Constructing Constraints

It is well-known that there is a trade-oo between the expressive power of a constraint language and the tractability of the problems it can express. But how can you determine the expressive power of a given constraint language, and how can you tell if problems expressed in that language are tractable? In this paper we discuss some general approaches to these questions. We show that for languages over a nite domain the concept of anìn-dicator problem' gives a universal construction for any constraint within the expressive power of a language. We also discuss the fact that all known tractable languages over nite domains are characterised by the presence of a particular solution to a corresponding indicator problem, and raise the question of whether this is a universal property of tractable languages. 1 What is a constraint? A constraint is a way of specifying that a certain relationship must hold between the values taken by certain variables. There are very few \textbook" deenitions of this concept (because there are very few textbooks), but the deenition given in 22] is that a constraint is a set of labelings, and a labeling is a set of variable-value pairs. For example, the constraint that says that the variables X and Y must be assigned diierent values from the set fR; G; Bg would be expressed by the following set of labelings This interpretation of the notion of constraint is convenient for some types of analysis, but for our purposes, it is important to separate out more clearly two aspects of a constraint which are rather mixed together in this deenition. These two aspects are the relation which must hold between the values, and the particular variables over which that relation must hold. We therefore prefer to use the following deenition of constraints (which is similar to the deenition used by many authors, see, for example, 1, 4]). Deenition 1. A constraint C is a pair (s; R), where s is a tuple of variables of length m, called the constraint scope, and R is a relation of arity m, called the constraint relation.