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Solving Problems by Searching What is Problem-Solving?, State and Search Space - State graph, - Search ( State + Search), - Search Space . State and Search Space, - Eight Puzzle, - Robot Assembly, - Towers of Hanoi, - Rubik’s Cube, Searching Trees Methods, - Search Trees, - AND/OR Trees, - Game Trees. Search Methods, - Breadth-First Search, - Depth-First Search, Explore: Topics based Research Areas: @Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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Page 1: Solving Problems by Searching - portals.au.edu.pkportals.au.edu.pk/imc/Content/course/lecs/Lecture-3 (Solving Proble… · - Between state and goal, there are intermediate states

Solving Problems by Searching

What is Problem-Solving?,

State and Search Space

- State graph,

- Search ( State + Search),

- Search Space .

State and Search Space,

- Eight Puzzle,

- Robot Assembly,

- Towers of Hanoi,

- Rubik’s Cube,

Searching Trees Methods,

- Search Trees,

- AND/OR Trees,

- Game Trees.

Search Methods,

- Breadth-First Search,

- Depth-First Search,

Explore: Topics based

Research Areas:

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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1. What is Problem-Solving?

Problem-solving agents decide;

- what to do by finding sequences of

actions that lead to desirable states. (e.g.,

multi-ways to reach at destination).

- how the agent can formulate an

appropriate view of the problem.

The problem type that results from

the formulation process will

depend on the knowledge

available to the agent:

SimpleProblemSolvingAgent(percept)

state = UpdateState(state, percept)

if sequence is empty then

goal = FormulateGoal(state)

problem = FormulateProblem(state, goal)

sequence = Search(problem)

action = First(sequence)

sequence = Rest(sequence)

Return action

How an agent can decide what to do by systematically considering

the outcomes of various sequences of actions that it might take.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space

State:-

To define a problem

- visualize a solution, or select a search method having various states

as input.

States often took help to express the problem in graphical terms.

Drawing a graph of the problem showing ;

- various problem states and how they are interconnected.

The resulting graph is usually referred to a state graph or state space.

INITIAL

STATE

PROCEDURES GOAL (S)

CONTROL

STRATEGY

Figure: The relationship between the initial states, procedures, and goal(s) in the search process@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)

State:- State graph: Description

The problem is to just reach the destination.

- Between state and goal, there are intermediate states or nodes are often called subgoals.

The nodes in a state graph are interconnected by arcs or links.

- these arcs usually have arrows showing the direction from one state to the next.

The number on the arcs represent the distance between nodes.

Problem:-

It is difficult to represent a state graph in software form (as shown in

Figure).

- Goal is just reaching destination, no value for the least amount of time

covering the shortest distance.

- Also, it facilitates as non-parallel processing.

Due to above difficulty, state graph is converted into a search tree.@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)

State:- State graph

Figure shows a state graph for a simple problem;

- attempting to find the best path from one city, the source (S), to another city, the

destination or goal (G).

The state graph is a map showing the various possible intermediate

towns and cities to reach the “desired destination”.

Figure: A state graph showing alternate routines from the start state (S) to the goal (G).

F

S B

A

C

D

E

H

G3

4

9

7

5

6

86

2

4

5

Alphabets = Names of cities,

Numbers = distance (Kms)

Covering distance = 20 km/min

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space

Search ( State + Search):-

Search expands the importance of AI.

What choices are we searching through?

– Problem solving

Action combinations (move 1, then move 3, then move 2...).

– Natural language

Ways to map words to parts of speech.

– Computer vision

Ways to map features to object model.

– Machine learning

Possible concepts that fit examples seen so far.

– Motion planning

Sequence of moves to reach goal destination.

An intelligent agent is trying to find a set or sequence of actions to

achieve a goal.

This is a goal-based agent.@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space

Search ( State + Search):- (Example)

Formulate goal: Be in Bucharest.

Formulate problem: states are

cities, operators drive between

pairs of cities

Find solution: Find a sequence of

cities (e.g., Arad, Sibiu, Fagaras,

Bucharest) that leads from the

current state to a state meeting the

goal condition

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space

Search :- (Example)

• A search problem is defined by the

1. Initial state (e.g., Arad)

2. Operators (e.g., Arad -> Zerind, Arad -> Sibiu, etc.)

3. Goal test (e.g., at Bucharest)

4. Solution cost (e.g., path cost)

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search SpaceSearch Space Definitions:-

Major factors used in searching process are;

• State

– A description of a possible state of the world. (e.g., total states).

– Includes all features of the world that are pertinent to the problem.

• Initial state

– Description of all pertinent aspects of the state in which the agent starts the

search.

• Goal test

– Conditions the agent is trying to meet (e.g., have $1M, reaching destination).

• Goal state

– Any state which meets the goal condition. (e.g., total combinations to reach

goal)

– Thursday, have $1M, live in baseball match.

– Friday, have $1M, live in TV show.

