A.8.9 Core relations and search

Using CLP(FD) constraints to solve combinatorial tasks typically consists of two phases:

1. First, all relevant constraints are stated.
2. Second, if the domain of each involved variable is finite, then enumeration predicates can be used to search for concrete solutions.

It is good practice to keep the modeling part, via a dedicated predicate called the core relation, separate from the actual search for solutions. This lets us observe termination and determinism properties of the core relation in isolation from the search, and more easily try different search strategies.

As an example of a constraint satisfaction problem, consider the cryptoarithmetic puzzle SEND + MORE = MONEY, where different letters denote distinct integers between 0 and 9. It can be modeled in CLP(FD) as follows:

puzzle([S,E,N,D] + [M,O,R,E] = [M,O,N,E,Y]) :-
        Vars = [S,E,N,D,M,O,R,Y],
        Vars ins 0..9,
        all_different(Vars),
                  S*1000 + E*100 + N*10 + D +
                  M*1000 + O*100 + R*10 + E #=
        M*10000 + O*1000 + N*100 + E*10 + Y,
        M #\= 0, S #\= 0.

Notice that we are not using labeling/2 in this predicate, so that we can first execute and observe the modeling part in isolation. Sample query and its result (actual variables replaced for readability):

?- puzzle(As+Bs=Cs).
As = [9, A2, A3, A4],
Bs = [1, 0, B3, A2],
Cs = [1, 0, A3, A2, C5],
A2 in 4..7,
all_different([9, A2, A3, A4, 1, 0, B3, C5]),
91*A2+A4+10*B3#=90*A3+C5,
A3 in 5..8,
A4 in 2..8,
B3 in 2..8,
C5 in 2..8.

From this answer, we see that this core relation terminates and is in fact deterministic. Moreover, we see from the residual goals that the constraint solver has deduced more stringent bounds for all variables. Such observations are only possible if modeling and search parts are cleanly separated.

Labeling can then be used to search for solutions in a separate predicate or goal:

?- puzzle(As+Bs=Cs), label(As).
As = [9, 5, 6, 7],
Bs = [1, 0, 8, 5],
Cs = [1, 0, 6, 5, 2] ;
false.

In this case, it suffices to label a subset of variables to find the puzzle's unique solution, since the constraint solver is strong enough to reduce the domains of remaining variables to singleton sets. In general though, it is necessary to label all variables to obtain ground solutions.