# Search problems by pengtt

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```									        Search Problems

Russell and Norvig:
Chapter 3, Sections 3.1 – 3.3
CS121 – Winter 2003
Problem-Solving Agent
sensors

?
environment
agent

actuators

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Problem-Solving Agent
sensors

?
environment
agent

actuators
• Actions
• Initial state
• Goal test
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State Space and Successor Function

state space

successor function

• Actions
• Initial state
• Goal test
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Initial State
state space

successor function

• Actions
• Initial state
• Goal test
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Goal Test
state space

successor function

• Actions
• Initial state
• Goal test
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Example: 8-puzzle

8    2                             1   2   3

3    4    7                        4   5   6

5    1    6                        7   8

Initial state                     Goal state

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Example: 8-puzzle
8      2       7

3      4

8   2          5      1       6

3   4   7
5   1   6      8              2       8   2

3      4       7   3   4   7

5      1       6   5   1   6

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Example: 8-puzzle

Size of the state space = 9!/2 = 181,440

15-puzzle  .65 x 1012
0.18 sec
6 days
24-puzzle  .5 x 1025
12 billion years

10 millions states/sec

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Search Problem

State space
Initial state
Successor function
Goal test
Path cost

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Search Problem
State space
   each state is an abstract representation of
the environment
   the state space is discrete
Initial state
Successor function
Goal test
Path cost

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Search Problem
State space
Initial state:
   usually the current state
   sometimes one or several hypothetical
states (“what if …”)
Successor function
Goal test
Path cost

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Search Problem
State space
Initial state
Successor function:
   [state  subset of states]
   an abstract representation of the possible
actions
Goal test
Path cost

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Search Problem

State space
Initial state
Successor function
Goal test:
   usually a condition
   sometimes the description of a state
Path cost

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Search Problem
State space
Initial state
Successor function
Goal test
Path cost:
   [path  positive number]
   usually, path cost = sum of step costs
   e.g., number of moves of the empty tile

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Search of State Space

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Search of State Space

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Search State Space

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Search of State Space

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Search of State Space

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Search of State Space

 search tree
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Simple Agent Algorithm
Problem-Solving-Agent
3. successor  select/read action models
4. problem  (initial-state, goal, successor)
5. solution  search(problem)
6. perform(solution)

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Example: 8-queens
Place 8 queens in a chessboard so that no two queens
are in the same row, column, or diagonal.

A solution                     Not a solution

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Example: 8-queens
Formulation #1:
• States: any arrangement of
0 to 8 queens on the board
• Initial state: 0 queens on the
board
queen in any square
• Goal test: 8 queens on the
board, none attacked

 648 states with 8 queens

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Example: 8-queens
Formulation #2:
• States: any arrangement of
k = 0 to 8 queens in the k
leftmost columns with none
attacked
• Initial state: 0 queens on the
board
queen to any square in the
leftmost empty column such
that it is not attacked
by any other queen
 2,067 states   • Goal test: 8 queens on the
board
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실제 n-queen 문제
Neural, Genetic 또는 Heuristic 방법으로 잘
해결
   최악의 경우에는 처리 불가능
   실제 n이 커지면 답이 매우 많으므로 간단한
Heuristics로도 답을 쉽게 찾음
 따라서 n이 커도 답을 잘 찾는다고 해서 인공지능
접근방법이 문제를 해결한다는 증거는 아님
 그러나 많은 실제 문제는 알고리즘에서 이야기하는
최악의 경우로는 잘 가지 않음
 더구나 대부분 우리가 원하는 답은 최적이 아니라 실제
활용해서 도움이 되는, feasible solution을 원하므로
인공지능 기법이 효과적으로 이용될 수 있음

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What is the state space?
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Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2
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Cost of one step = ???
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Cost of one step: length of segment
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Example: Assembly Planning

Initial state

Complex function:
it must find if a collision-free                      Goal state
merging motion exists

Successor function:
• Merge two subassemblies

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Example: Assembly Planning

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Example: Assembly Planning

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Assumptions in Basic Search

The   environment is static
The   environment is discretizable
The   environment is observable
The   actions are deterministic

 open-loop solution

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Search Problem Formulation

Real-world environment  Abstraction

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Search Problem Formulation

Real-world environment  Abstraction
   Validity:
 Can the solution be executed?

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Search Problem Formulation

Real-world environment  Abstraction
   Validity:
 Can the solution be executed?
 Does the state space contain the solution?

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Search Problem Formulation

Real-world environment  Abstraction
   Validity:
 Can the solution be executed?
 Does the state space contain the solution?
   Usefulness
 Is the abstract problem easier than the real-
world problem?

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Search Problem Formulation
Real-world environment  Abstraction
   Validity:
 Can the solution be executed?
 Does the state space contain the solution?
   Usefulness
 Is the abstract problem easier than the real-
world problem?
Without abstraction an agent would be
swamped by the real world

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Search Problem Variants

One or several initial states
One or several goal states
The solution is the path or a goal node
   In the 8-puzzle problem, it is the path to a
goal node
   In the 8-queen problem, it is a goal node

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Problem Variants

One or several initial states
One or several goal states
The solution is the path or a goal node
Any, or the best, or all solutions

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Important Parameters

Number of states in state space

8-puzzle  181,440              8-queens  2,057
15-puzzle  .65 x 1012          100-queens  1052
24-puzzle  .5 x 1025

There exist techniques to solve
N-queens problems efficiently!

Stating a problem as a search problem
is not always a good idea!
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Important Parameters

Number of states in state space
Size of memory needed to store a state

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Important Parameters

Number of states in state space
Size of memory needed to store a state
Running time of the successor function

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Applications

Route finding: airline travel,
telephone/computer networks
Pipe routing, VLSI routing
Pharmaceutical drug design
Robot motion planning
Video games

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Task Environment             Observable    Deterministic    Episodic      Static    Discrete    Agents
Crossword puzzle                Fully      Deterministic Sequential       Static    Discrete     Single
Chess with a clock              Fully         Strategic    Sequential     Semi      Discrete     Multi
Poker                          Partially      Strategic    Sequential     Static    Discrete     Multi
Backgammon                      Fully        Stochastic    Sequential     Static    Discrete     Multi
Taxi driving                   Partially     Stochastic    Sequential   Dynamic    Continuous    Multi
Medical diagnosis              Partially     Stochastic    Sequential   Dynamic    Continuous    Single
Image-analysis                  Fully      Deterministic    Episodic      Semi     Continuous    Single
Part-picking robot             Partially     Stochastic     Episodic    Dynamic    Continuous    Single
Refinery controller            Partially     Stochastic    Sequential   Dynamic    Continuous    Single
Interactive English tutor      Partially     Stochastic    Sequential   Dynamic     Discrete     Multi
Figure 2.6      Examples of task environments and their characteristics.

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Summary

Problem-solving agent
State space, successor function, search
Examples: 8-puzzle, 8-queens, route
planning
Assumptions of basic search
Important parameters

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Future Classes
Search strategies
   Blind strategies
   Heuristic strategies

Extensions
   Uncertainty in state sensing
   Uncertainty action model
   On-line problem solving

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