Local Search, CSP, Game Search

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Local Search, CSP, Game Search

• Artificial Intelligence A Modern Approach
• (Chapter 4.3, 5, 6)
• Juntae Kim
• Department of Computer Engineering
• Dongguk University

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Local Search

• For problems such that
• Goal state itself is a solution (Exgt optimization
  problem)
• Path is irrelevant
• Keep single current state, and try to improve
  it
• (iterative improvement)
• Hill Climbing
• Record only current state
• Continuously select best node among the children

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Local Search

• Examples
• Minimize Y f(X1, X2, Xn) (exgt y x12x22)
• From (X1, X2) move to (X1 1, X2), (X1, X21),
  (X1-1, X2), (X1, X2-1)
• TSP
• From any complete tour, perform pairwise exchange
  
• 8-queens
• From any configuration, move 1

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Hill Climbing

• function Hill-Climbing ( )
• current Make-Node(Initial-State)
• loop do
• next highest value successor of current
• if (Valuenext ? Valuecurrent)) then
  return State(current)
• current neighbor

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Problems of Hill Climbing

• Problem
• Local maxima ? random-restart

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Simulated Annealing

• Idea
• Allow to take some down hill steps (bad moves) to
  escape from local maxima
• But gradually decrease their size and frequency
• Algorithm
• 1. Select random child as next state
• 2. If next is better than current, move to next
• 3. Else move to next with probability e D E/T
• D E value(current) - value(next)
• T a parameter that decreases as time goes

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Simulated Annealing

• function Simulated-Annealing ( )
• current Make-Node(Initial-State)
• for t 1 to ? do
• Decrease(T)
• if (T 0) then return State(current)
• next a random successor of current
• ?E Valuenext - Valuecurrent
• if (?E gt 0) then current next
• else current next with probability e DE
  / T

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Simulated Annealing

• Concept
• If T decrease sufficiently slowly, the algorithm
  will find a global optimum

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Genetic Algorithms

• Evolutionary programming
• Finding best solutions by evolutionary process
• For an optimization problem with huge solution
  space
• Definitions
• Individual(chromosome) - one solution
• Gene - Each element in individual
• Population - set of individual

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Genetic Algorithms

• Fitness function
• f(i) evaluation of individual solution (how good
  is it?)
• Selection
• Select part of the population for reproduction
• With probability P(i) f(i) / ?f(i)
• Reproduction
• Cross-over
• Mutation

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Genetic Algorithms

• function Genetic-Algorithm ( )
• population Generate initial population (size
  N)
• repeat
• new-population empty set
• evaluate population by fitness function
• for i 1 to N
• x, y Random-Selection(population)
• child Reproduce(x, y)
• child Mutate(child) with small
  probability
• add child to new-population
• until some individual is fit enough, or enough
  time has elapsed
• return best individual

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GA Example
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GA CNF-Satisfaction

• Find T,F assignment that makes the expression T
• NP-complete
• Genetic algorithm
• Representation of individual - 6 bit string
• aF, bT, cF, dF, eT, fT ? 010011
• Fitness function true clauses
• f(010011) 3 (0 ? f(i) ? 5)
• Crossover - divides at random point
• Mutation - flip one bit

(a ? c) ? (a ? c ? e) ? (b ? c ? d ? e) ? (a
? b ? c) ? (e ? f)
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GA Timetabling

• Find the best timetable
• Genetic algorithm
• Representation
• 2 dimensional matrix of classes (Time vs. Room)
• Fitness function Score of the timetable
• Weighted sum of penalties f(i) -?(WjPj)
• Crossover - between independent room sets
• Mutation - change the position of a class

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Game Search

• Two person game
• Introduces uncertainty
• Search space
• Search space of interesting game is too large
• Chess 35100
• Baduk 361! gt 180180 gt 10360
• Goal
• Finding good decision (not optimal) in a given
  time

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Example - TicTacToe
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Minimax Search

• In two persons game
• MAX - my move
• ? value of node max of children node
• MIN - opponents move
• ? value of node min of children node

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Minimax Search

• Algorithm
• 1. Generate game tree
• 2. Evaluate leaf nodes
• 3. Run depth-limited DFS
• - Compute values of nodes by minimax
  computation
• 4. Choose node with highest value
• Evaluation functions
• Determine the quality of program
• Weighted linear function
• f w1f1 w2f2 wnfn
• fk features of the particular position (Exgt each
  piece)

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Alpha-Beta Pruning

• Use current upper/lower bound to reduce search
• a - current max value of MAX node (always
  increase)
• b - current min value of MIN node (always
  decrease)
• Prune a subtree if
• b of MIN node lt a of parent MAX
• a of MAX node lt b of parent MIN

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Complexity of Minimax

• Time complexity O(bm)
• Space complexity O(bm)
• Time complexity with a-b pruning O(bm/2)
• Double the search depth in given time
• Chess
• b ? 35, m ? 100 ? full search is impossible
• If time limit is 100 secs, explore104 nodes/sec
• ? search 106 nodes per move ? 354
• ? search depth 4 ahead using simple minimax
• ? search depth 8 ahead using a-b pruning

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Expected Minimax

• When chance node (probability) is introduced
• Exgt Backgammon
• Introduce chance node
• Value of chance node expected value computed
  according to probability
• Evaluation function value affects the decision
• Order-preserving is not sufficient

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Expected Minimax
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Constraint Satisfaction Problems

• CSP
• Variables X1, X2, , Xn ? Values Di (domain)
• Constraints C1, C2, , Cm
• State a consistent (regal) assignment
• Goal complete assignment that satisfies all
  constraints
• General purpose heuristics can be defined
• Example
• Timetabling
• Airline scheduling
• VLSI layout
• Floor planning

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Example Graph Coloring

• Variables WA, NT, SA, Q, NSW, V, SA, T
• Domain red, green, blue
• Adjacent ? different (WA ? NT, )

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Example Graph Coloring

• Solution (complete assignment that satisfies all
  constraints)

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Constraint Graph

• Node variables
• Arc constraints

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CSP as a Standard Search

• State space
• States a regal assignment
• Successor function assigning a new value
• Initial state empty assignment
• Goal test complete assignment
• Properties
• Every solution appears at depth n
• (n of variables)
• Possible complete assignment
• dn (d of values)
• DFS can be used

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Heuristics for CSP

• MRV (Minimum remaining value) heuristic
• Choose the variable with the fewest regal values

2
3
2
2
1
3
3
Early pruning of the search tree (pick a variable
that is most likely to cause a failure)
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Heuristics for CSP

• Degree heuristic
• When MRV are same, choose the variable with
• the most constraint on other unassigned variables

3
2
1
3
2
0
1
2
5
3
2
1
2
MRV 3
MRV 1
MRV 2
Reduce branching factor of the search
tree (reduce the number of future choice)
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Heuristics for CSP

• Least constraining value heuristic
• Choose the value that rules out fewest choices
  for the neighboring variables

1
0
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Constraint Propagation

• Forward checking
• Keep track of remaining regal values
• Terminate search when any variable has no regal
  values

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Constraint Propagation

• Arc consistency
• Provides early detection for failure
• X ? Y is consistent if
• for every x ? X, there is some allowed y ? Y

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Constraint Propagation

• Propagation
• If X looses a value due to the consistency,
• neighbors of X need to be rechecked