- Lambda calculus
In

mathematical logic andcomputer science ,**lambda calculus**, also written as**λ-calculus**, is aformal system designed to investigate function definition, function application andrecursion . It was introduced byAlonzo Church andStephen Cole Kleene in the 1930s as part of an investigation into thefoundations of mathematics , but has emerged as a useful tool in the investigation of problems in computability or recursion theory, and forms the basis of a paradigm of computer programming calledfunctional programming . [*Henk Barendregt, "The Impact of the Lambda Calculus in Logic and Computer Science." "The Bulletin of Symbolic Logic", Volume*]**3**, Number 2, June 1997.The lambda calculus can be thought of as an idealized, minimalistic programming language. It is capable of expressing any

algorithm , and it is this fact that makes the model offunctional programming an important one. Functional programs are stateless and deal exclusively with functions that accept and return data (including other functions), but they produce no side effects in 'state' and thus make no alterations to incoming data. Modern functional languages, building on the lambda calculus, include Erlang, Haskell, Lisp, ML, and Scheme.The lambda calculus continues to play an important role in mathematical foundations, through the

Curry-Howard correspondence . However, as a naïve foundation for mathematics, the untyped lambda calculus is unable to avoid set-theoretic paradoxes (see theKleene-Rosser paradox ).This article deals with the "untyped lambda calculus" as originally conceived by Church. Most modern applications concern typed lambda calculi.

**Informal description**In lambda calculus, "every" expression is a

unary function , i.e. a function with only one input (known as its argument). When an expression is applied to another expression ('called' with the other expression as its argument), it returns a single value (known as its result).Since "every" expression is a unary function, and every argument and result are functions too, and as such lambda calculus is quite interesting and unique within both

computation andmathematics .A function is anonymously defined by a lambda expression which expresses the function's action on its argument. For instance, the "add-two" function "f" such that

`"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 parameter is immaterial) and the application of the function`"f"(3)`would be written as`(λ "x". "x" + 2) 3`. Note that part of what makes this description "informal" is that the expression`"x" + 2`(or even the number 2) is not part of lambda calculus; an explanation of how numbers and arithmetic can be represented in lambda calculus is below. Function application is left associative:`"f" "x" "y" = ("f" "x") "y"`. Consider the function which takes a function as an argument and applies it to the number`3`as follows:`λ "f". "f" 3`. This latter function could be applied to our earlier "add-two" function as follows:`(λ "f". "f" 3) (λ "x". "x" + 2)`. The three expressions:

*`(λ "f". "f" 3) (λ "x". "x" + 2)`

*`(λ "x". "x" + 2) 3`

*`3 + 2`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"`. A common convention is to abbreviate curried functions as, in this example,`λ "x" "y". "x" - "y"`. While it is not part of the formal definition of the language,:`λ "x"`is used as an abbreviation for:_{1}"x"_{2}… "x"_{n}. expression`λ "x"`Not every lambda expression can be reduced to a definite value like the ones above; consider for instance:_{1}. λ "x"_{2}. … λ "x"_{n}. expression`(λ "x". "x" "x") (λ "x". "x" "x")`or :`(λ "x". "x" "x" "x") (λ "x". "x" "x" "x")`and try to visualize what happens when you start to apply the first function to its argument.`(λ "x". "x" "x")`is also known as the ωcombinator ;`((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.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 example, in the formula below, the letter`"y"`is used first as a formal parameter, then as a free variable::`(λ "x" "y". "y" "x") (λ "x".`.To reduce the expression, we rename the first identifier "z" so that the reduction does not mix up the names::**"y**")`(λ "x" "z". "z" "x") (λ "x". "y")`the reduction is then:`λ "z". "z" (λ "x". "y")`.If one only formalizes the notion of function application and replaces the use of lambda expressions by the use of "combinators", one obtains

combinatory logic .**Formal definition****Definition**Lambda expressions are composed of :variables v

_{1}, v_{2}, . . . v_{n}:the abstraction symbols λ and .:parentheses ( )The set of lambda expressions, Λ, can be defined recursively:

#If x is a variable, then x $in$ Λ

#If x is a variable and M $in$ Λ, then ( λ x . M ) $in$ Λ

#If M, N $in$ Λ, then ( M N ) $in$ ΛInstances of 2 are known as abstractions and instances of 3, applications. [*Citation*]

last = Barendregt

first = Hendrik Pieter

author-link =

last2 =

first2 =

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title = The Lambda Calculus: Its Syntax and Semantics

