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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 (symbol) that specifies places in an expression where substitution 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 symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol.

- Examples
- Variable-binding operators
- Formal explanation
- Function expressions
- Natural language
- See also
- References

In computer programming, the term **free variable** refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context.

A **bound variable** is a variable that was previously *free*, but has been *bound* to a specific value or set of values called domain of discourse or universe. For example, the variable x becomes a **bound variable** when we write:

For all x, (

x+ 1)^{2}=x^{2}+ 2x+ 1.

or

There exists x such that

x^{2}= 2.

In either of these propositions, it does not matter logically whether x or some other letter is used. 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.

The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use can create an ambiguity with the definition of dummy variables 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 quantifiers; consequently the logical value 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 of the proofs, we use bound variables. For example, the following proof shows that every square of even integer is divisible by

- Let be a positive even integer. Then there is an integer such that . Since , we have divisible by

not only *k* but also *n* have been used as bound variables as a whole in the proof.

The following

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

for differentiation.

Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science. In all cases, however, they are purely syntactic properties of expressions and variables in them. For this section we can summarize syntax by identifying an expression with a tree 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 logical operators 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 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*.

In the lambda calculus, `x`

is a bound variable in the term `M = λx. T`

and a free variable in the term `T`

. We say `x`

is bound in `M`

and free in `T`

. If `T`

contains a subterm `λx. U`

then `x`

is rebound in this term. This nested, inner binding of `x`

is said to "shadow" the outer binding. Occurrences of `x`

in `U`

are free occurrences of the new `x`

.^{ [1] }

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

where *t* is an expression. *t* may contain some, all or none of the *x*_{1}, …, *x*_{n} and it may contain other variables. In this case we say that function definition binds the variables *x*_{1}, …, *x*_{n}.

In this manner, function definition expressions of the kind shown above can be thought of as *the* variable binding operator, analogous to the lambda expressions of lambda calculus. Other binding operators, like the summation sign, can be thought of as higher-order functions applying to a function. So, for example, the expression

could be treated as a notation for

where is an operator with two parameters—a one-parameter function, and a set to evaluate that function over. The other operators listed above can be expressed in similar ways; for example, the universal quantifier can be thought of as an operator that evaluates to the logical conjunction of the boolean-valued function *P* applied over the (possibly infinite) set *S*.

When analyzed in formal semantics, natural languages can be seen to have free and bound variables. In English, personal pronouns like *he*, *she*, *they*, etc. can act as free variables.

*Lisa found***her**book.

In the sentence above, the possessive pronoun *her* is a free variable. It may refer to the previously mentioned *Lisa* or to any other female. In other words, *her book* could be referring to Lisa's book (an instance of coreference) or to a book that belongs to a different female (e.g. Jane's book). Whoever the referent of *her* is can be established according to the situational (i.e. pragmatic) context. The identity of the referent can be shown using coindexing subscripts where *i* indicates one referent and *j* indicates a second referent (different from *i*). Thus, the sentence *Lisa found her book* has the following interpretations:

*Lisa*(interpretation #1:_{i}found her_{i}book.*her*= of*Lisa*)*Lisa*(interpretation #2:_{i}found her_{j}book.*her*= of a female that is not Lisa)

The distinction is not purely of academic interest, as some languages do actually have different forms for *her _{i}* and

English does allow specifying coreference, but it is optional, as both interpretations of the previous example are valid (the ungrammatical interpretation is indicated with an asterisk):

*Lisa*(interpretation #1:_{i}found her_{i}own book.*her*= of*Lisa*)- *
*Lisa*(interpretation #2:_{i}found her_{j}own book.*her*= of a female that is not Lisa)

However, reflexive pronouns, such as *himself*, *herself*, *themselves*, etc., and reciprocal pronouns, such as *each other*, act as bound variables. In a sentence like the following:

*Jane hurt***herself**.

the reflexive *herself* can only refer to the previously mentioned antecedent, in this case *Jane*, and can never refer to a different female person. In this example, the variable *herself* is bound to the noun *Jane* that occurs in subject position. Indicating the coindexation, the first interpretation with *Jane* and *herself* coindexed is permissible, but the other interpretation where they are not coindexed is ungrammatical:

*Jane*(interpretation #1:_{i}hurt herself_{i}.*herself*=*Jane*)- *
*Jane*(interpretation #2:_{i}hurt herself_{j}.*herself*= a female that is not Jane)

Note that the coreference binding can be represented using a lambda expression as mentioned in the previous Formal explanation section. The sentence with the reflexive could be represented as

- (λ
*x*.*x*hurt*x*)Jane

in which *Jane* is the subject referent argument and *λx.x hurt x* is the predicate function (a lambda abstraction) with the lambda notation and *x* indicating both the semantic subject and the semantic object of sentence as being bound. This returns the semantic interpretation *JANE hurt JANE* with *JANE* being the same person.

