Lambda calculus (also written as λcalculus) is a formal system in mathematical logic for expressing computability based on function abstraction and application using variable Name 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.
Lambda calculus consists of constructing lambda terms and performing reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules:

$x$: A variable is a character or string representing a parameter.

$(\backslash lambda\; x.M)$: A lambda abstraction is a function definition, taking as input the bound variable $x$ (between the λ and the punctum/dot .) and returning the body $M$.

$(M\backslash \; N)$: An application, applying a function $M$ to an argument $N$. Both $M$ and $N$ are lambda terms.
The reduction operations include:

$(\backslash lambda\; x.Mx)\backslash rightarrow(\backslash lambda\; y.My)$ , renaming the bound variables in the expression. Used to avoid .

$((\backslash lambda\; x.M)\backslash \; N)\backslash rightarrow\; (Mx:=N)$ , replacing the bound variables with the argument expression in the body of the abstraction.
If De Bruijn indexing is used, then αconversion is no longer required as there will be no name collisions. If repeated application of the reduction steps eventually terminates, then by the Church–Rosser theorem it will produce a βnormal form.
Variable names are not needed if using a universal lambda function, such as Iota and Jot, which can create any function behavior by calling it on itself in various combinations.
Explanation and applications
Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any
Turing machine.
Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
Lambda calculus may be untyped or typed. In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can. On the other hand, typed lambda calculi allow more things to be proven. For example, in the simply typed lambda calculus it is a theorem that every evaluation strategy terminates for every simply typed lambdaterm, whereas evaluation of untyped lambdaterms need not terminate (see below). One reason there are many different typed lambda calculi has been the desire to do more (of what the untyped calculus can do) without giving up on being able to prove strong theorems about the calculus.
Lambda calculus has applications in many different areas in mathematics, philosophy, linguistics,
and
computer science.
[.]
Lambda calculus has played an important role in the development of the theory of programming languages. Functional programming languages implement lambda calculus. Lambda calculus is also a current research topic in
category theory.
History
The lambda calculus was introduced by mathematician
Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.
The original system was shown to be
Consistency in 1935 when
Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox.
Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus.
Until the 1960s when its relation to programming languages was clarified, the lambda calculus was only a formalism. Thanks to Richard Montague and other linguists' applications in the semantics of natural language, the lambda calculus has begun to enjoy a respectable place in both linguistics
and computer science.
Origin of the λ symbol
There is some uncertainty over the reason for Church's use of the Greek letter
lambda (λ) as the notation for functionabstraction in the lambda calculus, perhaps in part due to conflicting explanations by Church himself. According to Cardone and Hindley (2006):
By the way, why did Church choose the notation “λ”? In an he stated clearly that it came from the notation “$\backslash hat\{x\}$” used for classabstraction by Whitehead and Russell, by first modifying “$\backslash hat\{x\}$” to “$\backslash land\; x$” to distinguish functionabstraction from classabstraction, and then changing “$\backslash land$” to “λ” for ease of printing.
This origin was also reported in Rosser,. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and λ just happened to be chosen.
Dana Scott has also addressed this question in various public lectures.
[Dana Scott, " Looking Backward; Looking Forward", Invited Talk at the Workshop in honour of Dana Scott’s 85th birthday and 50 years of domain theory, 7–8 July, FLoC 2018 (talk 7 July 2018). The relevant passage begins at 32:50. (See also this extract of a May 2016 talk at the University of Birmingham, UK.)]
Scott recounts that he once posed a question about the origin of the lambda symbol to Church's former student and soninlaw John W. Addison Jr., who then wrote his fatherinlaw a postcard:
Dear Professor Church,
Russell had the iota operator, Hilbert had the epsilon operator. Why did you choose lambda for your operator?
According to Scott, Church's entire response consisted of returning the postcard with the following annotation: "eeny, meeny, miny, moe".
Informal description
Motivation
Computable functions are a fundamental concept within computer science and mathematics. The lambda calculus provides simple semantics for computation which are useful for formally studying properties of computation. The lambda calculus incorporates two simplifications that make its semantics simple.
The first simplification is that the lambda calculus treats functions "anonymously;" it does not give them explicit names. For example, the function
 $\backslash operatorname\{square\backslash \_sum\}(x,\; y)\; =\; x^2\; +\; y^2$
can be rewritten in
anonymous form as
 $(x,\; y)\; \backslash mapsto\; x^2\; +\; y^2$
(which is read as "a
tuple of and is
maplet to
$x^2\; +\; y^2$"). Similarly, the function
 $\backslash operatorname\{id\}(x)\; =\; x$
can be rewritten in anonymous form as
 $x\; \backslash mapsto\; x$
where the input is simply mapped to itself.
The second simplification is that the lambda calculus only uses functions of a single input. An ordinary function that requires two inputs, for instance the $\backslash operatorname\{square\backslash \_sum\}$ function, can be reworked into an equivalent function that accepts a single input, and as output returns another function, that in turn accepts a single input. For example,
 $(x,\; y)\; \backslash mapsto\; x^2\; +\; y^2$
can be reworked into
 $x\; \backslash mapsto\; (y\; \backslash mapsto\; x^2\; +\; y^2)$
This method, known as
currying, transforms a function that takes multiple arguments into a chain of functions each with a single argument.
Function application of the $\backslash operatorname\{square\backslash \_sum\}$ function to the arguments (5, 2), yields at once
 $((x,\; y)\; \backslash mapsto\; x^2\; +\; y^2)(5,\; 2)$
 $=\; 5^2\; +\; 2^2$
 $=\; 29$,
whereas evaluation of the curried version requires one more step
 $\backslash Bigl(\backslash bigl(x\; \backslash mapsto\; (y\; \backslash mapsto\; x^2\; +\; y^2)\backslash bigr)(5)\backslash Bigr)(2)$
 $=\; (y\; \backslash mapsto\; 5^2\; +\; y^2)(2)$ // the definition of $x$ has been used with $5$ in the inner expression. This is like βreduction.
 $=\; 5^2\; +\; 2^2$ // the definition of $y$ has been used with $2$. Again, similar to βreduction.
 $=\; 29$
to arrive at the same result.
The lambda calculus
The lambda calculus consists of a language of
lambda terms, that are defined by a certain formal syntax, and a set of transformation rules for manipulating the lambda terms. These transformation rules can be viewed as an equational theory or as an operational definition.
As described above, having no names, all functions in the lambda calculus are anonymous functions. They only accept one input variable, so currying is used to implement functions of several variables.
Lambda terms
The syntax of the lambda calculus defines some expressions as valid lambda calculus expressions and some as invalid, just as some strings of characters are valid C programs and some are not. A valid lambda calculus expression is called a "
lambda term".
The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms:

