There are two motivations for supporting complex constraints. One is to simplify problem modelling. It is shorter, and more natural, to use a single alldistinct constraint on N variables than to use n*(n-1)/2 (pairwise) disequalities!
The second motivation is to achieve specialised constraint propagation
alldistinct constraint on N variables, has the same
semantics as n*(n-1)/2 (pairwise) disequalities, but it can also achieve better
propagation than would be possible with the disequalities.
For example if any M of the variables have the same domain, and its size is
than M, then the
alldistinct constraint can immediately fail.
However if two variables X and Y have the same domain, with M>1
elements, the constraint X
## Y can achieve no
propagation at all.
Thus the pairwise disequalities are unable to achieve the same
propagation as the alldistinct constraint.
The constraint atmost(Number, List, Val) constrains atmost Number of the variables in the list List to take the value Val. This is a difficult constraint to express using logic. One way is to constrain each sublist of length Number+1 to contain a variable with value different from Val, but the resulting number of constraints can be very large!
A more natural way is to constrain all the variables to take a value
different from Val, and to allow the constraint to be violated up to
The fd library supports such a facility with the constraint
#=(T1, T2, B).
This constraint makes B=1 if T1=T2 and B=0 otherwise.
It is possible to express atmost by imposing the constraint
for each variable in the list and
then adding the constraint
The built-in atmost constraint is essentially implemented like
The other fd constraints (
#>, etc.) can be
extended with an extra 0/1 variable in the same way.
The fd library includes a great variety of facilities, which are best explored by obtaining the ECLiPSe extensions manual [Be97] and looking at the programming examples in the section on the fd library there.