untyped lambda calculus

Lambda calculus (also written as ''λ''-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.

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:

* $x$ – variable, a character or string representing a parameter or mathematical/logical value.
* $(\lambda x.M)$ – abstraction, function definition ($M$ is a lambda term). The variable $x$ becomes bound in the expression.
* $(M\ N)$ – application, applying a function $M$ to an argument $N$. $M$ and $N$ are lambda terms.

The reduction operations include:

* (\lambda x.M \rightarrow(\lambda y.M – α-conversion, renaming the bound variables in the expression. Used to avoid name collisions.
* ((\lambda x.M)\ E)\rightarrow (M :=E – β-reduction, replacing the bound variables with the argument expression in the body of the abstraction.

If De Bruijn indexing is used, then α-conversion 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 β-normal 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, but on the other hand typed lambda calculi allow more things to be proven; in the simply typed lambda calculus it is, for example, a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms §need not terminate. 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 logically inconsistent 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 lambda symbol

There is some uncertainty over the reason for Church's use of the Greek letter lambda (λ) as the notation for function-abstraction 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 n unpublished 1964 letter to Harald Dicksonhe stated clearly that it came from the notation “$\hat$” used for class-abstraction by Whitehead and Russell, by first modifying “$\hat$” to “$\land x$” to distinguish function-abstraction from class-abstraction, and then changing “$\land$” to “λ” for ease of printing.

This origin was also reported in osser, 1984, p.338 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.
Scott recounts that he once posed a question about the origin of the lambda symbol to Church's former student and son-in-law John W. Addison Jr., who then wrote his father-in-law 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
: $\operatorname(x, y) = x^2 + y^2$
can be rewritten in ''anonymous form'' as
: $(x, y) \mapsto x^2 + y^2$
(which is read as "a tuple of and is mapped to $x^2 + y^2$"). Similarly, the function
: $\operatorname(x) = x$
can be rewritten in anonymous form as
: $x \mapsto x$
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 $\operatorname$ 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,
: $(x, y) \mapsto x^2 + y^2$
can be reworked into
: $x \mapsto (y \mapsto x^2 + y^2)$
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 $\operatorname$ function to the arguments (5, 2), yields at once
: $((x, y) \mapsto x^2 + y^2)(5, 2)$
: $= 5^2 + 2^2$
: $= 29$,
whereas evaluation of the curried version requires one more step
: $\Bigl(\bigl(x \mapsto (y \mapsto x^2 + y^2)\bigr)(5)\Bigr)(2)$
: $= (y \mapsto 5^2 + y^2)(2)$ // the definition of $x$ has been used with $5$ in the inner expression. This is like β-reduction.
: $= 5^2 + 2^2$ // the definition of $y$ has been used with $2$. Again, similar to β-reduction.
: $= 29$
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 $(\lambda x.t)$ 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.

An abstraction $\lambda x.t$ denotes an § anonymous function that takes a single input and returns . For example, $\lambda x.x^2+2$ 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.
$(\lambda x.t)$ binds the variable in the term . The definition of a function with an abstraction merely "sets up" the function but does not invoke it.

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 $\lambda x.x+y$ (i.e. $f(x) = x + y$), in lambda calculus is a variable that is not yet defined. The abstraction $\lambda x.x+y$ is syntactically valid, and represents a function that adds its input to the yet-unknown .

Parentheses may be used and may be needed to disambiguate terms. For example,
#$\lambda x.((\lambda x.x)x)$ which is of form $\lambda x.B$ —an abstraction, and
#$((\lambda x.x)x)$ which is of form $M N$ —an application. The examples 1 and 2 denote different terms; however example 1 is a function definition, while example 2 is an application.

Here, example 1 defines a function $\lambda x.B$, where $B$ is $((\lambda x.x)x)$, the result of applying $(\lambda x.x)$ to x, while example 2 is $M N$; $M$ is the lambda term $(\lambda x.x)$ to be applied to the input N. Both examples 1 and 2 would evaluate to the identity function $\lambda x.x$.

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, $\lambda x.x$ represents the identity function, $x \mapsto x$, and $(\lambda x.x)y$ represents the identity function applied to $y$. Further, $(\lambda x.y)$ represents the constant function $x \mapsto y$, the function that always returns $y$, no matter the input. In lambda calculus, function application is regarded as left-associative, so that $stx$ means $(st)x$.