variable is itself a valid lambda term.

if is a lambda term, and is a variable, then $(\backslash lambda\; x.t)$ is a lambda term (called an abstraction);

if and are lambda terms, then $(t$ $s)$ is a lambda term (called an application).
Nothing else is a lambda term. Thus a lambda term is valid if and only if it can be obtained by repeated application of these three rules. However, some parentheses can be omitted according to certain rules. For example, the outermost parentheses are usually not written.
See §Notation, below for when to include parentheses
An ''abstraction'' $\backslash lambda\; x.t$ denotes an § anonymous function that takes a single input and returns . For example, $\backslash lambda\; x.(x^2+2)$ is an abstraction for the function $f(x)\; =\; x^2\; +\; 2$ using the term $x^2+2$ for . The name $f(x)$ is superfluous when using abstraction.
$(\backslash lambda\; x.t)$ binds the variable in the term . The definition of a function with an abstraction merely "sets up" the function but does not invoke it.
See §Notation below for usage of parentheses
An ''application'' $t$ $s$ represents the application of a function to an input , that is, it represents the act of calling function on input to produce $t(s)$.
There is no concept in lambda calculus of variable declaration. In a definition such as $\backslash lambda\; x.(x+y)$ (i.e. $f(x)\; =\; (x\; +\; y)$), in lambda calculus is a variable that is not yet defined. The abstraction $\backslash lambda\; x.(x+y)$ is syntactically valid, and represents a function that adds its input to the yetunknown .
Parentheses may be used and might be needed to disambiguate terms. For example,

$\backslash lambda\; x.((\backslash lambda\; x.x)x)$ which is of form $\backslash lambda\; x.B$ — an abstraction, and

$(\backslash lambda\; x.(\backslash lambda\; x.x))$ $x$ which is of form $M$ $N$ — an application.
The examples 1 and 2 denote different terms; except for the scope of the parentheses they would be the same. But example 1 is a function definition, while example 2 is function application. Lambda variable is a placeholder in both examples.
Here, example 1 defines a function $\backslash lambda\; x.B$, where $B$ is $(\backslash lambda\; x.x)x$, an anonymous function $(\backslash lambda\; x.x)$, with input $x$; while example 2, $M$ $N$, is M applied to N, where $M$ is the lambda term $(\backslash lambda\; x.(\backslash lambda\; x.x))$ being applied to the input $N$ which is $x$. Both examples 1 and 2 would evaluate to the identity function $\backslash lambda\; x.x$.
Functions that operate on functions
In lambda calculus, functions are taken to be 'first class values', so functions may be used as the inputs, or be returned as outputs from other functions.
For example, the lambda term $\backslash lambda\; x.x$ represents the identity function, $x\; \backslash mapsto\; x$. Further, $\backslash lambda\; x.y$ represents the constant function $x\; \backslash mapsto\; y$, the function that always returns $y$, no matter the input. As an example of a function operating on functions, the function composition can be defined as $\backslash lambda\; f.\; \backslash lambda\; g.\; \backslash lambda\; x.\; (f\; (\; g\; x))$.
There are several notions of "equivalence" and "reduction" that allow lambda terms to be "reduced" to "equivalent" lambda terms.
Alpha equivalence
A basic form of equivalence, definable on lambda terms, is alpha equivalence. It captures the intuition that the particular choice of a bound variable, in an abstraction, does not (usually) matter.
For instance,
$\backslash lambda\; x.x$ and
$\backslash lambda\; y.y$ are alphaequivalent lambda terms, and they both represent the same function (the identity function).
The terms
$x$ and
$y$ are not alphaequivalent, because they are not bound in an abstraction.
In many presentations, it is usual to identify alphaequivalent lambda terms.
The following definitions are necessary in order to be able to define βreduction:
Free variables
The
free variables
of a term are those variables not bound by an abstraction. The set of free variables of an expression is defined inductively:

