Cardinality Constraints in Disjunctive Deductive Databases
We investigate cardinality constraints of the form M ↪θ K, where M is a set and θ is one of the comparison operators “=”, “≤”, or “≥”; such a constraint states that “exactly”, “at most”, or “at least”, respectively, K elements out of the set M have to be chosen.
We show how a set C of constraints can be represented by means of a positive-disjunctive deductive database P C , such that the models of P C correspond to the solutions of C. This allows for embedding cardinality constraints into applications dealing with incomplete knowledge.
We also present a sound calculus represented by a definite logic program P cc , which allows for directly reasoning with sets of exactly-cardinality constraints (i.e., where θ is “=”). Reasoning with P cc is very efficient, and it can be used for performance reasons before P C is evaluated. For obtaining completeness, however, P C is necessary, since we show the theoretical result that a sound and complete calculus for exactly- cardinality constraints does not exist.
Keywordsdisjunctive logic programming constraint logic programming cardinality constraints hyperresolution
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