Free variables and bound variables

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In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol.

The variable x becomes a bound variable, for example, when we write

or

In either of these propositions, it does not matter logically whether we use x or some other letter. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.

In computer programming, a free variable is a variable referred to in a function that is not a local variable or an argument of that function.[1] An upvalue is a free variable that has been bound (closed over) with a closure. Note that variable "freeness" only applies in lexical scoping: there is no distinction, and hence no closures, when using dynamic scope.

The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use creates an ambiguity with the definition of dummy variables in regression analysis.

Contents

Examples

Before stating a precise definition of free variable and bound variable, the following are some examples that perhaps make these two concepts clearer than the definition would:

In the expression

n is a free variable and k is a bound variable; consequently the value of this expression depends on the value of n, but there is nothing called k on which it could depend.

In the expression

y is a free variable and x is a bound variable; consequently the value of this expression depends on the value of y, but there is nothing called x on which it could depend.

In the expression

x is a free variable and h is a bound variable; consequently the value of this expression depends on the value of x, but there is nothing called h on which it could depend.

In the expression

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