Set Based Search Modeling Examples

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Set Based Search Modeling Examples

• Andrew Kuipers
• amkuiper_at_cpsc.ucalgary.ca
• Please include CPSC433 in the subject line of
  any emails regarding this course.

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Example Tic Tac Toe

• To model Tic Tac Toe as a set based search model,
  we need to define
• What a fact is
• Which combinations of facts constitute legal
  states
• What the extension rules are that we can use to
  move from one set of facts (state) into another

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Example Tic Tac Toe

• Facts
• A fact will simply be that a certain mark is made
  at a specific location on the tic-tac-toe board
• let R 1, , 3
• let M X, O, B (note were also modeling
  blanks as marks)
• F (i, j, m) ? i, j ?R, ? m ?M

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Example Tic Tac Toe

• States
• A set of facts satisfying the following
• For each location on the board, there exists a
  fact marking that location either X, O or blank
• Each location on the board may have only one fact
  associated with it (only one mark per location)
• The play order is preserved
• ie by forcing X to always have the first move,
  the of Xs should never exceed the of Os by
  more than 1, and the of Os should never exceed
  the of Xs

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Example Tic Tac Toe

• States
• S F ? F Covered(F) ? AllUnique(F)
• ? PlayOrderPreserved(F)
• Covered(F) ? ?i,j?R ?m?M (i, j, m) ? F
• AllUnique(F) ? ?i,j?R ?m,m?M
• ((i,j,m)?F?(i,j,m)?F) ? (m m)

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Example Tic Tac Toe

• PlayOrderPreserved(F) ?
• 0 ? (Count(X, F) Count(O, F)) ? 1
• Count(m, F) (i, j, m) ? F, ?i, j?R
• ie (1,1,X), (1,2,B), (1,3,B),
• (2,1,O), (2,2,O), (2,3,B),
• (3,1,X), (3,2,B), (3,3,B)

X
O
O
X
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Example Tic Tac Toe

• Extension Rules
• Now we need to model what a valid move is in
  the game
• Can only change blank marks to either an X or an
  O we dont allow changing Xs or Os
• Only one change per move
• The result of the move must itself be a valid
  state, such that the conditions of Coverage,
  Uniqueness and Play Order defined above are
  preserved

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Example Tic Tac Toe

• Extension Rules
• Ext A ? B ?s?S ? A ? S (s-A)?B?S
• ? SingleMove(A,B)
• SingleMove(A, B) ?
• A (i, j, B) ? B (i, j, p) s.t. p?
  X, O

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Example Tic Tac Toe

• ie
• Which of the following are in Ext?
• (1,2,B) ? (1,2,O)
• (3,3,B) ? (3,3,X)
• (2,1,O) ? (2,1,X)
• (2,3,B) ? (3,2,X)

X
O
O
X
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Example Tic Tac Toe

• What now?
• We have defined the search model
• How can we define a search instance?
• What would a search control for the problem look
  like?