Constraint Logic Programming Using ECLiPSe

6.4 Constraint satisfaction problems: examples 97
where x ∈ {S, E, N, D, M, O, R, Y }. In turn, the all different constraint
is expressed as the set of ten constraints
∑
is x,d ≤ 1,
x∈{S,E,N,D,M,O,R,Y }
where d ∈ {0, 1, . . ., 9}. The 0/1 variables is x,d relate back to the original
variables used in the equality constraint(s) via eight constraints
9∑
d · is x,d = x,
d=0
where x ∈ {S, E, N, D, M, O, R, Y }.
The disadvantage of this approach is of course that we need to introduce
80 new variables.
The problem under consideration has a unique solution depicted by the
following sum:
That is, the assignment
9567
+ 1085
10652
{(S, 9), (E, 5), (N, 6), (D, 7), (M, 1), (O, 0), (R, 8), (Y, 2)}
is a unique solution of the original CSP and each other discussed representation
of the problem has a unique solution, as well.
6.4.3 The n-queens problem
This is probably the most well-known CSP. The problem is to place n queens
on the n×n chess board, where n ≥ 3, so that they do not attack each other.
We discuss two representations of this problem as a CSP.
Representation using 0/1 variables This representation uses n 2 0/1
variables x i,j , where i ∈ [1..n] and j ∈ [1..n], each of them representing one
field of the chess board. The appropriate constraints can then be written
using sums of relevant variables.
The following constraints formalise the problem:
• ∑ n
j=1 x i,j = 1 for i ∈ [1..n] (exactly one queen per row),
• ∑ n
j=1 x j,i = 1 for i ∈ [1..n] (exactly one queen per column),