Some operations on constraints produce a new constraint that is a consequence of them. Constraint composition operates on a pair of binary constraints and with a common variable. The composition of such two constraints is the constraint that is satisfied by every evaluation of the two non-shared variables for which there exists a value of the shared variable such that the evaluation of these three variables satisfies the two original constraints and .
Constraint projection restricts the effects of a constraint to some of its variables. Given a constraint its projection to a subset of its variables is the constraint that is satisfied by an evaluation if this evaluation can be extended to the other variables in such a way the original constraint is satisfied.
Extended composition is similar in principle to composition, but allows for an arbitrary number of possibly non-binary constraints; the generated constraint is on an arbitrary subset of the variables of the original constraints. Given constraints and a list of their variables, the extended composition of them is the constraint where an evaluation of satisfies this constraint if it can be extended to the other variables so that are all satisfied.