Bags , sometimes called multisets, are like sets except that duplicate elements are allowed. B. It is possible to define operations like union, additive union and intersection for bags in an obvious way. In general BCH solution is a superset of the traditional constraint hierarchy solution set . FGH is responsible for removing such valuations from the BCH solution which are unacceptable to traditional constraint hierarchies. For this purpose FGH uses an extra mathematical construct filter that is a function from bags to bags.
Filters correspond to the comparators in traditional constraint hierarchies.
Note, that 0 is the minimum element of the ordering and 1 is the maximum element of the ordering. More precisely, for each tuple of values for the involved variables, a corresponding element of the semiring domain A is given. This element can be intrepreted as the tuple's weight, or cost, or level of confindence, or else. Now, the multiplicative operation is used to combine the values of the tuples of each constraint to get the value of a tuple for all the variables, and the additive operation is used to obtain the value of the tuples of the variables in the type of problem. That is, for each valuation of the variables a combined value in A is computed by multiplying tuple's weights of individual constraints, and such valuations that have maximum combined weight are selected to the solution.