More Formally

• If the partial solution violates constraints, we know any completion of it will also violate constraints.
• Otherwise, we can optimistically assume that the best completion of the partial solution will satisfy the constraints. Further, we can assume that we can choose the highest-scoring (2-consistent) choice for each unassigned variable. This is an optimistic estimate of the maximum score.

Then, if the optimistic estimate is bad enough, we can possibly prune away the partial solution.