Chapt 3 Problem Solving

8/7/2019 Chapt 3 Problem Solving

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Introduction to

Artificial Intelligence

Problem Solving and Strategies

Solving problems by searchingSearch Strategies

Blind searchInformed search

Game play searchP roblem reduction paradigm

Search graph AND/OR graph

No. 3



Introduction to

Artificial IntelligenceProblem as a State Space Search

To build a system to solve a particular problem, we need to:D efine the problem : must include precise specifications ~initial solution & final solution.A nalyze the problem : select the most important features thatcan have an immense impact.Isolate and represent : convert these important features intoknowledge representation.Identify problem solving technique(s) : choose the besttechnique and apply it to particular problem.



Introduction to

Artificial IntelligenceThe Quest

Typical questions that need to be answered:Is the problem solver guaranteed to find a solution?W ill the system always terminate or caught in a infinite loop?If the solution is found, it is optimal ?W hat is the complexity of searching process?H ow the system be able to reduce searching complexity?H ow it can effectively utilize the representation paradigm?



Introduction to

Artificial IntelligenceImportant Terms

Search space possible conditions and solutions.Initial state state where the searching processstarted.Goal state the ultimate aim of searching process.Problem space ³ what to solve´Searching strategy strategy for controlling thesearch.Search tree tree representation of search space,showing possible solutions from initial state.



Introduction to

Artificial IntelligenceExample: Travelling Salesperson Problem

B

C

D

E

75

50

100

100125

125125

75

50

A

Suppose a salesperson has five cities to visit and theymust return home. Goal find the shortest path to travel.



Introduction to


A

A A A

C

C

C

C

D

D

D

D

The possible search space for TS P .

Example: Traveling Salesperson Problem (cont)



Introduction to


The 8 P uzzle Games contains eight tiles that can bemoved around in nine spaces.

A im : to arrange the tiles sequence using defined legalmovement (Eg: move the blank board).

1 2 3

4 6

7 8

5

Example: Eight Puzzle Games





Introduction to

Artificial IntelligenceSearching for Solution

Search through state space (explicitly using searching tree).

Node expansion :- generating new node related to previous nodes.

Concepts:

State :- conditions in which the node corresponds.P arent node :- the superior node

P ath cost :- the cost, from initial to goal state.

Depth:- number of steps along the path from initial state



Introduction to

Artificial IntelligenceNode expansion



Introduction to

Artificial IntelligenceNode expansion (initial)



Introduction to

Artificial IntelligenceNode expansion (expanding A rad)



Introduction to

Artificial IntelligenceNode expansion (expanding Sibiu)



Introduction to

Artificial IntelligenceMeasuring Searching Performance

The output from problem-solving (searching) algorithm is either FAILURE or SOLUTION.

Four ways:

Completeness : is guaranteed to find a solution?Optimality : does it find optimal solution ?

Time complexity : how long?

Space complexity : how much memory?

Complexity : branching factor ( b ), depth ( d ), and max.depth ( m )



Introduction to

Artificial IntelligenceSearching Strategies

Heuristic search searchprocess takes place bytraversing search space withapplied rules (information).

Techniques : Greedy Best FirstSearch, A* A lgorithm

There is no guarantee thatsolution is found.

Blind search traversing thesearch space until the goalnodes is found (might be doingexhaustive search).

T echniques : Breadth FirstUniform Cost , D epth first,Interactive D eepeningsearch .

Guarantees solution.



Introduction to

Artificial IntelligenceBlind Search : Breadth First Search (BFS)

Strategy ~ search all the nodes expanded at given depthbefore any node at next level.

Concept : First In First Out (FIFO) queue.Complete ?: Yes with finite b (branch).

Space : similar to complexity ± keep nodes in everymemory

Optimal ? = Yes (if cost =1)



Introduction to

Artificial IntelligenceBlind Search : Breadth First Search

1 2

43



Introduction to

Artificial IntelligenceBlind Search : Uniform-Cost Search (UCS)

Strategy ~ search all the nodes expanded with the lowestpath cost.

If all path cost equals identical to BFS.Complete ?: Yes (if finite depth and total cost is small)

Optimality ? : Yes (proportion to cost function)



Introduction to

Artificial IntelligenceBlind Search : Depth First Search (DFS)

Strategy ~ search all the nodes expanded in deepest path.

Last In First Out concept.

Complete ?: No

Optimality ? : No



Introduction to

Artificial IntelligenceBlind Search : D epth First Search ( D FS)

1 2 3

4 5

««.

N+1



Introduction to

Artificial IntelligenceHeuristic Search :

Important aspect: formationof heuristic function ( h(n)).

Heuristic functionadditional knowledge to

guide searching strategy(short cut).

Distance: heuristic functioncan be straight linedistance (SLD)

A*

B C*

D

E

h(n)=0

h(n)= 34

h(n)=2 4

h(n)=67

h(n)=12h(n)=9



Introduction to

Artificial IntelligenceHeuristic Search : Heuristic Function



Introduction to


Heuristic Search :Greedy-Best Search

Tries to expand the node that is closest to the goal.