The free variables of $x$ are just $x$

The Set theory of free variables of $\backslash lambda\; x.t$ is the set of free variables of $t$, but with $x$ removed

The Set theory of free variables of $t$ $s$ is the union of the set of free variables of $t$ and the set of free variables of $s$.
For example, the lambda term representing the identity $\backslash lambda\; x.x$ has no free variables, but the function $\backslash lambda\; x.\; y$ $x$ has a single free variable, $y$.
Captureavoiding substitutions
Suppose
$t$,
$s$ and
$r$ are lambda terms, and
$x$ and
$y$ are variables.
The notation
$tx$ indicates substitution of
$r$ for
$x$ in
$t$ in a
captureavoiding manner. This is defined so that:

$xx\; =\; r$ ; with $r$ substituted for $x$, $x$ becomes $r$

$yx\; =\; y$ if $x\; \backslash neq\; y$ ; with $r$ substituted for $x$, $y$ (which is not $x$) remains $y$

$(t$ $s)x\; =\; (tx)(sx)$ ; substitution distributes to both sides of an application

$(\backslash lambda\; x.t)x\; =\; \backslash lambda\; x.t$ ; a variable bound by an abstraction is not subject to substitution; substituting such variable leaves the abstraction unchanged

$(\backslash lambda\; y.t)x\; =\; \backslash lambda\; y.(tx)$ if $x\; \backslash neq\; y$ and $y$ does not appear among the free variables of $r$ ($y$ is said to be "fresh variable" for $r$) ; substituting a variable which is not bound by an abstraction proceeds in the abstraction's body, provided that the abstracted variable $y$ is "fresh" for the substitution term $r$.
For example, $(\backslash lambda\; x.x)y\; =\; \backslash lambda\; x.(xy)\; =\; \backslash lambda\; x.x$, and $((\backslash lambda\; x.y)x)x\; =\; ((\backslash lambda\; x.y)x)(xx)\; =\; (\backslash lambda\; x.y)y$.
The freshness condition (requiring that $y$ is not in the free variables of $r$) is crucial in order to ensure that substitution does not change the meaning of functions.
For example, a substitution that ignores the freshness condition could lead to errors: $(\backslash lambda\; x.y)y\; =\; \backslash lambda\; x.(yy)\; =\; \backslash lambda\; x.x$. This erroneous substitution would turn the constant function $\backslash lambda\; x.y$ into the identity $\backslash lambda\; x.x$.
In general, failure to meet the freshness condition can be remedied by alpharenaming first, with a suitable fresh variable.
For example, switching back to our correct notion of substitution, in $(\backslash lambda\; x.y)y$ the abstraction can be renamed with a fresh variable $z$, to obtain $(\backslash lambda\; z.y)y\; =\; \backslash lambda\; z.(yy)\; =\; \backslash lambda\; z.x$, and the meaning of the function is preserved by substitution.
βreduction
The βreduction rule states that an application of the form
$(\; \backslash lambda\; x\; .\; t)\; s$ reduces to the term
$t$. The notation
$(\; \backslash lambda\; x\; .\; t\; )\; s\; \backslash to\; t$ is used to indicate that
$(\; \backslash lambda\; x\; .t\; )\; s$ βreduces to
$t$.
For example, for every
$s$,
$(\; \backslash lambda\; x\; .\; x\; )\; s\; \backslash to\; x\; =\; s$. This demonstrates that
$\backslash lambda\; x\; .\; x$ really is the identity.
Similarly,
$(\; \backslash lambda\; x\; .\; y\; )\; s\; \backslash to\; y\; =\; y$, which demonstrates that
$\backslash lambda\; x\; .\; y$ is a constant function.
The lambda calculus may be seen as an idealized version of a functional programming language, like Haskell or Standard ML. Under this view, βreduction corresponds to a computational step. This step can be repeated by additional βreductions until there are no more applications left to reduce. In the untyped lambda calculus, as presented here, this reduction process may not terminate. For instance, consider the term $\backslash Omega\; =\; (\backslash lambda\; x\; .\; xx)(\; \backslash lambda\; x\; .\; xx\; )$.
Here $(\; \backslash lambda\; x\; .\; xx)(\; \backslash lambda\; x\; .\; xx)\; \backslash to\; (\; xx\; )\; =\; (\; x\; )(\; x\; )\; =\; (\; \backslash lambda\; x\; .\; xx)(\; \backslash lambda\; x\; .\; xx\; )$.
That is, the term reduces to itself in a single βreduction, and therefore the reduction process will never terminate.
Another aspect of the untyped lambda calculus is that it does not distinguish between different kinds of data. For instance, it may be desirable to write a function that only operates on numbers. However, in the untyped lambda calculus, there is no way to prevent a function from being applied to , strings, or other nonnumber objects.
Formal definition
Definition
Lambda expressions are composed of:

variables v_{1}, v_{2}, ...;

the abstraction symbols λ (lambda) and . (dot);

parentheses ().
The set of lambda expressions, , can be defined inductively:

If x is a variable, then

If x is a variable and then

If then
Instances of rule 2 are known as abstractions and instances of rule 3 are known as applications.
[ftp://ftp.cs.ru.nl/pub/CompMath.Found/ErrataLCalculus.pdf.]
Notation
To keep the notation of lambda expressions uncluttered, the following conventions are usually applied:

Outermost parentheses are dropped: M N instead of ( M N).

