Cardinality constraints in disjunctive deductive databases

We investigate cardinality constraints of the form M -theta K, where M is a set and theta is one of the comparison operators or "greater than or equal to"; 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 theta 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.