The **lambda calculus** is a formal system designed to investigate function definition, function application and recursion. It was introduced by Alonzo Church and Stephen Kleene in the 1930s; Church used the lambda calculus in 1936 to give a negative answer to the Entscheidungsproblem. The calculus can be used to cleanly define what a "computable function" is. The question of whether two lambda calculus expressions are equivalent cannot be solved by a general algorithm, and this was the first question, even before the halting problem, for which undecidability could be proved. Lambda calculus has greatly influenced functional programming languages, especially LISP.

This article deals with the "untyped lambda calculus" as originally conceived by Church. Since then, some typed lambda calculi have been developed.

Originally, Church had tried to construct a complete formal system for the foundations of mathematics; when the system turned out to be susceptible to the analog of Russell's paradox, he separated out the lambda calculus and used it to study computability, culminating in his negative answer to the Entscheidungsproblem.

- (λ
*x*. λ*y*.*x*-*y*) 7 2 and (λ*y*. 7 -*y*) 2 and 7 - 2

Not every lambda expression can be reduced to a definite value like the ones above; consider for instance

- (λ
*x*.*x**x*) (λ*x*.*x**x*)

- (λ
*x*.*x**x**x*) (λ*x*.*x**x**x*)

While the lambda calculus itself does not contain symbols for integers or addition, these can be defined as abbreviations within the calculus and arithmetic can be expressed as we will see below.

Lambda calculus expressions may contain *free variables*, i.e. variables not bound by any λ. For example, the variable *y* is free in the expression (λ *x*. *y*), representing a function which always produces the result *y*. Occasionally, this necessitates the renaming of formal arguments, for instance in order to reduce

- (λ
*x*. λ*y*.*y**x*) (λ*x*.*y*) to λ*z*.*z*(λ*x*.*y*)

Formally, we start with a countably infinite set of identifiers, say {a, b, c, ..., x, y, z, x_{1}, x_{2}, ...}. The set of all lambda expressions can then be described by the following context-free grammar in BNF:

- <expr> → <identifier>
- <expr> → (λ <identifier> . <expr>)
- <expr> → (<expr> <expr>)

Lambda expressions such as λ *x*. (*x* *y*) do not define a function because the occurrence of the variable *y* is *free*, i.e., it is not *bound* by any λ in the expression. The binding of occurrences of variables is (with induction upon the structure of the lambda expression) defined by the following rules:

- In an expression of the form
*V*where*V*is a variable this*V*is the single free occurrence. - In an expression of the form λ
*V*.*E*the free occurrences are the free occurrences in*E*except those of*V*. In this case the occurrences of*V*in*E*are said to be bound by the λ before*V*. - In an expresssion of the form (
*E**E'*) the free occurrences are the free occurrences in*E*and*E'*.

The beta-reduction rule expresses the idea of function application. It states that

if all free occurrences inThe relation == is then defined as the smallest equivalence relation that satisfies these two rules.

A more operational definition of the equivalence relation can be given by applying the rules only from left to right. A lambda expression which does not allow any beta reduction, i.e., has no subexpression of the form ((λ *V*. *E*) *E' *), is called a *normal form*. Not every lambda expression is equivalent to a normal form, but if it is, then the normal form is unique up to naming of the formal arguments. Furthermore, there is an algorithm for computing normal forms: keep replacing the first (left-most) formal argument with its corresponding concrete argument, until no further reduction is possible. This algorithm halts if and only if the lambda expression has a normal form. The Church-Rosser theorem then states that two expressions result in the same normal form up to renaming of the formal arguments if and only if they are equivalent.

