1 Constraint Programming in Constraint

1.1 Abstract We view constraints as relations and constraint satisfaction as a dynamic process of approaching the solution set of the constraints. We have developed a semantic model for dynamic systems, called Constraint Nets, to provide a real-time programming semantics and to model and analyze dynamic systems. In this paper, we explore the relationship between constraint satisfaction and constraint nets by showing how to implement various constraint methods on constraint nets. In particular, we examine discrete and continuous methods for discrete and continuous domain constraint satisfaction problems. Hard and soft constraints within the framework of unconstrained and constrained optimization are considered. Finally, we present an application of this on-line constraint satisfaction framework to the design of robot control systems. Constraints are relations among entities. Constraint satisfaction can be viewed in two diierent ways. In a logical deductive view, a constraint system is a structure hD; `i where D is a set of constraints andìs an entailment relation between constraints 20]. In this view, constraint satisfaction is seen as a process involving multiple agents concurrently interacting on a store-as-constraint system by checking entailment and consistency relations and reening the system monotonically. This approach is useful in database or knowledge-based systems, and can be embedded in logic programming languages 2, 5, 9]. Characteristically, the global constraint is not explicitly represented, though for any given relation tuple the system is able to check whether or not the relation tuple is entailed. In an alternative view, which in our opinion is more appropriate for real-time embedded systems, the constraint satisfaction problem is formulated as nding a relation tuple that is entailed by a given set of constraints 12]. In this paper, we present a new approach in which constraint satisfaction is a dynamic process with the solution set as an attractor of the process. \Monotonicity" is characterized by a Liapunov