A Theorem-proving Approach to Job- Shop Scheduling

In this paper we formulate the job-shop scheduling problem in the theorem-proving framework described in 12, 5]. In the theorem-proving framework two kinds of predicates are allowed: ordinary predicates, deened by if-and-only-if-deenitions 3], and primitive predicates, \deened" by axioms that can be thought of as integrity constraints. The framework seeks to unify constraint logic programming (CLP) and abductive logic programming (ALP). The class of the primitive predicates includes constraint predicates like equality and inequality as well as abducible predicates. The \integrity constraints" deening the primitive predicates consist of axioms for equality and inequality in the case of constraint predicates and more conventional integrity constraints in the case of abducibles. The framework diiers from conventional CLP primarily in the fact that constraint handling is performed by explicit execution of integrity constraints written in logical form. In the theorem-proving framework, the initial goal is stated as a fact. Then, if-and-only-if deenitions are executed backward, in conventional logic programming style, to unfold ordinary atoms. Integrity constraints are executed forward, both to propagate consequences of primitive atoms and to delete redundancies among primitive atoms. In this sense, integrity constraints are similar to the constraint handling rules in 6]. Finally, logical simpliication, by using logical axioms, and splitting, i.e. distribution of disjunction over conjunction, can be applied to goals. Every goal obtained in this way from the initial goal is equivalent to the initial goal. Furthermore, every such goal is logically equivalent to a disjunction of conjunctions (if put into disjunctive normal form). One disjunct in the disjunctive normal form of a goal is an answer if and only if it consists only of primitives and it satisses the integrity constraints as in ALP 10]. In this theorem-proving framework, we simulate the branch and bound method for job-shop scheduling proposed by Carlier and Pinson 2], that rst solved the notorious 10 10 job-shop problem proposed in 15]. The logic of the Carlier and Pinson algorithm is simpliied and improved in the simulation. The contribution of the paper is twofold. One the one hand, it shows that forward execution of integrity constraints in the theorem-proving framework provides an elegant and potentially eecient approach to job-shop scheduling. On the other hand, as a case study, it indicates the potential suitability of the framework for CLP in general.