Applications are assumed to be left associative: M N P may be written instead of (( M N) P).

When all variables are singleletter, the space in applications may be omitted: MNP instead of M N P.

The body of an abstraction extends as far right as possible: λ x. M N means λ x.( M N) and not (λ x. M) N.

A sequence of abstractions is contracted: λ x.λ y.λ z. N is abbreviated as λ xyz. N.
[
]
Free and bound variables
The abstraction operator, λ, is said to bind its variable wherever it occurs in the body of the abstraction. Variables that fall within the scope of an abstraction are said to be
bound. In an expression λ
x.
M, the part λ
x is often called
binder, as a hint that the variable
x is getting bound by prepending λ
x to
M. All other variables are called
free. For example, in the expression λ
y.
x x y,
y is a bound variable and
x is a free variable. Also a variable is bound by its nearest abstraction. In the following example the single occurrence of
x in the expression is bound by the second lambda: λ
x.
y (λ
x.
z x).
The set of free variables of a lambda expression, M, is denoted as FV( M) and is defined by recursion on the structure of the terms, as follows:

FV( x) = { x}, where x is a variable.

FV(λ x. M) = FV( M) \ { x}.
An expression that contains no free variables is said to be closed. Closed lambda expressions are also known as combinators and are equivalent to terms in combinatory logic.
Reduction
The meaning of lambda expressions is defined by how expressions can be reduced.
There are three kinds of reduction:

αconversion: changing bound variables;

βreduction: applying functions to their arguments;

