Solving Problems By Searching and Constraint Satisfaction Problem

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Solving Problems By Searching andConstraint
Satisfaction Problem
CS570 Artificial Intelligence

• 2001. 4. 18.
• 20015087 ???
• 20013460 ???

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Overview

• Solving Problem by Searching
• Problem solving agents
• Problem types
• Problem formulation
• Example problems
• Searching Strategies(Basic search algorithms)
• Constraint Satisfaction Problem
• Introduction
• Solving Techniques
• Applications

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Problem-Solving Agents

• Problem-Solving Agents
• one kind of goal-based agent
• finding sequences of actions that lead to
  desirable states.
• Steps
• Goal Formulation
• limiting the objectives
• Problem Formulation
• deciding what actions and states to consider
• Search
• looking for the possible action sequence
• Execution

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Formulating Problem - Example

• The eight possible states of the simplified
  vacuum world

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Properties of environment

• Accessible vs. Inaccessible
• Deterministic vs. Nonderterministic
• Episodic vs. Nonepisodic
• Static vs. Dynamic
• Discrete vs. Continuaous

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Knowledge and Problem Types (1)

• Single-state problem
• accessible - the agents sensor knows
• which state it is in.
• deterministic - the agent knows
• what each of its actions does
• Action sequence can be completely planned.
• example Right, Suck

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Knowledge and Problem Types (2)

• Multiple-state problem
• inaccessible
• limited access to the world state
• deterministic
• The agent must reason about sets of states that
  it mignt get to.
• Example
• Right, Suck, Left, Suck

reach a Goal
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Knowledge and Problem Types (3)

• Contingency problem
• inaccessible
• non-deterministic
• sensing during the execution phase.
• most of the real, physical world problems
• Keep your eyes open while walking!
• The agent must calculate a whole tree of actions,
  rather than a single action sequence.

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Knowledge and Problem Types (4)

• Exploration problem
• unknown state space (no map, no sensor)
• The agent must experiment, gradually discovering
  what its actions do and what sorts of states
  exist.

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Well-defined Problems and Solutions
single-state problem

• Problem
• a collection of information that the agent will
  use to decide what to do
• the basic elements of a problem definition
• initial state
• operator (or successor function S)
• goal test
• path cost function(g)

path
state space
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Problem-Solving Agents - Example

• A simplified road map of Romania

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Problem-Solving Agents Example

• Situation
• On holiday in Romania currently in Arad.
• Flight leaves tomorrow from Bucharest.
• Formulate goal
• be in Bucharest
• Formulate problem
• initial state be in Arad
• state various cities
• operators driving between cities
• Find solution
• sequence of cities, e.g., Arad, Sibiu, Fagaras,
  Bucharest

Abstraction
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Measuring Problem-Solving Performance

• Effectiveness of a search
• Does it find a solution at all?
• Is it a good solution (one with low path cost)?
• What is the search cost associated with the time
  and memory required to find a solution?
• total cost path cost search cost

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Example Problems

• Toy Problems
• concise and exact description
• abstract version of real problem
• Real-World Problem
• no single agreed-upon description

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The 8-puzzle

• problem formulation
• State the location of each of the eight tiles
  in one of the nine squares
• Operators blank moves left, right, up, or down
• Goal test state matches the right figure
• Path cost each step costs 1, that is the length
  of the path

Start state
Goal state
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The 8-queens problem(1)

• problem formulation
• Goal test 8 queens on board, none attacked
• Path cost zero
• States any arrangement of 0 to 8 queens on
  board
• Operators add a queen to any square
• States arrangements of 0 to 8 queens with none
  attacked
• Operators place a queen in the left-most empty
  column such that is not attacked by any other
  queen

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The 8-queens problem(2)

• States arrangements of 8queens, one in each
  column
• Operators move any attacked queen to another
  square in the same column

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Cryptarithmetic

• problem formulation
• State a cryptarithmetic puzzle with some
  letters replaced by digits
• Operators replace all occurrences of a letter
  with a digit not already appearing in the puzzle
• Goal test puzzle contains only digits, and
  represents a correct sum
• Path cost zero

