Horn cardinality rules

We address the problem of finding a "tight" representation of Horn cardinality rules in a mixed integer programming model by describing a convex hull of it. A cardinality Horn rule asserts that if at least k of the propositions A\,..., Am are true, then B is true. We also show that Horn cardinality rules have properties analogous to ordinary Horn rules. 1 I n t r o d u c t i o n As rule-based systems and other types of logic modeling grow in popularity, logical rules and propositions can play an increasingly important role in mathematical programming models. Such simple logical constraints as "if A is produced, then either B or C must be produced" have long been a part of mathematical programming. But much more complex logic models are now being formulated, and they can also be embedded in mathematical programming models. Propositional Horn formulas have some very attractive properties that account for their popularity in rule-based systems. A Horn inference problem are solvable by the linear programming relaxation, (see, for example, [2], [7]) That is, one can determine the satisfiability of a set of Horn rules simply by checking whether the corresponding LP relaxation is feasible. A Horn clause *The first author is partially supported by AFOSR grant 91-0287 and ONR grant N00014-92-J-1028. Both authors are partially supported by NSF grant 1-55093.