ηreduction: which captures a notion of extensionality.
We also speak of the resulting equivalences: two expressions are αequivalent, if they can be αconverted into the same expression. βequivalence and ηequivalence are defined similarly.
The term redex, short for reducible expression, refers to subterms that can be reduced by one of the reduction rules. For example, (λ x. M) N is a βredex in expressing the substitution of N for x in M. The expression to which a redex reduces is called its reduct; the reduct of (λ x. M) N is M x.
If x is not free in M, λ x. M x is also an ηredex, with a reduct of M.
αconversion
αconversion (
alphaconversion), sometimes known as αrenaming,
allows bound variable names to be changed. For example, αconversion of λ
x.
x might yield λ
y.
y. Terms that differ only by αconversion are called
αequivalent. Frequently, in uses of lambda calculus, αequivalent terms are considered to be equivalent.
The precise rules for αconversion are not completely trivial. First, when αconverting an abstraction, the only variable occurrences that are renamed are those that are bound to the same abstraction. For example, an αconversion of λ x.λ x. x could result in λ y.λ x. x, but it could not result in λ y.λ x. y. The latter has a different meaning from the original. This is analogous to the programming notion of variable shadowing.
Second, αconversion is not possible if it would result in a variable getting captured by a different abstraction. For example, if we replace x with y in λ x.λ y. x, we get λ y.λ y. y, which is not at all the same.
In programming languages with static scope, αconversion can be used to make name resolution simpler by ensuring that no variable name masks a name in a containing scope (see αrenaming to make name resolution trivial).
In the De Bruijn index notation, any two αequivalent terms are syntactically identical.
Substitution
Substitution, written
M x, is the process of replacing all
free occurrences of the variable
x in the expression
M with expression
N. Substitution on terms of the lambda calculus is defined by recursion on the structure of terms, as follows (note: x and y are only variables while M and N are any lambda expression):
 x x = N
 y x = y, if x ≠ y
 ( M_{1} M_{2}) x = M_{1} x M_{2} x
 (λ x. M) x = λ x. M
 (λ y. M) x = λ y.( M x), if x ≠ y and y ∉ FV( N) See above for the FV
To substitute into an abstraction, it is sometimes necessary to αconvert the expression. For example, it is not correct for (λ x. y) y to result in λ x. x, because the substituted x was supposed to be free but ended up being bound. The correct substitution in this case is λ z. x, up to αequivalence. Substitution is defined uniquely up to αequivalence. See Captureavoiding substitutions above
βreduction
βreduction (
beta reduction) captures the idea of function application. βreduction is defined in terms of substitution: the βreduction of (λ
x.
M)
N is
M x.
For example, assuming some encoding of 2, 7, ×, we have the following βreduction: (λ n. n × 2) 7 → 7 × 2.
βreduction can be seen to be the same as the concept of local reducibility in natural deduction, via the Curry–Howard isomorphism.
ηreduction
ηreduction (
eta reduction) expresses the idea of
extensionality,
[Luke Palmer (29 Dec 2010) Haskellcafe: What's the motivation for η rules?] which in this context is that two functions are the same if and only if they give the same result for all arguments. ηreduction converts between λ
x.
f x and
f whenever
x does not appear free in
f.
ηreduction can be seen to be the same as the concept of local completeness in natural deduction, via the Curry–Howard isomorphism.
Normal forms and confluence
For the untyped lambda calculus, βreduction as a
rewrite system is neither strongly normalising nor weakly normalising.
However, it can be shown that βreduction is confluent when working up to αconversion (i.e. we consider two normal forms to be equal if it is possible to αconvert one into the other).
Therefore, both strongly normalising terms and weakly normalising terms have a unique normal form. For strongly normalising terms, any reduction strategy is guaranteed to yield the normal form, whereas for weakly normalising terms, some reduction strategies may fail to find it.
Encoding datatypes
The basic lambda calculus may be used to model
arithmetic, booleans, data structures, and recursion, as illustrated in the following subsections
i,
ii,
iii, and
§ iv.
Arithmetic in lambda calculus
There are several possible ways to define the
in lambda calculus, but by far the most common are the
, which can be defined as follows:
and so on. Or using the alternative syntax presented above in
Notation:
A Church numeral is a higherorder function—it takes a singleargument function , and returns another singleargument function. The Church numeral is a function that takes a function as argument and returns the th composition of , i.e. the function composed with itself times. This is denoted and is in fact the th power of (considered as an operator); is defined to be the identity function. Such repeated compositions (of a single function ) obey the laws of exponents, which is why these numerals can be used for arithmetic. (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 impossible.)
One way of thinking about the Church numeral , which is often useful when analysing programs, is as an instruction 'repeat n times'. For example, using the and functions defined below, one can define a function that constructs a (linked) list of n elements all equal to x by repeating 'prepend another x element' n times, starting from an empty list. The lambda term is
By varying what is being repeated, and varying what argument that function being repeated is applied to, a great many different effects can be achieved.
We can define a successor function, which takes a Church numeral and returns by adding another application of , where '(mf)x' means the function 'f' is applied 'm' times on 'x':
Because the th composition of composed with the th composition of gives the th composition of , addition can be defined as follows:
can be thought of as a function taking two natural numbers as arguments and returning a natural number; it can be verified that
and
are βequivalent lambda expressions. Since adding to a number can be accomplished by adding 1 times, an alternative definition is:
[; A note (accessed 2017) at the original location suggests that the authors consider the work originally referenced to have been superseded by a book.]
Similarly, multiplication can be defined as
Alternatively
since multiplying and is the same as repeating the add function times and then applying it to zero.
Exponentiation has a rather simple rendering in Church numerals, namely
The predecessor function defined by for a positive integer and is considerably more difficult. The formula
can be validated by showing inductively that if
T denotes , then for . Two other definitions of are given below, one using conditionals and the other using pairs. With the predecessor function, subtraction is straightforward. Defining
 ,
yields when and otherwise.
Logic and predicates
By convention, the following two definitions (known as Church booleans) are used for the boolean values and :
Then, with these two lambda terms, we can define some logic operators (these are just possible formulations; other expressions could be equally correct):
We are now able to compute some logic functions, for example:
 :
 :
and we see that is equivalent to .
A predicate is a function that returns a boolean value. The most fundamental predicate is , which returns if its argument is the Church numeral , but if its argument were any other Church numeral:
The following predicate tests whether the first argument is lessthanorequalto the second:
 ,
and since , if and , it is straightforward to build a predicate for numerical equality.
