Constraint Programming in Constraint Nets

We view constraints as relations and constraint satisfaction as a dynamic process of approaching a stable equilibrium. We have developed an algebraic model of dynamics, 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. 1 Motivation Constraints are relations among entities. Constraint satisfaction can be viewed in two diierent ways. First, in the 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 the 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, even though for any given relation tuple the system is able to check whether or not it is entailed. An alternative view, more appropriate for real-time embedded systems, is to formulate the constraint satisfaction problem as nding a relation tuple that is entailed by a given set of constraints 12]. In this paper, we present an approach to this problem. In this approach, constraint satisfaction is a dynamic process with each solution as a stable equilibrium, and the solution set as an attractor of the process. \Monotonicity" is characterized by a Liapunov function, representing the \distance" to the set of solutions over time. Moreover, soft as well as hard constraints can be represented and solved. This approach has been taken in neural nets 18], optimization, graphical simulation 16] and robot control 15]; however, it has not yet been investigated seriously in the area of constraint programming. We have developed and implemented an algebraic model of dynamics, called Constraint Nets (CN), to provide a real-time programming semantics 22] and to model and analyze robotic systems 23]. Here we investigate the relationship between constraint satisfaction and constraint nets. The rest of this paper is organized as follows. Section 2 describes some basic concepts of dynamic systems. Section 3 introduces Constraint Nets. Section 4 presents various constraint methods for solving global consistency and unconstrained optimization problems. Section 5 discusses embedded constraint solvers and some implementation issues. …