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
3
4
5
1
1
2
3
7
4
5
6
6
7
8
Initial state
Goal state
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Example: 8-puzzle
8
2
3
4
7
5
1
6
8
2
3
4
5
1
8
7
6
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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State space
Initial state
Successor function
Goal test
Path cost
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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
1. initial-state  sense/read state
2. goal  select/read goal
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
• Successor function: add a
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
 2,067 states
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
• Successor function: add a
queen to any square in the
leftmost empty column such
that it is not attacked
by any other queen
• Goal test: 8 queens on the
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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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Example: Robot navigation
What is the state space?
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Example: Robot navigation
Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2
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Example: Robot navigation
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Example: Robot navigation
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Example: Robot navigation
Cost of one step = ???
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Example: Robot navigation
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Example: Robot navigation
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Example: Robot navigation
Cost of one step: length of segment
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Example: Robot navigation
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Example: Assembly Planning
Initial state
Complex function:
it must find if a collision-free
merging motion exists
Goal state
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
The
The
The
environment is static
environment is discretizable
environment is observable
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
15-puzzle  .65 x 1012
24-puzzle  .5 x 1025
8-queens  2,057
100-queens  1052
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
Static
Discrete
Agents
Crossword puzzle
Fully
Static
Discrete
Single
Chess with a clock
Fully
Strategic
Sequential
Semi
Discrete
Multi
Partially
Strategic
Sequential
Static
Discrete
Multi
Fully
Stochastic
Sequential
Static
Discrete
Multi
Taxi driving
Partially
Stochastic
Sequential
Dynamic
Continuous
Multi
Medical diagnosis
Partially
Stochastic
Sequential
Dynamic
Continuous
Single
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
Poker
Backgammon
Image-analysis
Figure 2.6
Deterministic
Episodic
Deterministic Sequential
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
finding, robot navigation, assembly
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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