FORTY Solution 29786 F2, O9, R7, etc.
TEN 850 TEN 850 ------------ --
------- SIXTY 31486
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The vaccum world

• problem formulation ( a single state problem)
• State one of the eight states shown in Figure
• Operators move left, move right, suck
• Goal test no dirt left in any square
• Path cost each action costs 1

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Missionaries and cannibals

• problem formulation
• State the number of missionaries, cannibals and
  boats on the bank of the river from which they
  started(3,3,1)
• Operators either 1 missionary, 1 cannibal, 2
  missionaries, 2 cannibals, or one of each across
  in the boat.
• Goal test 3 missionaries and 3 cannibals in the
  other side of the river (0,0,0)
• Path cost the number of crossings

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Example Real-World Problems

• Route finding
• Touring and travelling salesperson problems
• VLSI layout
• Robot navigation
• Assembly sequencing

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Searching for Solutions

• Partial search tree for route finding from Arad
  to Bucharest.

(a) The initial state (search node)
(b) After expanding Arad
(c) After expanding Sibiu
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Search Strategies

• Criteria
• Completeness
• Time complexity
• Space complexity
• Optimality
• Classification
• Uninformed search ( blind search)
• have no information about the number of steps or
  the path cost from the current state to the goal
• Informed search ( heuristic search)
• have some information
• example Bucharest is southeast of Arad.

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Breadth-first Search
Searching Strategies

• Properties
• Complete Yes (if b is finite)
• Time complexity 1bb2bd O(bd)
• Space complexity O(bd) (keeps every node in
  memory)
• Optimal Yes (if cost1 per step) not optimal in
  general
• where b is branching factor and
• d is the depth of the search tree

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Uniform cost Search (1)
Searching Strategies

• Expand least-cost unexpanded node
• the breadth-first search is just uniform cost
  search with g(n)DEPTH(n)

G
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Depth-first Search (1)
Searching Strategies

• Depth-first Search

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Depth-first Search (2)
Searching Strategies

• Properties
• Space complexity O(bm)
• i.e., linear space
• Time complexity O(bm)
• where m is the maximum depth
• Complete, Optimal No
• fails in infinite-depth spaces, spaces with loops
• Modify to avoid repeated states along path
• ? complete in finite spaces

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Depth-limited Search
Searching Strategies

• Depth-first search with depth limit l
• Properties
• Complete Yes
• Time complexity O(bl)
• where l is the depth limit
• Space complexity O(bl)
• Optimal No

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Iterative Deepening Search (1)
Searching Strategies

• Choose the best depth limit by trying all
  possible depth limit

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Iterative Deepening Search (1)
Searching Strategies

• Limit0
• Limit1
• Limit2
• Limit3

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Iterative Deepening Search (2)
Searching Strategies

• Choose the best depth limit by trying all
  possible depth limit
• Breadth-first search Depth-first search
• Properties
• Optimal , Complete Yes
• Time complexity
• (d1)b0db(d-1)b21bd O(bd)
• Space complexity O(bd)

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Bidirectional Search (1)
Searching Strategies

• Simultaneously search
• forward from the initial state
• backward from the goal
• stop when the two searches meet in the middle.

Goal
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Bidirectional Search (2)
Searching Strategies

• Several issues
• Operators are reversible.
• Many goal states ? multiple state search
• Properties
• Complete Yes
• Time complexity O(2b(d/2)) O(b(d/2))
• Space complexity O(b(d/2))
• Optimal Yes, if step cost 1
• Can be modified to explore uniform-cost tree

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Comparison Search Strategies
Searching Strategies

• Evaluation of search strategies.
• b is the branching factor
• d is the depth of solution
• m is the maximum depth of the search tree
• l is the depth limit.

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Avoiding Repeated States

• Three ways to deal with repeated states
• Do not return to the state you just came from.
• Do not create paths with cycles in them.
• Do not generate any state that was ever generated
  before.