The availability of predicates and the above definition of and make it convenient to write "ifthenelse" expressions in lambda calculus. For example, the predecessor function can be defined as:
which can be verified by showing inductively that is the add − 1 function for > 0.
Pairs
A pair (2tuple) can be defined in terms of and , by using the Church encoding for pairs. For example, encapsulates the pair (,), returns the first element of the pair, and returns the second.
A linked list can be defined as either NIL for the empty list, or the of an element and a smaller list. The predicate tests for the value . (Alternatively, with , the construct obviates the need for an explicit NULL test).
As an example of the use of pairs, the shiftandincrement function that maps to can be defined as
which allows us to give perhaps the most transparent version of the predecessor function:
Additional programming techniques
There is a considerable body of programming idioms for lambda calculus. Many of these were originally developed in the context of using lambda calculus as a foundation for programming language semantics, effectively using lambda calculus as a lowlevel programming language. Because several programming languages include the lambda calculus (or something very similar) as a fragment, these techniques also see use in practical programming, but may then be perceived as obscure or foreign.
Named constants
In lambda calculus, a library would take the form of a collection of previously defined functions, which as lambdaterms are merely particular constants. The pure lambda calculus does not have a concept of named constants since all atomic lambdaterms are variables, but one can emulate having named constants by setting aside a variable as the name of the constant, using abstraction to bind that variable in the main body, and apply that abstraction to the intended definition. Thus to use to mean
N (some explicit lambdaterm) in
M (another lambdaterm, the "main program"), one can say
 M N
Authors often introduce
syntactic sugar, such as , to permit writing the above in the more intuitive order
 N M
By chaining such definitions, one can write a lambda calculus "program" as zero or more function definitions, followed by one lambdaterm using those functions that constitutes the main body of the program.
A notable restriction of this is that the name be not defined in N, for N to be outside the scope of the abstraction binding ; this means a recursive function definition cannot be used as the N with . The construction would allow writing recursive function definitions.
Recursion and fixed points
Recursion is the definition of a function using the function itself. A definition containing itself inside itself, by value, leads to the whole value being of infinite size. Other notations which support recursion natively overcome this by referring to the function definition
by name. Lambda calculus cannot express this: all functions are anonymous in lambda calculus, so we can't refer by name to a value which is yet to be defined, inside the lambda term defining that same value. However, a lambda expression can receive itself as its own argument, for example in . Here
E should be an abstraction, applying its parameter to a value to express recursion.
Consider the factorial function recursively defined by
 .
In the lambda expression which is to represent this function, a parameter (typically the first one) will be assumed to receive the lambda expression itself as its value, so that calling it – applying it to an argument – will amount to recursion. Thus to achieve recursion, the intendedasselfreferencing argument (called here) must always be passed to itself within the function body, at a call point:
 :: with to hold, so and
The selfapplication achieves replication here, passing the function's lambda expression on to the next invocation as an argument value, making it available to be referenced and called there.
This solves it but requires rewriting each recursive call as selfapplication. We would like to have a generic solution, without a need for any rewrites:
 :: with to hold, so and
 where
 :: so that
Given a lambda term with first argument representing recursive call (e.g. here), the fixedpoint combinator will return a selfreplicating lambda expression representing the recursive function (here, ). The function does not need to be explicitly passed to itself at any point, for the selfreplication is arranged in advance, when it is created, to be done each time it is called. Thus the original lambda expression is recreated inside itself, at callpoint, achieving selfreference.
In fact, there are many possible definitions for this operator, the simplest of them being:
In the lambda calculus, is a fixedpoint of , as it expands to:
Now, to perform our recursive call to the factorial function, we would simply call , where n is the number we are calculating the factorial of. Given n = 4, for example, this gives:
Every recursively defined function can be seen as a fixed point of some suitably defined function closing over the recursive call with an extra argument, and therefore, using , 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.
Standard terms
Certain terms have commonly accepted names:
is the identity function. and form complete combinator calculus systems that can express any lambda term  see
the next section. is , the smallest term that has no normal form. is another such term.
is standard and defined above, and can also be defined as , so that . and defined above are commonly abbreviated as and .
Abstraction elimination
If
N is a lambdaterm without abstraction, but possibly containing named constants (combinators), then there exists a lambdaterm
T(,
N) which is equivalent to
N but lacks abstraction (except as part of the named constants, if these are considered nonatomic). This can also be viewed as anonymising variables, as
T(,
N) removes all occurrences of from
N, while still allowing argument values to be substituted into the positions where
N contains an . The conversion function
T can be defined by:
 T(, ) := I
 T(, N) := K N if is not free in N.
 T(, M N) := S T(, M) T(, N)
In either case, a term of the form
T(,
N)
P can reduce by having the initial combinator
I,
K, or
S grab the argument
P, just like βreduction of
N P would do.
I returns that argument.
K throws the argument away, just like
N would do if has no free occurrence in
N.
S passes the argument on to both subterms of the application, and then applies the result of the first to the result of the second.
The combinators B and C are similar to S, but pass the argument on to only one subterm of an application ( B to the "argument" subterm and C to the "function" subterm), thus saving a subsequent K if there is no occurrence of in one subterm. In comparison to B and C, the S combinator actually conflates two functionalities: rearranging arguments, and duplicating an argument so that it may be used in two places. The W combinator does only the latter, yielding the B, C, K, W system as an alternative to SKI combinator calculus.
Typed lambda calculus
A
typed lambda calculus is a typed formalism that uses the lambdasymbol (
$\backslash lambda$) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see Kinds of typed lambda calculi). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and
untyped lambda calculus a special case with only one type.
[Types and Programming Languages, p. 273, Benjamin C. Pierce]
Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here typability usually captures desirable properties of the program, e.g. the program will not cause a memory access violation.
Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry–Howard isomorphism and they can be considered as the internal language of classes of category theory, e.g. the simply typed lambda calculus is the language of Cartesian closed categories (CCCs).
Reduction strategies
