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Constraints as (Extended) Equations

Constraints can be actually viewed as equations: in both cases, variables are related by properties, and solving a set of equations amounts to finding which assignment of values to variables meets all the equations. Mathematical equations can be solved if appropriate methods are known, and the same happens with constraints. But constraint tools usually provide domains which are not commonly treated by classical mathematics; or, at least, constraint systems for which solving methods are not a central point of the usual mathematical background.

Using the appropriate domain for each problem is essential: constraint domains have specific characteristics and solving methods which make them more appropriate than others for some problems. Fortunately, deciding which constraint system has to be used is often not difficult: in most cases the problem itself strongly suggests which constraint system to use. In general, the process of solving a problem is a combination of propagation (a general term to refer to equation solving) and search, when an incomplete solution is found.

But looking at constraints as a kind of extended equations does not allow the perception of the whole scenario: equations (even in their extended constraint-like version) suffer from the same drawbacks as OR: lack of modularity (the whole problem is a big set of interrelated equations), lack of dynamic creation of equations, sometimes lack of power to solve completely the equation system proposed, or the solution, as returned by the solver (assignments of values to variables) not coming out in the appropriate format (which, for example, might have to be shared with other tools).

Solutions to these problems can be worked out by coupling constraints and programming.    Next: Why Constraints and Programming? Up: What is Constraint (Logic) Previous: Constraints: Representation and Solving
MCL
1998-12-03