This article deals with the "untyped lambda calculus" as originally conceived by Church. Since then, some typed lambda calculi have been developed, which are not (yet) covered here.

In lambda calculus, every expression stands for a function with a single argument; the argument of the function is in turn a function with a single argument, and the value of the function is another function with a single argument. Functions are anonymously defined by a lambda expression which expresses the function's action on its argument. For instance, the "addtwo" function f(x) = x + 2 would be expressed in lambda calculus as λ x. x + 2 (or equivalently as λ y. y + 2; the name of the formal argument is immaterial) and the number f(3) would be written as (λ x. x + 2) 3. Function application is left associative: f x y = (f x) y. Consider the function which takes a function as argument and applies it to the argument 3: λ x. x 3. This latter function could be applied to our earlier "add2" function as follows: (λ x. x 3) (λ x. x+2). It is clear that the three expressions
Not every lambda expression can be reduced to a definite value like the ones above; consider for instance
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
If one only formalizes the notion of function application and does not allow lambda expressions, one obtains combinatory logic.
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 contextfree grammar in BNF:
The first two rules generate functions, while the third describes the application of a function to an argument. Usually the brackets for lambda abstraction (rule 2) and function application (rule 3) are omitted if there is no ambiguity under the assumptions that (1) function application is leftassociative, and (2) a lambda binds to the entire expression following it. For example, the expression ((λ 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:
Over the set of lambda expressions an equivalence relation (here denoted as ==) is defined that captures the intuition that two expressions denote the same function. This equivalence relation is defined by the socalled alphaconversion rule and the betareduction rule.
The alphaconversion rule is intended to express the idea that the names of the bound variables are unimportant; for example that λx.x and λy.y are the same function. However, the rule is not as simple as it first appears. There are a number of restrictions on when one bound variable may be replaced with another.
The alphaconversion rule states that if V and W are variables, E is a lambda expression and E[V/W] means the expression E with every free occurrence of V in E replaced with W then
The betareduction rule expresses the idea of function application. It states that
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 (leftmost) 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 ChurchRosser 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 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:
By convention, the following two definitions are used for the boolean values TRUE and FALSE:
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
One may view the righthand side of this definition as a function 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 fixedpoint of 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 ChurchTuring thesis for a discussion of other approaches and their equivalence.
There is no algorithm which takes as input two lambda expressions and output "YES" or "NO" depending on whether or not the two expressions are equivalent. This was historically the first problem for which the unsolvability could be proven. Of course, in order to do so, the notion of "algorithm" has to be cleanly defined; Church used a definition via recursive functions, which is now known to be equivalent to all other reasonable definitions of the notion.
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.
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.
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.
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