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




                   Search Problems                 2
Problem-Solving Agent
         sensors


     ?
                                     environment
    agent

            actuators
                             • Actions
                             • Initial state
                             • Goal test
                   Search Problems                 3
State Space and Successor Function

           state space

             successor function




                          • Actions
                          • Initial state
                          • Goal test
                Search Problems             4
Initial State
          state space

            successor function




                          • Actions
                          • Initial state
                          • Goal test
                Search Problems             5
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
                Search Problems   21
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

           Search Problems                   24
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
 2,067 states   • Goal test: 8 queens on the
                   board
                  Search Problems                   25
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

                 Search Problems          26
Example: Robot navigation




   What is the state space?
                 Search Problems   27
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 = ???
                   Search Problems   31
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




           Search Problems   35
  Example: Assembly Planning

                                      Initial state


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


Successor function:
• Merge two subassemblies

                             Search Problems                 36
Example: Assembly Planning




           Search Problems   37
Example: Assembly Planning




           Search Problems   38
Assumptions in Basic Search

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

 open-loop solution


                  Search Problems      39
Search Problem Formulation

  Real-world environment  Abstraction




               Search Problems       40
Search Problem Formulation

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




                     Search Problems     41
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?




                      Search Problems                    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?
   Without abstraction an agent would be
  swamped by the real world

                      Search Problems                    48
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



                    Search Problems             49
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




                Search Problems        50
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!
                     Search Problems                   51
Important Parameters

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




               Search Problems        52
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


                 Search Problems    54
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.




                                             Search Problems                                    55
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

               Search Problems         56
Future Classes
  Search strategies
     Blind strategies
     Heuristic strategies


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



                       Search Problems   57

								
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