeat

1. If queue is empty, return failure

2. Dequeue a node

3. If node contains goal state, return solution

4. Expand node

• Search algorithms differ in their queuing function!

Search Strategies

S

G

d

b

p

q

c

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Adapted from UC Berkeley’s CS188 Course

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Search Strategies

S

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dp

a

c

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q c G

a

qe

p

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q c G

a 24

Depth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand deepest node first

Implementation: LIFO stack

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S

a

b

d p

a

c

e

p

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f

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q c G

a

qe

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q c G

a

Depth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand deepest node first

Implementation: LIFO stack

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S

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b

d p

a

c

e

p

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q c G

a

qe

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q c G

a

Key Properties

•Completeness: Is the alg. guaranteed to find a

solution if the solution exists?

•Optimality: Does the alg. find the optimal solution?

•Time complexity

•Space complexity (size of the fringe) …

b1 node

b nodes

b2 nodes

bm nodes

m tiers

b: branching factor

m: maximum depth

d: depth of shallowest goal node

Number of nodes in tree? 1+b+b2+…+bm=O(bm)27

DFS Properties

•Complete?

•Optimal?

•Time complexity

•Space complexity

…

b

1 node

b nodes

b2 nodes

bm nodes

m tiers

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Breadth-First Search

S

G

d

b

p q

c

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h

a

f

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Strategy: Expand shallowest node first

Implementation: FIFO queue

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S

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dp

a

c

e

p

h

f

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q

q c G

a

qe

p

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q

q c G

a

Breadth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand shallowest node first

Implementation: FIFO queue

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S

a

b

dp

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

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q

q c G

a

BFS Properties

•Complete?

•Optimal?

•Time complexity

•Space complexity

…

b

1 node

b nodes

b2 nodes

bm nodes

m tiers

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depth d

Quiz: DFS vs BFS

…b

…b

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Iterative Deepening Search

• Can we combine search methods to take advantage

of DFS space complexity and BFS

completeness/shallow solution advantage?

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IDS Properties•Complete?

•Optimal?

•Time complexity

•Space complexity

…b

1 node

b nodes

b2 nodes

bm nodes

m tiers

Wasteful? Most nodes found in

lowest level of search so not too

bad

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Cost-Sensitive Search

START

GOAL

d

b

pq

c

e

h

a

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r

2

9 2

81

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2

3

2

4

4

15

1

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2

Recall that BFS was only optimal under some conditions

(i.e. we only cared about number of actions taken). What

can we do if actions have different costs? 35

Uniform Cost Search

Strategy: Expand node with lowest path cost first

Implementation: Priority queue

START

GOAL

d

b

pq

c

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h

a

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2

9 2

81

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2

3

2

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15

1

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2

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UCS Properties

•Complete?

•Optimal?

•Time complexity

•Space complexity

…

b

C*/𝜀 tiers

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Summary

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Criteria DFS BFS IDS UCS

Optimal No Yes Yes Yes

Complete No Yes Yes Yes

Time O(bm) O(bd) O(bd) O(b+1+floor(C*/ε))

Space O(bm) O(bd) O(bd) O(b+1+floor(C*/ε))

• Assumes no knowledge about the problem

- This makes them general but expensive

Figure 3.21 (Russell & Norvig)

Summary

• These algorithms are basically the

same except for the order in which they

expand nodes

• Basically all priority queues with different

ways to determining priorities

• How successful the search is depends

heavily on your model!

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Questions?

• Next class: Informed search

“Can we make use of additional information?”

Why take a bus to London if we are going to Ottawa?

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