Evaluates using only heuristic function : f (n) = h(n)

P ossibly lead to the solution very fast.

P roblem ? ~ can end up in sub-optimal solutions(doesn¶t take notice of the distance it travels).

Complexity and time:Complete & optimal ? : No (stuck in infinite loop)



Introduction to


Heuristic Search :Greedy-Best Search

1

2



Introduction to


Searching comparison (D

FS, BFS, GBFS)

A :5

CB

FED

H

G

J

:4:5

:4 :2 :6 :4

:0 :3

D iscussion:



Introduction to

Artificial IntelligenceH euristic Search :Greedy-Best Search

3



Introduction to

Artificial IntelligenceH euristic Search : A* Algorithm

W idely known algorithm ± (pronounced as ³ A star´search).

Evaluates nodes by combining g(n)³cost to reachthe node´ and h(n) ³ cost to get to the goal´

f (n) = g(n) + h(n), f (n) estimated cost of thecheapest solution.

Complete and optimal ~ since evaluates all paths.Time ? ~ a bit time consuming

Space ? ~ lot of it!



Introduction to


Heuristic Search :A*

A

lgorithm

S

GE

DA

G¶ H

:10

:8 :9

:0:4 :0 :3

2 3

25 13

Path cost for S- D -G

f (S) = g(S) + h(S)

= 0 + 10 10 f (D) = (0+3) + 9 12

f (G) = (0+3+3) + 0 6

Total path cost = f (S)+ f (D)+ f (G) 28

Path cost for S- A -G¶

f (S) = 0 + 10 10 f (A) = (0+2) + 8 10 f (G¶) = (0+2+2) + 0 4

Total path cost = f (S)+ f (A)+ f (G¶) 24

* P ath S-A-G is chosen= Lowest cost



Introduction to

Artificial IntelligenceHeuristic Search : A* A lgorithm



Introduction to




Introduction to




Introduction to




Introduction to




Introduction to


Issues in Heuristic SearchSearching using heuristic function does not solely ondirected solution but the best algorithm to findshortest path towards goal.

Admissible attempt to find possible shortest pathto a goal whenever it exists.

Informedness question in what sense theheuristic function is better than another.

Monotonicity question if the best state isdiscovered by heuristic search, is there anyguarantee that the same state won¶t be found later at lowest searching cost?



Introduction to


Games andA

I ?

Game playing is a good problem for AI research: All the information is available (human and computer have equal information.Non-trivial (requires decision making + need todisplay ³ intelligence´).Can be played in a controlled environmentCan compare humans and computers ability directly(percentage of wins/losses).For fun ?





Introduction to

Artificial IntelligenceGames ..

Backgammon Checkers Reversi



Introduction to

Artificial IntelligenceSo what ?

³ Deep Blue´ (IBM)parallel processor, 32 nodeseach chip can search 200 million configurations/secondmemorizes starts, end-gamespower based on speed and memory: no common sense

Kasparov vs. Deep Blue, May 19976 game full-regulation chess match (sponsored by ACM)

Kasparov lost the match (2.5 to 3.5)a historic achievement for computer chess: the first time acomputer is the best chess-player on the planet



Introduction to

Artificial IntelligenceSo what ?

Checkers/ D raughtscurrent world champion is Chinook,can beat any human + uses alpha-beta search

Othello (Reversi)computers can easily beat the world experts

Backgammon

system which learns is ranked in the top 3 in the worlduses neural networks to learn from playing manymany games against itself



Introduction to




Introduction to


8 8 P P u u z z

z z l l ee

Initial State

Goal State

Solution:

a p c p f p j p l p m

Search Space



Introduction to

Artificial Intelligence8 8- -puzzle : puzzle : RulesRules

Start

Goal



Introduction to


H euristic I:Number of tiles out of place

Current State Goal State

8 8- -puzzle : puzzle : HeuristicsHeuristics

Number of tiles out place: 5 @ h( n) = 5



Introduction to


Depth-First SearchBreath-First SearchBest-First Search

A* Algorithm

8 8- -puzzle : puzzle : StrategiesStrategies



Introduction to


H euristic II:Sum of distances out of place

Current State Goal State


Tile D istances out of place

2 1

8 2

1 1

Sum of distances out of place: 4@

h(n)=

4



Introduction to


Goal State

H euristic




Introduction to


Exercise:

2 8 3

7 51 6 4

1 2 3

8 47 6 5Goal StateInitial State

H euristic: sum of distances out of place

Consider the following 8-p uzzle problem:

Construct a state space for the above problem usingGreedy Best-First strategy



Introduction to


References

Cawsey, A. (1998). The Essence of ArtificialIntelligence, P rentice H all.Russell, S. and Norvig, P . (2003). Artificial

Intelligence: A Modern Approach, P rentice- H all2nd Edition.

Documents

Chapt 3 Problem Solving