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# The concept of a generalized constraint

Constraints are ubiquitous. A typical constraint is an expression of the form X e C, where X is the constrained variable and C is the set of values which X is allowed to take. A typical constraint is hard (inelastic) in the sense that if u is a value of X then u satisfies the constraint if and only if u e C.

The problem with hard constraints is that most real-world constraints are not hard, meaning that most real-world constraints have some degree of elasticity. For example, the constraints "check-out time is 1 p.m." and "speed limit is 100 km/hr" are, in reality, not hard. How can such constraints be defined? The concept of a generalized constraint is motivated by questions of this kind.

Real-world constraints may assume a variety of forms. They may be simple in appearance and yet have a complex structure. Reflecting this reality, a generalized constraint, GC(X ), is defined as an expression of the form.

where X is the constrained variable; R is a constraining relation which, in general, is non-bivalent; and r is an indexing variable which identifies the modality of the constraint, that is, its semantics. The constrained variable, X, may assume a variety of forms. In particular,

• X is an n-ary variable, X = (X1,...,X„),
• X is a proposition, for example, X = Leslie is tall,
• X is a function,
• X is a function of another variable, X = f(Y),
• X is conditioned on another variable, X/Y,
• X has a structure, for example, X = Location(Residence(Carol)),

X is a group variable. In this case, there is a group, G[A]; with each member of the group, Name,, i = 1,...,n, associated with an attribute- value, A;. A, may be vector valued. Symbolically,

Basically, G[A] is a relation.

X is a generalized constraint, X = Y isr R.

A generalized constraint is associated with a test-score function, ts(u) (Zadeh, 1981a; 1981b) which associates with each object, u, to which the constraint is applicable the degree to which u satisfies the constraint. Usually, ts(u) is a point in the unit interval. However, if necessary, the test-score may be a vector, an element of a semiring (Rossi and Codognet, 2003), an element of a lattice (Goguen, 1969) or, more generally, an element of a partially ordered set, or a bimodal distribution—a constraint which will be described later. The test-score function defines the semantics of the constraint with which it is associated.

The constraining relation, R, is, or is allowed to be, non-bivalent (fuzzy). The principal modalities of generalized constraints are summarized in the following.

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