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, $\lambda x.x$ and $\lambda y.y$ are alpha-equivalent lambda terms, and they both represent the same function (the identity function).
The terms $x$ and $y$ are not alpha-equivalent, because they are not bound in an abstraction.
In many presentations, it is usual to identify alpha-equivalent lambda terms.

The following definitions are necessary in order to be able to define β-reduction:

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 of free variables of $\lambda x.t$ is the set of free variables of $t$, but with $x$ removed
* The set 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 $\lambda x.x$ has no free variables, but the function $\lambda x. y$ $x$ has a single free variable, $y$.

Capture-avoiding substitutions

A systematic change in variables to avoid capture of a free variable can introduce error, in a functional programming language where functions are first class citizens.

Suppose $t$, $s$ and $r$ are lambda terms and $x$ and $y$ are variables.
The notation t := r/math> indicates substitution of $r$ for $x$ in $t$ in a ''capture-avoiding'' manner. This is defined so that:
* x := r= r; $x$ substituted for $r$ is just $r$
* y := r= y if $x \neq y$; $x$ substituted for $r$ when dealing with $y$ is just $y$
* $(t$ s) := r= (t := r(s := r; substitution distributes to further application of the variable
* (\lambda x.t) := r= \lambda x.t; although $x$ has been mapped to $r$, subsequently mapping all $x$ to $t$ will not change the lambda function $(\lambda x.t)$
* (\lambda y.t) := r= \lambda y.(t := r if $x \neq y$ and $y$ is not in the free variables of $r$. The variable $y$ is said to be "fresh" for $r$.

For example, (\lambda x.x) := y= \lambda x.(x := y = \lambda x.x, and ((\lambda x.y)x) := y= ((\lambda x.y) := y(x := y = (\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 can lead to errors: (\lambda x.y) := x= \lambda x.(y := x = \lambda x.x. This substitution turns the constant function $\lambda x.y$ into the identity $\lambda x.x$ by substitution.

In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable.
For example, switching back to our correct notion of substitution, in (\lambda x.y) := x/math> the abstraction can be renamed with a fresh variable $z$, to obtain (\lambda z.y) := x= \lambda z.(y := x = \lambda z.x, and the meaning of the function is preserved by substitution.

β-reduction

The β-reduction rule states that an application of the form $( \lambda x . t) s$ reduces to the term t x := s/math>. The notation ( \lambda x . t ) s \to t x := s is used to indicate that $( \lambda x .t ) s$ β-reduces to t x := s .
For example, for every $s$, ( \lambda x . x ) s \to x x := s = s . This demonstrates that $\lambda x . x$ really is the identity.
Similarly, ( \lambda x . y ) s \to y x := s = y , which demonstrates that $\lambda x . y$ 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, β-reduction corresponds to a computational step. This step can be repeated by additional β-reductions 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 $\Omega = (\lambda x . xx)( \lambda x . xx )$.
Here ( \lambda x . xx)( \lambda x . xx) \to ( xx ) x := \lambda x . xx = ( x x := \lambda x . xx )( x x := \lambda x . xx ) = ( \lambda x . xx)( \lambda x . xx ).
That is, the term reduces to itself in a single β-reduction, 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 truth values, strings, or other non-number 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''.

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 single-letter, 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 appending 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'') = , where ''x'' is a variable.
# FV(λ''x''.''M'') = FV(''M'') \ .
#

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:
* α-conversion: changing bound variables;
* β-reduction: applying functions to their arguments;
* η-reduction: which captures a notion of extensionality.

We also speak of the resulting equivalences: two expressions are ''α-equivalent'', if they can be α-converted into the same expression. β-equivalence and η-equivalence 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 β-redex 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'' := ''N''

If ''x'' is not free in ''M'', λ''x''.''M x'' is also an η-redex, with a reduct of ''M''.

α-conversion

α-conversion, sometimes known as α-renaming, allows bound variable names to be changed. For example, α-conversion of λ''x''.''x'' might yield λ''y''.''y''. Terms that differ only by α-conversion are called ''α-equivalent''. Frequently, in uses of lambda calculus, α-equivalent terms are considered to be equivalent.

