1. <PI> In a reduction system, a closure is a data structure that holds an expression and an environment of variable bindings in which that expression is to be evaluated. The variables may be local or global. Closures are used to represent unevaluated expressions when implementing functional programming languages with lazy evaluation. In a real implementation, both expression and environment are represented by pointers. A suspension is a closure which includes a flag to say whether or not it has been evaluated. The term "thunk" has come to be synonymous with "closure" but originated outside functional programming.
2. <theory> In domain theory, given a partially ordered set, D and a subset, X of D, the upward closure of X in D is the union over all x in X of the sets of all d in D such that x <= d. Thus the upward closure of X in D contains the elements of X and any greater element of D. A set is "upward closed" if it is the same as its upward closure, i.e. any d greater than an element is also an element. The downward closure (or "left closure") is similar but with d <= x. A downward closed set is one for which any d less than an element is also an element.
a. closure of a system
A system with at least one closed term is closed iff |- At/x implies |- (x)Ax for each closed term t and each 1-wff A. (A 1-wff is a wff with exactly one free variable.) Informally, the system is closed if the fact that A is a theorem when its free variable is replaced with any closed term in the language implies the universal generalization (x)A. It follows from closure that every member of the domain is named by some closed term; hence the domain is countable.
See omega-completeness, term, closed, open wff
b. closure of a wff
In predicate logic, the binding of all free variables from a wff by placing them within the scope of suitable quantifiers. A closed wff is considered its own closure. The closure of a closure of a wff A is considered a closure of A. Notation: A^c (a closure of wff A). Closed wffs are also called sentences.
See bound variable, generalization, instantiation
[Glossary of First-Order Logic]
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