Linear Constraints

PyLog supports linear arithmetic relations over integers and enforces them by domain propagation.

Relations

• Equality/disequality: #=, #\=
• Ordering: #<, #=<, #>, #>=

Examples

?- X in 1..10, Y in 1..10, X + Y #= 10.
true.  % domains are narrowed internally

?- X in 1..10, Y in 1..10, 2*X + 3*Y #=< 20.
true.

?- X in 1..10, Y in 1..10, X - Y #> 3.
true.

Propagation insight

Constraints narrow variable domains before any search. For X+Y #= 10 with X,Y in 1..10, both domains become 1..9 since a variable cannot be 10 unless the other is 0 (outside the domain).

Difference constraints

Constraints of the form X - Y #=< C (and relatives) propagate efficiently and are useful to encode precedence and spacing.

?- S1 in 0..10, S2 in 0..10, E1 #= S1 + 3, E2 #= S2 + 2, E1 #=< S2.
true.

Linearity requirement

All terms must be linear combinations of variables and integers. Non‑linear terms such as X*Y or X*X are rejected.

?- X in 1..9, Y in 1..9, X*Y #= 6.
% error: non‑linear term not supported

Modeling sums

Sum constraints can be modeled by a chain of linear relations.

sum_list([], 0).
sum_list([X|Xs], S) :- sum_list(Xs, T), S #= X + T.

% Helper for posting domains to multiple variables
ins([], _).
ins([V|Vs], Dom) :- V in Dom, ins(Vs, Dom).

?- Vs = [X,Y,Z], ins(Vs, 1..3), sum_list(Vs, 6), label(Vs).
X = 1, Y = 2, Z = 3 ; ...

Tips

• Post constraints as equations/inequalities; avoid mixing with is/2 (which evaluates) in constraint models.
• Normalize expressions to make linearity obvious (e.g., 2*X + 3*Y #=< 20).