Constraint inference

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

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

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