Whether a term is normalising or not, and how much work needs to be done in normalising it if it is, depends to a large extent on the reduction strategy used. Common lambda calculus reduction strategies include:
 Normal order
 The leftmost, outermost redex is always reduced first. That is, whenever possible the arguments are substituted into the body of an abstraction before the arguments are reduced.
 Applicative order
 The leftmost, innermost redex is always reduced first. Intuitively this means a function's arguments are always reduced before the function itself. Applicative order always attempts to apply functions to normal forms, even when this is not possible.
 Full βreductions
 Any redex can be reduced at any time. This means essentially the lack of any particular reduction strategy—with regard to reducibility, "all bets are off".
Weak reduction strategies do not reduce under lambda abstractions:
 Call by value
 A redex is reduced only when its right hand side has reduced to a value (variable or abstraction). Only the outermost redexes are reduced.
 Call by name
 As normal order, but no reductions are performed inside abstractions. For example, is in normal form according to this strategy, although it contains the redex .
Strategies with sharing reduce computations that are "the same" in parallel:
 Optimal reduction
 As normal order, but computations that have the same label are reduced simultaneously.
 Call by need
 As call by name (hence weak), but function applications that would duplicate terms instead name the argument, which is then reduced only "when it is needed".
Computability
There is no algorithm that takes as input any two lambda expressions and outputs or depending on whether one expression reduces to the other.
[ More precisely, no computable function can decision problem the question. This was historically the first problem for which undecidability could be proven. As usual for such a proof, computable means computable by any model of computation that is Turing complete. In fact computability can itself be defined via the lambda calculus: a function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F( x)= y if and only if f =_{β} , where and are the corresponding to x and y, respectively and =_{β} meaning equivalence with βreduction. See the Church–Turing thesis for other approaches to defining computability and their equivalence.
]
Church's proof of uncomputability first reduces the problem to determining whether a given lambda expression has a normal form. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and constructing a Gödel numbering for lambda expressions, he constructs a lambda expression that closely follows the proof of Gödel's first incompleteness theorem. If is applied to its own Gödel number, a contradiction results.
Complexity
The notion of computational complexity for the lambda calculus is a bit tricky, because the cost of a βreduction may vary depending on how it is implemented.
To be precise, one must somehow find the location of all of the occurrences of the bound variable in the expression , implying a time cost, or one must keep track of the locations of free variables in some way, implying a space cost. A naïve search for the locations of in is O( n) in the length n of . were an early approach that traded this time cost for a quadratic space usage. More generally this has led to the study of systems that use explicit substitution.
In 2014, it was shown that the number of βreduction steps taken by normal order reduction to reduce a term is a reasonable time cost model, that is, the reduction can be simulated on a Turing machine in time polynomially proportional to the number of steps.
This was a longstanding open problem, due to
size explosion, the existence of lambda terms which grow exponentially in size for each βreduction. The result gets around this by working with a compact shared representation. The result makes clear that the amount of space needed to evaluate a lambda term is not proportional to the size of the term during reduction. It is not currently known what a good measure of space complexity would be.
An unreasonable model does not necessarily mean inefficient. Optimal reduction reduces all computations with the same label in one step, avoiding duplicated work, but the number of parallel βreduction steps to reduce a given term to normal form is approximately linear in the size of the term. This is far too small to be a reasonable cost measure, as any Turing machine may be encoded in the lambda calculus in size linearly proportional to the size of the Turing machine. The true cost of reducing lambda terms is not due to βreduction per se but rather the handling of the duplication of redexes during βreduction. It is not known if optimal reduction implementations are reasonable when measured with respect to a reasonable cost model such as the number of leftmostoutermost steps to normal form, but it has been shown for fragments of the lambda calculus that the optimal reduction algorithm is efficient and has at most a quadratic overhead compared to leftmostoutermost.[ In addition the BOHM prototype implementation of optimal reduction outperformed both Caml and Haskell on pure lambda terms.][
]
Lambda calculus and programming languages
As pointed out by
Peter Landin's 1965 paper "A Correspondence between ALGOL 60 and Church's Lambdanotation",
sequential procedural programming languages can be understood in terms of the lambda calculus, which provides the basic mechanisms for procedural abstraction and procedure (subprogram) application.
Anonymous functions
For example, in Python the "square" function can be expressed as a lambda expression as follows:
(lambda x: x**2)
The above example is an expression that evaluates to a firstclass function. The symbol lambda creates an anonymous function, given a list of parameter names, x – just a single argument in this case, and an expression that is evaluated as the body of the function, x**2. Anonymous functions are sometimes called lambda expressions.
For example, Pascal and many other imperative languages have long supported passing subprograms as arguments to other subprograms through the mechanism of function pointers. However, function pointers are not a sufficient condition for functions to be first class datatypes, because a function is a first class datatype if and only if new instances of the function can be created at runtime. And this runtime creation of functions is supported in Smalltalk, JavaScript and Wolfram Language, and more recently in Scala, Eiffel ("agents"), C# ("delegates") and C++11, among others.
Parallelism and concurrency
The Church–Rosser property of the lambda calculus means that evaluation (βreduction) can be carried out in
any order, even in parallel. This means that various nondeterministic evaluation strategies are relevant. However, the lambda calculus does not offer any explicit constructs for parallelism. One can add constructs such as Futures to the lambda calculus. Other
process calculi have been developed for describing communication and concurrency.
Semantics
The fact that lambda calculus terms act as functions on other lambda calculus terms, and even on themselves, led to questions about the semantics of the lambda calculus. Could a sensible meaning be assigned to lambda calculus terms? The natural semantics was to find a set
D isomorphic to the function space
D →
D, of functions on itself. However, no nontrivial such
D can exist, by
cardinality constraints because the set of all functions from
D to
D has greater cardinality than
D, unless
D is a
singleton set.
In the 1970s, Dana Scott showed that if only Scott continuity were considered, a set or Domain theory D with the required property could be found, thus providing a Model theory for the lambda calculus.[ Written 1969, widely circulated as an unpublished manuscript.]
This work also formed the basis for the denotational semantics of programming languages.
Variations and extensions
These extensions are in the
lambda cube:

Typed lambda calculus – Lambda calculus with typed variables (and functions)

System F – A typed lambda calculus with typevariables

Calculus of constructions – A typed lambda calculus with type system as firstclass values
These are extensions of lambda calculus that are not in the lambda cube:

Binary lambda calculus – A version of lambda calculus with binary I/O, a binary encoding of terms, and a designated universal machine.

Lambdamu calculus – An extension of the lambda calculus for treating classical logic
These formal systems are variations of lambda calculus:
These formal systems are related to lambda calculus:

Combinatory logic – A notation for mathematical logic without variables

SKI combinator calculus – A computational system based on the S, K and I combinators, equivalent to lambda calculus, but reducible without variable substitutions
See also

Applicative computing systems – Treatment of objects in the style of the lambda calculus

Cartesian closed category – A setting for lambda calculus in category theory

Categorical abstract machine – A model of computation applicable to lambda calculus

Clojure, programming language

Curry–Howard isomorphism – The formal correspondence between programs and proofs

De Bruijn index – notation disambiguating alpha conversions

De Bruijn notation – notation using postfix modification functions

Deductive lambda calculus – The consideration of the problems associated with considering lambda calculus as a Deductive system.

Domain theory – Study of certain posets giving denotational semantics for lambda calculus

Evaluation strategy – Rules for the evaluation of expressions in programming languages

Explicit substitution – The theory of substitution, as used in βreduction

Functional programming

Harrop formula – A kind of constructive logical formula such that proofs are lambda terms

Interaction nets

Kleene–Rosser paradox – A demonstration that some form of lambda calculus is inconsistent

Knights of the Lambda Calculus – A semifictional organization of LISP and Scheme hackers

Krivine machine – An abstract machine to interpret callbyname in lambda calculus

Lambda calculus definition – Formal definition of the lambda calculus.

Let expression – An expression closely related to an abstraction.

Minimalism (computing)

Rewriting – Transformation of formulæ in formal systems

SECD machine – A virtual machine designed for the lambda calculus

Scott–Curry theorem – A theorem about sets of lambda terms

To Mock a Mockingbird – An introduction to combinatory logic

Universal Turing machine – A formal computing machine that is equivalent to lambda calculus

Unlambda – An esoteric functional programming language based on combinatory logic
Further reading

Abelson, Harold & Gerald Jay Sussman. Structure and Interpretation of Computer Programs. The MIT Press. .

Henk Barendregt Introduction to Lambda Calculus.

Barendregt, Hendrik Pieter, The Impact of the Lambda Calculus in Logic and Computer Science. The Bulletin of Symbolic Logic, Volume 3, Number 2, June 1997.

Barendregt, Hendrik Pieter, The Type Free Lambda Calculus pp1091–1132 of Handbook of Mathematical Logic, NorthHolland (1977)

Cardone and Hindley, 2006. History of Lambdacalculus and Combinatory Logic . In Gabbay and Woods (eds.), Handbook of the History of Logic, vol. 5. Elsevier.

Church, Alonzo, 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.

()

Kleene, Stephen, 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.

Peter Landin, A Correspondence Between ALGOL 60 and Church's LambdaNotation, Communications of the ACM, vol. 8, no. 2 (1965), pages 89–101. Available from the ACM site. A classic paper highlighting the importance of lambda calculus as a basis for programming languages.

Larson, Jim, An Introduction to Lambda Calculus and Scheme. A gentle introduction for programmers.


Schalk, A. and Simmons, H. (2005) '' An introduction to λcalculi and arithmetic with a decent selection of exercises. Notes for a course in the Mathematical Logic MSc at Manchester University.

A paper giving a formal underpinning to the idea of 'meaningisuse' which, even if based on proofs, it is different from prooftheoretic semantics as in the Dummett–Prawitz tradition since it takes reduction as the rules giving meaning.

Hankin, Chris, An Introduction to Lambda Calculi for Computer Scientists,
 Monographs/textbooks for graduate students

Sørensen, Morten Heine and Urzyczyn, Paweł (2006), Lectures on the Curry–Howard isomorphism, Elsevier, is a recent monograph that covers the main topics of lambda calculus from the typefree variety, to most typed lambda calculi, including more recent developments like pure type systems and the lambda cube. It does not cover subtyping extensions.

covers lambda calculi from a practical type system perspective; some topics like dependent types are only mentioned, but subtyping is an important topic.
 Documents
Notes
Some parts of this article are based on material from FOLDOC, used with permission.
External links

Graham Hutton, Lambda Calculus, a short (12 minutes) Computerphile video on the Lambda Calculus

Helmut Brandl, Step by Step Introduction to Lambda Calculus

David C. Keenan, To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction

L. Allison, Some executable λcalculus examples

Georg P. Loczewski, The Lambda Calculus and A++

Bret Victor, Alligator Eggs: A Puzzle Game Based on Lambda Calculus

Lambda Calculus on Safalra's Website

LCI Lambda Interpreter a simple yet powerful pure calculus interpreter

Lambda Calculus links on LambdatheUltimate

Mike Thyer, Lambda Animator, a graphical Java applet demonstrating alternative reduction strategies.

Implementing the Lambda calculus using C++ Templates

Shane SteinertThrelkeld, "Lambda Calculi", Internet Encyclopedia of Philosophy

Anton Salikhmetov, Macro Lambda Calculus