Pronouns can also behave in a different way. In the sentence below

*Ashley hit***her**.

the pronoun *her* can only refer to a female that is not Ashley. This means that it can never have a reflexive meaning equivalent to *Ashley hit herself*. The grammatical and ungrammatical interpretations are:

- *
*Ashley*(interpretation #1:_{i}hit her_{i}.*her*=*Ashley*) *Ashley*(interpretation #2:_{i}hit her_{j}.*her*= a female that is not Ashley)

The first interpretation is impossible. Only the second interpretation is permitted by the grammar.

Thus, it can be seen that reflexives and reciprocals are bound variables (known technically as anaphors) while true pronouns are free variables in some grammatical structures but variables that cannot be bound in other grammatical structures. The binding phenomena found in natural languages was particularly important to the syntactic government and binding theory (see also: Binding (linguistics)).

**First-order logic**—also known as **predicate logic**, **quantificational logic**, and **first-order predicate calculus**—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists*"* is a quantifier, while *x* is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

**Lambda 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 that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.

In programming languages, a **closure**, also **lexical closure** or **function closure**, is a technique for implementing lexically scoped name binding in a language with first-class functions. Operationally, a closure is a record storing a function together with an environment. The environment is a mapping associating each free variable of the function with the value or reference to which the name was bound when the closure was created. Unlike a plain function, a closure allows the function to access those *captured variables* through the closure's copies of their values or references, even when the function is invoked outside their scope.

**Combinatory logic** is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on **combinators** which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.

In mathematics and computer science in general, a *fixed point* of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a **fixed-point combinator** is a higher-order function that returns some fixed point of its argument function, if one exists.

In mathematics, an **expression** or **mathematical expression** is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations, and other aspects of logical syntax.

**System F**, also known as the (**Girard–Reynolds**) **polymorphic lambda calculus** or the **second-order lambda calculus**, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974).

In functional analysis, a branch of mathematics, the **Borel functional calculus** is a *functional calculus*, which has particularly broad scope. Thus for instance if *T* is an operator, applying the squaring function *s* → *s*^{2} to *T* yields the operator *T*^{2}. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential

The spectrum of a linear operator that operates on a Banach space consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard **decomposition** into three parts:

In mathematical logic, the **De Bruijn index** is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as that for syntactic equality. Each De Bruijn index is a natural number that represents an occurrence of a variable in a λ-term, and denotes the number of binders that are in scope between that occurrence and its corresponding binder. The following are some examples:

**Combinatory categorial grammar** (**CCG**) is an efficiently parsable, yet linguistically expressive grammar formalism. It has a transparent interface between surface syntax and underlying semantic representation, including predicate-argument structure, quantification and information structure. The formalism generates constituency-based structures and is therefore a type of phrase structure grammar.

In linguistics, **sloppy identity** is an interpretive property that is found with verb phrase ellipsis where the identity of the pronoun in an elided VP is not identical to the antecedent VP.

In mathematical logic, a **term** denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact.

**Logophoricity** is a phenomenon of binding relation that may employ a morphologically different set of anaphoric forms, in the context where the referent is an entity whose speech, thoughts, or feelings are being reported. This entity may or may not be distant from the discourse, but the referent must reside in a clause external to the one in which the logophor resides. The specially-formed anaphors that are morphologically distinct from the typical pronouns of a language are known as **logophoric pronouns,** originally coined by the linguist Claude Hagège. The linguistic importance of logophoricity is its capability to do away with ambiguity as to who is being referred to. A crucial element of logophoricity is the **logophoric context**, defined as the environment where use of logophoric pronouns is possible. Several syntactic and semantic accounts have been suggested. While some languages may not be purely logophoric, logophoric context may still be found in those languages; in those cases, it is common to find that in the place where logophoric pronouns would typically occur, non-clause-bounded reflexive pronouns appear instead.

In computer science, **partial application** refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function , we might fix the first argument, producing a function of type . Evaluation of this function might be represented as . Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept.

A **Hindley–Milner** (**HM**) **type system** is a classical type system for the lambda calculus with parametric polymorphism. It is also known as **Damas–Milner** or **Damas–Hindley–Milner**. It was first described by J. Roger Hindley and later rediscovered by Robin Milner. Luis Damas contributed a close formal analysis and proof of the method in his PhD thesis.

In applied mathematics, the **Atkinson–Mingarelli theorem**, named after Frederick Valentine Atkinson and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville differential operators.

**Computable topology** is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or *computational topology*, which studies the application of computation to topology.

In logic, a **quantifier** is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier in the first order formula expresses that everything in the domain satisfies the property denoted by . On the other hand, the existential quantifier in the formula expresses that there is something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.

Lambda calculus is a formal mathematical system based on lambda abstraction and function application. Two definitions of the language are given here: a standard definition, and a definition using mathematical formulas.

- ↑ Thompson 1991, p. 33.

- Thompson, Simon (1991).
*Type theory and functional programming*. Wokingham, England: Addison-Wesley. ISBN 0201416670. OCLC 23287456.

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