• Action

– Function that maps (transitions) from one state to another. (e.g., all

combination in 1 map)@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)Search Space Definitions:-

• Major factors used in searching process are;

• Problem formulation

– Describe a general problem as a search problem. (e.g., reach with safety).

• Solution

– Sequence of actions that transitions the world from the initial state to a goal

state.

• Solution cost (additive)

– Sum of the cost of operators.

– Alternative: sum of distances, number of steps, etc.

• Search

– Process of looking for a solution.

– Search algorithm takes problem as input and returns solution.

– We are searching through a space of possible states.

• Execution

– Process of executing sequence of actions (solution).@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Class Participation)

Apply all “10 major factors” used in searching process by considering following

examples:-

1) Initial state (e.g., Arad) , Goal test (e.g., at Bucharest)

2) Initial state (e.g., Arad) , Goal test (e.g., at Bucharest)

• State

• Initial state

• Goal test

• Goal state

• Action

• Problem

formulation

• Solution

• Solution cost

• Search

• Execution@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

Page 12: Solving Problems by Searching - portals.au.edu.pkportals.au.edu.pk/imc/Content/course/lecs/Lecture-3 (Solving Proble… · - Between state and goal, there are intermediate states

2. State and Search Space (Cont…)Search Space Definitions:-

• Overall block flow diagram of basic searching process as;

• State

• Initial state

• Goal test

• Goal state

• Action

• Problem

formulation

• Solution

• Solution cost

• Search

• Execution

OPERATOR (S)(Solution, Solution cost)

CONTROL

STRATEGY(Search, Execution)

DATA BASE

- Initial States

- State

- Goal

- Goal State

- Action

- Problem formulation

Figure: The basic search process.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)Search Space :- Example Problems– Eight Puzzle

States: tile locations.

Initial state: one specific tile

configuration.

Operators: move blank tile left, right, up,

or down.

Goal: tiles are numbered from one to

eight around the square.

Path cost: cost of 1 per move (solution

cost same as number of most or path

length).

Eight puzzle applet

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)Search Space :- Example Problems– Robot Assembly

States: real-valued coordinates of

• robot joint angles.

• parts of the object to be assembled.

Initial state: machine parts movements.

Operators: rotation of joint angles.

Goal: complete assembly.

Path cost: time to complete assembly.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)Search Space :- Example Problems– Towers of Hanoi

States: combinations of poles and disks.

Initial state: disks holding and shifting to

another pole.

Operators: move disk x from pole 1 to 2 to

pole 3 subject to constraints.

• cannot move disk on top of smaller disk.

• cannot move disk if other disks on top.

Goal: disks from largest (at bottom) to

smallest on goal pole.

Path cost: 1 per move.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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2. State and Search Space (Cont…)Search Space :- Example Problems– Rubik’s Cube

States: list of colors for each cell on each

face.

Initial state: one specific cube configuration.

Operators: rotate row x or column y on face

z direction.

Goal: configuration has only one color on

each face.

Path cost: 1 per move.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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3. Searching Trees Methods (Cont…)

Searching Using Trees :- (1. Search Trees)

A search tree based on the state graph.

- The new tree diagram states the same problem but in a slightly different format.

- The network thus formed is more like a hierarchy.

Nodes with no children are successors are called leaves.

The interconnecting arcs are referred to as branches.

S

CBA

GH

E AG

H

E

G

D F

G

LEVEL 0

(ROOT)

LEVEL 1

LEVEL 2

LEVEL 3

LEVEL 4

ROOT

NODE

DEAD

END

DEAD

END

DEAD

END

GOAL

GOAL

GOAL

GOAL

4

6

3

5

82

6

49

6

5 7

75

17

20

19

12

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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Searching Using Trees :- (2. AND/OR Trees)

One path or another could lead to the goal.

- we call these OR nodes because one branch, OR another,

OR another could be the path of the goal.

Successor modes might represent problem states

which must all the

- achieved or traversed before the goal is reached.

- These are referred to as AND nodes.

3. Searching Trees Methods (Cont…)

P

A B C

Figure: OR node

P

X Y Z

Figure: AND node

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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Searching Using Trees :- (2. AND/OR Trees)

To define complex problems, both AND and OR nodes must be combined.

In this AND/OR tree, there are three possible solution;

- These are paths as; (1) ABDEIJ, (2) ACFGHK, (3) ACFGHL

3. Searching Trees Methods (Cont…)

A

B C

Figure: AND/OR tree.

D E F HG

K LI J

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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Searching Using Trees :- (2. AND/OR Trees)

3. Searching Trees Methods (Class Participation)

1

23

Figure: AND/OR tree.

4 5 76

a8 9 b

d e f g h i

c

j

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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Searching Using Trees :- (3. Game Trees)

Game trees are a special kind of AND/OR tree.

- The tree expands with alternative AND and OR nodes, as the play continuous

and a win, lose, or draw goal is achieved.

