A Genetic Algorithm Approach to the Prisoners

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A Genetic Algorithm Approach to the Prisoners
Guards Puzzle

• CPSC 5185 Intro to Artificial Intelligence
• April 6, 2004

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Description of Prisoners Guards Puzzle
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Adjacent Squares
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Sample 3x3 Board

• 5 prisoners can we do better?

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A Better 3x3 Board

• 6 prisoners this is the maximum

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Some 4x4 Boards
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An Optimal 4x4 Board
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Finding Maximum No. Prisoners

• Characterize p(n), the max. no. prisoners on a
  valid n? n board.

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Larger Values of n

• Brute force doesnt work
• Trying an evolutionary approach

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Genetic Algorithm Overview

• Each generation has 500 chromosomes
• Spawn offspring through crossover and mutation
  operations
• 100 most fit in a generation retained in next
  generation

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GA Overview, continued

• Parents selected randomly, selection
  probabilities are proportional to fitness vals
• Two parents always produce two offspring
• Cataclysm prevents stagnation

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Converting Boards to Chromosomes

• Treat each board configuration as a chromosome
  (ignore placement rule)

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The Crossover Operation

• Crossover probability 0.7.

Parents
Children
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Mutation

• Mutation probability 0.001 for each board square
• Toggle value if mutation occurs

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Survival of the fittest

• Fitness function 1
  fitness total number of prisoners
• Fitness function 2
  fitness (no. pris.) x (validity)
  validity ( valid sqrs) / (total sqrs)

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Fitness Comp. Example
F1 6
F2 6 ( 5 / 9 ) 30 / 9
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Outline of the Algorithm

• See handout sheet

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Comparing Performances Using Two Fitness Functions

• See Excel printout

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My Best 5x5 Example
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A 6x6 Example
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Lower Bounds for p(n)

• For odd n, striping pattern yields
• For even n, goose pattern yields

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A Better 6x6
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Striping Not Best for 9x9
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Questions

• Is there a better lower bound for p(n)?
• What is a good upper bound for p(n)?