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Overview CSP

• Constraint Satisfaction Problem
• The Solution of CSP
• Solving Techniques
• Consistency techniques
• Search
• Example
• Applications

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Constraint Satisfaction Problem(1)

• CSP A special kind of problem that satisfies
  some additional structural properties beyond the
  basic requirements for problems in general
• CSP is defined as
• a set of variables X x1 , x2 , x3 xn
• for each variable, xi , a finite domain Di
• Discrete or continuous
• a set of constraints restricting the values that
  the variables can simultaneously take
• Unary Constraints
• Binary Constraints
• High order Constraints
• Absolute constraints vs. preference constraints

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Constraint Satisfaction Problem(2)

• Example 8 queens problem
• Variables the locations of each of the eight
  queens
• Domains squares on the board
• Constraints no two queens can be in the same
  row, column, or diagonal
• V1 the row that the
• first queen occupies
• V2 the row that the
• first queen occupies
• Domain 1,2,3,,8
• Constraints
• (1,3),(1,4),(2,4),(2,5),

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Q
Q
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The Solution of CSP

• If use DFS in 8 queens problem Search space
  will be 88
• Too huge!
• In CSP, the goal test is decomposed into a set of
  constraints on variables rather than being a
  black box.
• Solutions specifying values for all the
  variables such that the constraints are satisfied

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Solving Technology

• Consistency Techniques
• Systematic Search
• Generate-and-Test (GT)
• Generate each possible combination of the
  variables and then test to see if it satisfies
  all the constraints.
• The first combination that satisfies all the
  constraints is the solution.
• The number of combinations the size of the
  Cartesian product of all the variable domains
• Generator merged with tester
• Look back schema
• Look ahead schema

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Consistency Techniques

• Removing inconsistent values from variable domain
• Graph representation - constraint graph
• Consistency Techniques are not complete

Consistent pairs of values
removed by AC
a b c
a b c
Vi
Vj
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Systematic Search (1)

• Generator merged with tester
• Look back schema
• Backtracking, Backjumping, Backmarking,
  Backchecking
• Look ahead schema
• Forward checking (FC) , Look Ahead (LA)

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Backtracking
Look Back Schema

• Incrementally extending a partial solution(DFS)
• When conflict occurs, choose another value for
  inconsistent variable
• Drawbacks
• Thrashing
• Redundant work
• Late detection of collision
• Exponential time complexity

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Example - Backtracking

• There is no legal position available on the next
  row on the board below
• As a result the queen on the
• fifth row will be removed and
• a new position for it will be sought.
• If all legal positions on the row
• lead fail to a solution, the queen
• on the row above will be moved
• to a new legal position, and
• the search for a solution
• will proceed.

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Backjumping
Look Back Schema

• Avoid trashing
• most recent conflicting variable instead of
  immediate past variable

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3
1
3
4
1
2
3
2
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Backchecking, Backmarking
Look Back Schema

• Avoid redundant work

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Look Ahead Schema

• Prevent future conflict
• Examples of Look Ahead Schema
• Forward Checking
• When an assignment, X/a occurs, temporally delete
  all values from other variable domain
• Look Ahead

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Example Forward Checking
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AC Look Ahead

• When a queen is placed on the board all squares
  that are threatened by that queen are marked to
  be ignored
• When considering where to place the next queen it
  uses Arc Consistency to ensure that there are
  legitimate positions for each pair of queens to
  be added, and marks any positions that fail the
  test to be ignored.

Arc (Vi, Vj ) is arc consistent if for every
value x in the current domain of Vi there is some
value y in the domain of Vj such that Vi x and
Vj y is permitted by the binary constraint
between Vi and Vj
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not yet instantiated variables
already instantiated variables
partial look ahead full look ahead
checked by backtracking
forward checking
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Systematic Search (2)

• Smart generator
• Heuristic Method
• Hill-climbing
• Min-conflict (MC)
• Random walk
• Tabu search

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Application

• Traditional Operational Research (OR)
• Planning
• Scheduling
• Optimization
• NP-Hard