Search Problems

1
Search Problems

• Russell and Norvig Chapter 3, Sections 3.1
  3.3

2
Problem-Solving Agent
3
Problem-Solving Agent
4
State Space and Successor Function

• Actions
• Initial state
• Goal test

5
Initial State

• Actions
• Initial state
• Goal test

6
Goal Test
state space

• Actions
• Initial state
• Goal test

7
Example 8-puzzle
8
Example 8-puzzle
9
Example 8-puzzle
Size of the state space 9!/2
181,440 15-puzzle ? .65 x 1012 24-puzzle ?
.5 x 1025
10
Search Problem

• State space
• Initial state
• Successor function
• Goal test
• Path cost

11
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

12
Search Problem

• State space
• Initial state
• usually the current state
• sometimes one or several hypothetical states
  (what if )
• Successor function
• Goal test
• Path cost

13
Search Problem

• State space
• Initial state
• Successor function
• state ? subset of states
• an abstract representation of the possible
  actions
• Goal test
• Path cost

14
Search Problem

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

15
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

16
Search of State Space
17
Search of State Space
18
Search State Space
19
Search of State Space
20
Search of State Space
21
Search of State Space
? search tree
22
Simple Agent Algorithm

• Problem-Solving-Agent
• initial-state ? sense/read state
• goal ? select/read goal
• successor ? select/read action models
• problem ? (initial-state, goal, successor)
• solution ? search(problem)
• perform(solution)

23
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
24
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
25
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
• 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
• board

? 2,067 states
26
?? n-queen ??

• Neural, Genetic ?? Heuristic ???? ? ??
• ??? ???? ?? ???
• ?? n? ??? ?? ?? ???? ??? Heuristics?? ?? ?? ??
• ??? n? ?? ?? ? ???? ?? ???? ????? ??? ????? ???
  ??
• ??? ?? ?? ??? ?????? ????? ??? ???? ? ?? ??
• ??? ??? ??? ??? ?? ??? ??? ?? ???? ??? ??,
  feasible solution? ???? ???? ??? ????? ??? ? ??

27
Example Robot navigation
What is the state space?
28
Example Robot navigation
29
Example Robot navigation
30
Example Robot navigation
31
Example Robot navigation
Cost of one step ???
32
Example Robot navigation
33
Example Robot navigation
34
Example Robot navigation
35
Example Robot navigation
36
Example Assembly Planning
Initial state
Goal state
37
Example Assembly Planning
38
Example Assembly Planning
39
Assumptions in Basic Search

• The environment is static
• The environment is discretizable
• The environment is observable
• The actions are deterministic
• ? open-loop solution

40
Search Problem Formulation

• Real-world environment ? Abstraction

41
Search Problem Formulation

• Real-world environment ? Abstraction
• Validity
• Can the solution be executed?

42
Search Problem Formulation

• Real-world environment ? Abstraction
• Validity
• Can the solution be executed?
• Does the state space contain the solution?

43
(No Transcript)
44
(No Transcript)
45
(No Transcript)
46
(No Transcript)
47
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?

48
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

49
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

50
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

51
Important Parameters

• Number of states in state space

8-puzzle ? 181,44015-puzzle ? .65 x 1012
24-puzzle ? .5 x 1025
8-queens ? 2,057 100-queens ? 1052
Stating a problem as a search problem is not
always a good idea!
52
Important Parameters

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

53
Important Parameters

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

54
Applications

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

55
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. Figure 2.6 Examples of task environments and their characteristics. Figure 2.6 Examples of task environments and their characteristics. Figure 2.6 Examples of task environments and their characteristics. Figure 2.6 Examples of task environments and their characteristics. Figure 2.6 Examples of task environments and their characteristics. Figure 2.6 Examples of task environments and their characteristics.
56
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

57
Future Classes

• Search strategies
• Blind strategies
• Heuristic strategies
• Extensions
• Uncertainty in state sensing
• Uncertainty action model
• On-line problem solving