CS 482/682: ARTIFICIAL INTELLIGENCE
FALL 2009
Homework2
Constraint Satisfaction Problems  Logicbased Inference and Satisfiability  Adversarial Search
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Problem 1:
Consider the game Sudoku, where we try to fill a 9
×
9 grid of squares with numbers
subject to some constraints: every row must contain all of the digits 1
, . . . ,
9, every column must
contain all of the digits 1
, . . . ,
9, and each of the 9 different 3
×
3 boxes must also contain all of the
digits 1
, . . . ,
9. In addition, some of the boxes are filled with numbers already, indicating that the
solution to the problem must contain those assignments. Here is a sample board:
Figure 1: A sample Sudoku board
Each game is guaranteed to have a single solution. That is, there is only one assignment to the
empty squares which satisfies all the constraints. For the purposes of this homework, let’s use
n
,j
to refer to the number in row
, column
j
of the grid. Also, assume that
M
of the numbers have
been specified in the starting problem, where
M
=
29 for the problem shown above.
1. Incremental formulation approach.
(a) Formalize this problem using an incremental formulation. What are the start state, suc
cessor function, goal test, and path cost function?
(b) What is the branching factor, solution depth, and maximum depth of the search space?
What is the size of the state space?
(c) Assuming we don’t use a heuristic, which of the following would you recommend for solv
ing the incremental search formulation of this problem:
BFS
,
DFS
or Iterative Deepening
Algorithm?
(d) Given the incremental problem formulation, is heuristic search possible? If so, provide a
heuristic. If not, why not?
2. Constraint Satisfaction Problem formulation.
(a) What is the set of variables, and what is the domain of possible values for each? How do
the constrains look like?
(b) Which heuristic for backtracking search would you expect to work better for this problem,
the degree heuristic, or the minimum remaining values heuristic and why?
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(c) What, is the difference between “easy” and “hard” Sudoku problems? [Hint: There is a
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 Spring '10
 ABDELSALAM
 Max, Search algorithm, Minimax, Constraint satisfaction problem, alphabeta pruning

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