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.

## History

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.

## Informal description

are equivalent. A function of two variables is expressed in lambda calculus as a function of one argument which returns a function of one argument (see Currying). For instance, the function*f*(

*x*,

*y*) =

*x*-

*y*would be written as λ

*x*. λ

*y*.

*x*-

*y*. The three expressions

- (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*)

*x*.

*x*

*x*) is also known as the ω combinator; ((λ

*x*.

*x*

*x*) (λ

*x*.

*x*

*x*)) is known as Ω, (λ

*x*.

*x*

*x*

*x*) (λ

*x*.

*x*

*x*

*x*) as Ω

_{2}, etc.)

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*)

## Formal definition

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>)

*x*. (

*x*

*x*)) (λ

*y*.

*y*)) can be simply written as (λ

*x*.

*x*

*x*) λ

*y*.

*y*.

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'*.

*alpha-conversion rule*and the

*beta-reduction rule*.

### α-conversion

if*W*does not appear freely in

*E*and

*W*is not bound by a λ in

*E*whenever it replaces a

*V*. This rule tells us for example that λ

*x*. (λ

*x*.

*x*)

*x*is the same as λ

*y*. (λ

*x*.

*x*)

*y*.

### β-reduction

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

if all free occurrences in*E'*remain free in

*E*[

*V*/

*E'*].

The 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.

### η-conversion

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*

*g* *a* for all lambda expressions *a*, then in particular by taking *a* to be a variable *x* not appearing free in *f* we have *f* *x*

*g*

*x*and hence λ

*x*.

*f*

*x*

## λ *x* . *g* *x*, and so by eta-conversion *f*

*g*. So if we take eta-conversion to be valid, we find extensionality is valid.

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

*f* *y*, we have λ *x* . *f* *x*

*f*- i.e. eta-conversion is found to be valid.

## Arithmetic in lambda calculus

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*))

*n*in lambda calculus is a function that takes a function

*f*as argument and returns the

*n*-th power of

*f*. (Note that in Church's original lambda calculus, the formal parameter of a lambda expression was required to occur at least once in the function body, which made the above definition of 0 impossible.) Given this definition of the Church integers, we can define a successor function, which takes a number

*n*and returns

*n*+ 1:

- 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,

*m*and

*n*is the same as

*m*times adding

*n*to zero. Alternatively

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

*n*=

*n*- 1 of a positive integer

*n*is more difficult:

- 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

*g*1) (λ

*u*. PLUS (

*g*

*k*) 1)

*k*which evaluates to

*k*if

*g*(1) is zero and to

*g*(

*k*) + 1 otherwise.

## Logic and predicates

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

A*predicate*is a function which returns a boolean value. The most fundamental predicate is ISZERO which returns true if and only if its argument is zero:

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

## Recursion

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.

*g*which takes a function

*f*as an argument and returns another function

*g*(

*f*). Using the ISZERO predicate, the function

*g*can be defined in lambda calculus. The factorial function is then a

*fixed-point*of

*g*:

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

*every*function in lambda calculus has a fixed point, and the fixed point can be easily described: the fixed point of a function

*g*is given by

- (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*is a fixed point of

*g*, meaning that the two expressions

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

*Y*, every recursively defined function can be expressed as a lambda expression. In particular, we can now cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively.

## Computability and lambda calculus

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.

## Undecidability of 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.

## Lambda calculus and programming languages

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.

## References

- 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*

## External links

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

*Some parts of this article are based on material from FOLDOC, used with permission.*