The precise rules for α-conversion are not completely trivial. First, when α-converting an abstraction, the only variable occurrences that are renamed are those that are bound to the same abstraction. For example, an α-conversion 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, α-conversion 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, α-conversion can be used to make name resolution simpler by ensuring that no variable name masks a name in a containing scope (see α-renaming to make name resolution trivial).

In the De Bruijn index notation, any two α-equivalent terms are syntactically identical.

Substitution

Substitution, written ''M'' 'x'' := ''N'' 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''= ''N''
: ''y'' 'x'' := ''N''= ''y'', if ''x'' ≠ ''y''
: (''M''₁ ''M''₂) 'x'' := ''N''= ''M''₁ 'x'' := ''N''''M''₂ 'x'' := ''N'': (λ''x''.''M'') 'x'' := ''N''= λ''x''.''M''
: (λ''y''.''M'') 'x'' := ''N''= λ''y''.(''M'' 'x'' := ''N'', if ''x'' ≠ ''y'' and ''y'' ∉ FV(''N'') ''See above for the FV''

To substitute into an abstraction, it is sometimes necessary to α-convert the expression. For example, it is not correct for (λ''x''.''y'') 'y'' := ''x''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 α-equivalence. Substitution is defined uniquely up to α-equivalence.

β-reduction

β-reduction captures the idea of function application. β-reduction is defined in terms of substitution: the β-reduction of (λ''x''.''M'') ''N'' is ''M'' 'x'' := ''N''

For example, assuming some encoding of 2, 7, ×, we have the following β-reduction: (λ''n''.''n'' × 2) 7 → 7 × 2.

β-reduction can be seen to be the same as the concept of ''local reducibility'' in natural deduction, via the Curry–Howard isomorphism.

η-reduction

η-reduction (eta reduction) expresses the idea of extensionality,Luke Palme
(29 Dec 2010) Haskell-cafe: What's the motivation for η rules?
/ref> which in this context is that two functions are the same if and only if they give the same result for all arguments. η-reduction converts between λ''x''.''f'' ''x'' and ''f'' whenever ''x'' does not appear free in ''f''.

η-reduction 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, β-reduction as a rewriting rule is neither strongly normalising nor weakly normalising.

However, it can be shown that β-reduction is confluent when working up to α-conversion (i.e. we consider two normal forms to be equal if it is possible to α-convert 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 booleans, arithmetic, data structures and recursion, as illustrated in the following sub-sections.

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 numerals, which can be defined as follows:
:
:
:
:
and so on. Or using the alternative syntax presented above in ''Notation'':

:
:
:
:

A Church numeral is a higher-order function—it takes a single-argument function , and returns another single-argument 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 β-equivalent lambda expressions. Since adding to a number can be accomplished by adding 1 times, an alternative definition is:
:
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 are 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 , and if its argument is any other Church numeral:
:
The following predicate tests whether the first argument is less-than-or-equal-to 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 "if-then-else" 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 (2-tuple) 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 shift-and-increment 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 low-level 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 lambda-terms are merely particular constants. The pure lambda calculus does not have a concept of named constants since all atomic lambda-terms 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 lambda-term) in ''M'' (another lambda-term, 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 lambda-term 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. Lambda calculus cannot express this as directly as some other notations: all functions are anonymous in lambda calculus, so we can't refer to a value which is yet to be defined, inside the lambda term defining that same value. However, recursion can still be achieved by arranging for a lambda expression to receive itself as its argument value, for example in  .

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 intended-as-self-referencing argument (called here) must always be passed to itself within the function body, at a call point:

:
::: with   to hold, so   and
:

The self-application 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 re-writing each recursive call as self-application. We would like to have a generic solution, without a need for any re-writes:

:
::: with   to hold, so   and
:  where 
::: so that 

Given a lambda term with first argument representing recursive call (e.g. here), the ''fixed-point'' combinator will return a self-replicating lambda expression representing the recursive function (here, ). The function does not need to be explicitly passed to itself at any point, for the self-replication is arranged in advance, when it is created, to be done each time it is called. Thus the original lambda expression is re-created inside itself, at call-point, achieving self-reference.

In fact, there are many possible definitions for this operator, the simplest of them being:

:

In the lambda calculus,   is a fixed-point of , as it expands to:

:
:
:
:
:

Now, to perform our recursive call to the f