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A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer ...
A free parameter is a variable in a mathematical model which cannot be predicted precisely or constrained by the model [1] and must be estimated [2] experimentally or theoretically. A mathematical model, theory, or conjecture is more likely to be right and less likely to be the product of wishful thinking if it relies on few free parameters and ...
the occurrence of both x and y in C(y, x) is free, while the occurrence of x and y in B(y, x) is bound (i.e. non-free). Syntax tree of the formula ((,)) (,), illustrating scope and variable capture. Bound and free variable occurrences are colored in red and green, respectively.
An open formula can be transformed into a closed formula by applying a quantifier for each free variable. This transformation is called capture of the free variables to make them bound variables. For example, when reasoning about natural numbers , the formula " x +2 > y " is open, since it contains the free variables x and y .
Any variable can be classified as being either a free variable or a bound variable. For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined.
All other variables are called free. For example, in the expression λy.x x y, y is a bound variable and x is a free variable. Also a variable is bound by its nearest abstraction. In the following example the single occurrence of x in the expression is bound by the second lambda: λx.y (λx.z x).
It is the presence of a free variable, rather than the inconstant truth value, that is important; for example, even for complex numbers, where the formula is always true, it is still not considered a sentence. Such a formula may be called a predicate instead.
A substitution is called a ground substitution if it maps all variables of its domain to ground, i.e. variable-free, terms. The substitution instance tσ of a ground substitution is a ground term if all of t ' s variables are in σ ' s domain, i.e. if vars(t) ⊆ dom(σ).