Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and in other disciplines involving
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of sym ...
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, a free variable is a
In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefor ...
(symbol) that specifies places in an
Expression may refer to:
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, ...
may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different ...
that will later be replaced by some value), or a
In software, a wildcard character is a kind of placeholder represented by a single character, such as an asterisk (), which can be interpreted as a number of literal characters or an empty string. It is often used in file searches so the full n ...
that stands for an unspecified symbol.
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
, the term free variable refers to variables
used in a function
that are neither
In computer science, a local variable is a variable that is given ''local scope''. A local variable reference in the function or block in which it is declared overrides the same variable name in the larger scope. In programming languages with ...
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s of that function. The term non-local variable
is often a synonym in this context.
A bound variable, in contrast, is a variable that has been ''bound'' to a specific value or range of values in the
domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
The domain ...
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
. This may be achieved through the use of logical quantifiers, variable-binding operators, or an explicit statement of allowed values for the variable (such as, "…where
is a positive integer".) Examples are given in the next section. However it is done, the variable ceases to be an independent variable on which the value of the expression depends, whether that value be a truth value or the numerical result of a calculation, or, more generally, an element of an image set of a function. Note that while the domain of discourse in many contexts is understood, when an explicit range of values for the bound variable has not been given, it may be necessary to specify the domain in order to properly evaluate the expression. For example, consider the following expression in which both variables are bound by logical quantifiers:
This expression evaluates to ''false'' if the domain of
is the real numbers, but ''true'' if the domain is the complex numbers.
The term "dummy variable" is also sometimes used for a bound variable (more commonly in general mathematics than in computer science), but this should not be confused with the identically named but unrelated concept of dummy variable
as used in statistics, most commonly in regression analysis.
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
''z'' is a free variable and ''x'' and ''y'' are bound variables, associated with logical quantifier
s; consequently the
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
In some progr ...
of this expression depends on the value of ''z'', but there is nothing called ''x'' or ''y'' on which it could depend.
More widely, in most proofs, bound variables are used. For example, the following proof shows that all squares of positive even integers are divisible by
be a positive even integer. Then there is an integer
, we have
not only ''k'' but also ''n'' have been used as bound variables as a whole in the proof.
are some common variable-binding operators. Each of them binds the variable x for some set S.
Note that many of these are operators
which act on functions of the bound variable. In more complicated contexts, such notations can become awkward and confusing. It can be useful to switch to notations which make the binding explicit, such as
for sums or
Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science. In all cases, however, they are purely
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...
properties of expressions and variables in them. For this section we can summarize syntax by identifying an expression with a
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
whose leaf nodes are variables, constants, function constants or predicate constants and whose non-leaf nodes are logical operators. This expression can then be determined by doing an inorder traversal
of the tree. Variable-binding operators are
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s that occur in almost every formal language. A binding operator Q takes two arguments: a variable ''v'' and an expression ''P'', and when applied to its arguments produces a new expression Q(''v'', ''P''). The meaning of binding operators is supplied by the
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comp ...
of the language and does not concern us here.
Variable binding relates three things: a variable ''v'', a location ''a'' for that variable in an expression and a non-leaf node ''n'' of the form Q(''v'', ''P''). Note: we define a location in an expression as a leaf node in the syntax tree. Variable binding occurs when that location is below the node ''n''.
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...
is a bound variable in the term
M = λx. T
and a free variable in the term
. We say
is bound in
and free in
contains a subterm
is rebound in this term. This nested, inner binding of
is said to "shadow" the outer binding. Occurrences of
are free occurrences of the new
Variables bound at the top level of a program are technically free variables within the terms to which they are bound but are often treated specially because they can be compiled as fixed addresses. Similarly, an identifier bound to a recursive function
is also technically a free variable within its own body but is treated specially.
A ''closed term'' is one containing no free variables.
To give an example from mathematics, consider an expression which defines a function