In constraint satisfaction, constraint inference is a relationship between constraints and their consequences. A set of constraints D entails a constraint C if every solution to D is also a solution to C . In other words, if V is a valuation of the variables in the scopes of the constraints in D and all constraints in D are satisfied by V , then V also satisfies the constraint 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 ) and ( ( y , z ) , S ) with a common variable. The composition of such two constraints is the constraint ( ( 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 such that the evaluation of these three variables satisfies the two original constraints ( ( x , y ) , R ) and ( ( y , z ) , S ) .
Constraint projection restricts the effects of a constraint to some of its variables. Given a constraint ( t , R ) its projection to a subset t ′ of its variables is the constraint ( 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 ) 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 and a list A of their variables, the extended composition of them is the constraint ( A , R ) where an evaluation of A satisfies this constraint if it can be extended to the other variables so that C 1 , … , C m are all satisfied.