There is third rule, eta-conversion, which may be added to these two to form a new equivalence relation. Eta-conversion expresses the idea of extensionality, which in this context is that two functions are the same iff they give the same result for all arguments. Eta-conversion converts between λ *x* . *f* *x* and *f*, whenever *x* does not appear free in *f*. This can be seen to be equivalent to extensionality as follows:

If *f* and *g* are extensionally equivalent, i.e. if *f* *a*

Conversely if extensionality is taken to be valid, then since by beta-reduction for all *y* we have (λ *x* . *f* *x*) *y*

There are several possible ways to define the natural numbers in lambda calculus, but by far the most common are the Church integers, which can be defined as follows:

- 0 = λ
*f*. λ*x*.*x* - 1 = λ
*f*. λ*x*.*f**x* - 2 = λ
*f*. λ*x*.*f*(*f**x*) - 3 = λ
*f*. λ*x*.*f*(*f*(*f**x*))

- SUCC = λ
*n*. λ '\'f*. λ*x*.*f*(*n

- PLUS = λ
*m*. λ*n*. λ*f*. λ*x*.*m**f*(*n**f**x*)

- PLUS 2 3 and 5

- MULT = λ
*m*. λ*n*.*m*(PLUS*n*) 0,

- MULT = λ
*m*. λ*n*. λ*f*.*m*(*n**f*)

- PRED = λ
*n*. λ*f*. λ*x*.*n*(λ*g*. λ*h*.*h*(*g**f*)) (λ*u*.*x*) (λ*u*.*u*)

- PRED = λ
*n*.*n*(λ*g*. λ*k*. (*g*1) (λ*u*. PLUS (*g**k*) 1)*k*) (λ*l*. 0) 0

By convention, the following two definitions are used for the boolean values TRUE and FALSE:

A- ISZERO = λ
*n*.*n*(λ*x*. FALSE) TRUE

Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this. However, this impression is misleading. Consider for instance the factorial function *f*(*n*) recursively defined by

*f*(*n*) = 1, if*n*= 0; and*n*·*f*(*n*-1), if*n*>0.

*f*=*g*(*f*).

- (λ
*x*.*g*(*x**x*)) (λ*x*.*g*(*x**x*))

*Y*= λ*g*. (λ*x*.*g*(*x**x*)) (λ*x*.*g*(*x**x*))

*Y**g*and*g*(*Y**g*)

A function *F* : **N** → **N** of natural numbers is defined to be *computable* if there exists a lambda expression *f* such that for every pair of *x*, *y* in **N**, *F*(*x*) = *y* if and only if the expressions *f* *x* and *y* are equivalent. This is one of the many ways to define computability; see the Church-Turing thesis for a discussion of other approaches and their equivalence.

Church's proof first reduces the problem to determining whether a given lambda expression has a *normal form*. A normal form is an equivalent expression which cannot be reduced any further. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and utilizing Gödel's procedure of Gödel numbers for lambda expressions, he constructs a lambda expression *e* which closely follows the proof of Gödel's first incompleteness theorem. If *e* is applied to its own Gödel number, a contradiction results.

Most functional programming languages are equivalent to lambda calculus extended with constants and datatypes. LISP uses a variant of lambda notation for defining functions but only its purely functional subset is really equivalent to lambda calculus.

- Stephen Kleene,
*A theory of positive integers in formal logic*, American Journal of Mathematics, 57 (1935), pp 153 - 173 and 219 - 244. Contains the lambda calculus definitions of several familiar functions. - Alonzo Church,
*An unsolvable problem of elementary number theory*, American Journal of Mathematics, 58 (1936), pp 345 - 363. This paper contains the proof that the equivalence of lambda expressions is in general not decidable. - Jim Larson,
*An Introduction to Lambda Calculus and Scheme*. A gentle introduction for programmers. - Martin Henz,
*The Lambda Calculus*. Formally correct development of the Lambda calculus. - Henk Barendregt,
*The lambda calculus, its syntax and semantics*, North-Holland (1984), is*the*comprehensive reference on the (untyped) lambda calculus. - Amit Gupta and Ashutosh Agte,
*Untyped lambda-calculus, alpha-, beta- and eta- reductions and recursion*

- L. Allison,
*Some executable λ-calculus examples* - Georg P. Loczewski, ''The Lambda Calculus and A++