In constraint satisfaction, constraint inference is a relationship between constraints and their consequences. A set of constraints D {\displaystyle D} entails a constraint C {\displaystyle C} if every solution to D {\displaystyle D} is also a solution to C {\displaystyle C} . In other words, if V {\displaystyle V} is a valuation of the variables in the scopes of the constraints in D {\displaystyle D} and all constraints in D {\displaystyle D} are satisfied by V {\displaystyle V} , then V {\displaystyle V} also satisfies the constraint C {\displaystyle C} .

Some operations on constraints produce a new constraint that is a consequence of them. Constraint composition operates on a pair of binary constraints ( ( x , y ) , R ) {\displaystyle ((x,y),R)} and ( ( y , z ) , S ) {\displaystyle ((y,z),S)} with a common variable. The composition of such two constraints is the constraint ( ( x , z ) , Q ) {\displaystyle ((x,z),Q)} that is satisfied by every evaluation of the two non-shared variables for which there exists a value of the shared variable y {\displaystyle y} such that the evaluation of these three variables satisfies the two original constraints ( ( x , y ) , R ) {\displaystyle ((x,y),R)} and ( ( y , z ) , S ) {\displaystyle ((y,z),S)} .

Constraint projection restricts the effects of a constraint to some of its variables. Given a constraint ( t , R ) {\displaystyle (t,R)} its projection to a subset t ′ {\displaystyle t'} of its variables is the constraint ( t ′ , R ′ ) {\displaystyle (t',R')} that is satisfied by an evaluation if this evaluation can be extended to the other variables in such a way the original constraint ( t , R ) {\displaystyle (t,R)} 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 C 1 , … , C m {\displaystyle C_{1},\ldots ,C_{m}} and a list A {\displaystyle A} of their variables, the extended composition of them is the constraint ( A , R ) {\displaystyle (A,R)} where an evaluation of A {\displaystyle A} satisfies this constraint if it can be extended to the other variables so that C 1 , … , C m {\displaystyle C_{1},\ldots ,C_{m}} are all satisfied.