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# Constraint inference

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 \( C_1,\ldots,C_m \) 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 \(C_1,\ldots,C_m \) are all satisfied.

See also

Constraint satisfaction problem

References

Dechter, Rina (2003). Constraint processing. Morgan Kaufmann. ISBN 1-55860-890-7

Apt, Krzysztof (2003). Principles of constraint programming. Cambridge University Press. ISBN 0-521-82583-0

Marriott, Kim; Peter J. Stuckey (1998). Programming with constraints: An introduction. MIT Press. ISBN 0-262-13341-5

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