Example, tic-tac-toe game between players A and B.

- In 3x3 grid, there are 9 options for OR node at first step.

- Then, 2nd player has 8 possible choices with AND node.

3. Searching Trees Methods (Cont…)

Figure: Game tree.@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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Searching Using Trees :- (Search Problem Example)

Irrelevant nodes are accessing.

It makes heavy computational processing.

Uninformed search.

3. Searching Trees Methods (Cont…)

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods

“ Attempt the end, and never stand to doubt; Nothing’s

so hard but search will find it out.”

Searching method based on;

- guidance or direction.

Features by which to compare search strategies

– Completeness (always find solution)

– Cost of search (time and space)

– Cost of solution, optimal solution

– Make use of knowledge of the domain

• “uninformed search” vs. “informed search”.

Search strategies differ only in QueuingFunction.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods (Cont…)

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods (1. Breadth-First Search)

A breadth first search examines all of the nodes in a search tree;

- beginning with the root node.

Nodes in each level are completely examined before moving to the

next level.

Generate children of a state, QueueingFn adds the children to the

end of the open list.

Level-by-level search.

Order in which children are inserted on open list is arbitrary.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods (1. Breadth-First Search) (Cont…)

BFS Examples 1:

Figure: A breadth first search.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods (1. Breadth-First Search) (Cont…)

BFS Examples 2:In Figure, the search would end at node 7 as that is the goal.

BFS used downward /left-to-right procedure during nodes movements.

Figure: A breadth first search.

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods (1. Breadth-First Search) (Cont…)

Analysis:-

• Assume goal node at level d with constant branching factor b

• Time complexity (measured in #nodes generated)

1 (1st level ) + b (2nd level) + b2 (3rd level) + … + bd (goal level) + (bd+1 – b) = O(bd+1)

• Space complexity

At most majority of nodes at level d + majority of nodes at level d+1 = O(bd+1)

.

• Features

Simple to implement.

Complete.

Finds shortest solution (not necessarily least-cost unless all operators have equal cost).

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)

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4. Search Methods (2. Depth-First Search)

A depth first search begins at the root node;

- and works downward to successively deeper levels.

This process continuous until a solution is found;

- or backtracking is forced by reaching a dead end.

QueueingFn adds the children to the front of the open list.

BFS emulates FIFO queue.

DFS emulates LIFO stack

Net effect;

– Follow leftmost path to bottom, then backtrack.

– Expand deepest node first.

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4. Search Methods (2. Depth-First Search) (Cont…)

DFS Examples 1:

Figure: A depth-first search.

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4. Search Methods (2. Depth-First Search) (Cont…)

DFS Examples 2:In Figure, the search would end at node 13

as that is the goal.

Figure: A depth first search.

Figure Description

In Figure, when a dead end node

is discovered such as node 4;

- the search process backtracks so that

any additional branching alternative at the

next higher node level is attempted.

The search backs up to node 3.

- It has no alternate paths, so the search

backtracks to node 2.

Here, another path through node

5 is available.

The backtracking continues until

the goal is reached.

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4. Search Methods (2. Depth-First Search) (Cont…)

DFS Examples 2:

Advantages:-

a) The DFS guarantees a solution with complete survey.

b) Many different branches will have to be considered to maximum depth

before a solution is reached.

Problems:-

a) It can often lead into deeper and deeper subnetworks that are far from the

goal node.

b) A considerable amount of time and effort is wasted in searching all of

these possibilities.

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4. Search Methods (2. Depth-First Search) (Cont…)

Analysis:-

• Time complexity In the worst case, search entire space.

Goal may be at level d but tree may continue to level m, m>=d.

Particularly bad if tree is infinitely deep.

• Space complexity

Only need to save one set of children at each level.

1 + b + b + … + b (m levels total).

• Benefits May not always find solution.

Solution is not necessarily shortest or least cost.

If many solutions, may find one quickly (quickly moves to depth d).

Simple to implement.

Space often bigger constraint, so more usable than BFS for large problems.

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Avoiding Repeated States

• Do not return to parent or grandparent state

– In 8 puzzle, do not move up right after down

• Do not create solution paths with cycles

• Do not generate repeated states (need to store and check potentially

large number of states).

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6. Explore: Topics based Research Areas(1) Automatic Facial Age Estimation :-

Proposed architecture of Model Implemented results

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6. Explore: Topics based Research Areas(2) Mobile Robot Remote Path Planning and Motion Control in a Maze

Environment :-

Proposed architecture of Model Implemented results

BFS calculated the following path-vector 16: [4 3 2 7 8 13 12 17 16] and

DFS proposed a different path between the same fields: [4 5 10 9 14 19 18 13 12 17 16].

Level 1

Level 2

Level 3

Level 4

@Copyrights: Advanced Artificial Intelligence Organized by Dr. Ahmad Jalal (http://portals.au.edu.pk/imc/)