Set Based Search Modeling
Examples
Andrew Kuipers
[email protected]
Please include [CPSC433] in the subject line of any emails regarding
this course.
CPSC 433 Artificial Intelligence
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
CPSC 433 Artificial Intelligence
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: we’re also modeling blanks as marks)
F = { (i, j, m) |  i, j R,  m M }
CPSC 433 Artificial Intelligence
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 X’s
should never exceed the # of O’s by more than 1, and the #
of O’s should never exceed the # of X’s
CPSC 433 Artificial Intelligence
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’)
CPSC 433 Artificial Intelligence
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:
X
O
X
O
= { (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) }
CPSC 433 Artificial Intelligence
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 don’t allow changing X’s or O’s
– 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
CPSC 433 Artificial Intelligence
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 }
CPSC 433 Artificial Intelligence
Example: Tic Tac Toe
ie:
X
O
O
X
Which of the following are in Ext?
a) { (1,2,B) } → { (1,2,O) }
b) { (3,3,B) } → { (3,3,X) }
c) { (2,1,O) } → { (2,1,X) }
d) { (2,3,B) } → { (3,2,X) }
CPSC 433 Artificial Intelligence
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?
CPSC 433 Artificial Intelligence