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publisher = North Holland, Amsterdam. [ftp://ftp.cs.ru.nl/pub/CompMath.Found/ErrataLCalculus.pdf Corrections]

year = 1984

volume = 103

series = Studies in Logic and the Foundations of Mathematics

edition = Revised edition

url = http://www.elsevier.com/wps/find/bookdescription.cws_home/501727/description

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id =

isbn = 0-444-87508-5**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 means (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 are contracted: λ x λ y λ z. N is abbreviated as λ x y z . NCitation

first = Peter

last = Selinger

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last2 =

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editor-first =

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contribution-url =

title = Lecture Notes on the Lambda Calculus

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publisher = Department of Mathematics and Statistics, University of Ottawa

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id = ]**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 a lambda are said to be "bound". All other variables are called "free". For example in the following expression y is a bound variable and x is free:

:λ y . xxy

Also note that a variable binds to its "nearest" lambda. In the following expression one single occurrence of x 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}

# FV ( M N ) = FV ( M ) $cup$ FV ( N )Citation

last = Barendregt

first = Henk

author-link =

last2 = Barendsen

first2 = Erik

author2-link =

title = Introduction to Lambda Calculus

place =

publisher =

year = March 2000

volume =

edition =

url = ftp://ftp.cs.ru.nl/pub/CompMath.Found/lambda.pdf

doi =

id =

isbn = ]An expression which 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****α-conversion**Alpha conversion allows bound variable names to be changed. For example, an alpha conversion of

`λ"x"."x"`would be`λ"y"."y"`. Frequently in uses of lambda calculus, terms that differ only by alpha conversion are considered to be equivalent.The precise rules for alpha conversion are not completely trivial. First, when alpha-converting an abstraction, the only variable occurrences that are renamed are those that are bound to the same abstraction. For example, an alpha 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.Second, alpha 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.**ubstitution**Substitution, written

`E [V := E′]`, corresponds to the replacement of a variable`V`by expression`E′`every place it is free within`E`. The precise definition must be careful in order to avoid accidental variable capture. 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 on terms of the λ-calculus is defined by recursion on the structure of terms, as follows.:

`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 y. M) [x := N] ≡ λ y. (M [x := N] ), if x ≠ y and y∉fv(N)`Notice that substitution is defined uniquely up-to α-equivalence.

**β-reduction**Beta reduction expresses the idea of function application. The beta reduction of

`((λ "V". "E") "E′")`is simply`"E" ["V" := "E′"]`.**η-conversion**Eta conversion expresses the idea of

extensionality , which in this context is that two functions are the sameif and only if 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 conversion is not always appropriate when lambda expressions are interpreted as programs. Evaluation of

`λ "x". "f" "x"`can terminate even when evaluation of "f" does not.**Arithmetic in lambda calculus**There are several possible ways to define the

natural number s in lambda calculus, but by far the most common are theChurch numeral s, which can be defined as follows::`0 := λ "f" "x". "x"`:`1 := λ "f" "x". "f" "x"`:`2 := λ "f" "x". "f" ("f" "x")`:`3 := λ "f" "x". "f" ("f" ("f" "x"))`and so on. A Church numeral is ahigher-order function —it takes a single-argument function`"f"`, and returns another single-argument function. The Church numeral`"n"`is a function that takes a function`"f"`as argument and returns the`"n"`-th composition of`"f"`, i.e. the function`"f"`composed with itself`"n"`times. This is denoted`"f"`and is in fact the^{("n")}`"n"`-th power of`"f"`(considered as an operator);`"f"`is defined to be the identity function. Such repeated compositions (of a single function^{(0)}`"f"`) obey the laws of exponents, which is why these numerals can be used for arithmetic. Note that`1`returns`"f"`itself, i.e. it is essentially the identity function, and`0`"returns" the identity function. (Also 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.)We can define a successor function, which takes a number

`"n"`and returns`"n" + 1`by adding an additional application of`"f"`::`SUCC := λ "n" "f" "x". "f" ("n" "f" "x")`Because the`"m"`-th composition of`"f"`composed with the`"n"`-th composition of`"f"`gives the`"m+n"`-th composition of`"f"`, addition can be defined as follows::`PLUS := λ "m" "n" "f" "x". "n" "f" ("m" "f" "x")``PLUS`can be thought of as a function taking two natural numbers as arguments and returning a natural number; it can be verified that :`PLUS 2 3`and`5`are equivalent lambda expressions. Since adding`"m"`to a number,`"n"`can be accomplished by adding 1`"m"`times, an equivalent definition is::`PLUS := λ "n" "m". "m" SUCC "n"`[*cite book*] Similarly, multiplication can be defined as:

last = Felleisen

first = Matthias

authorlink =

coauthors = Matthew Flatt

title = Programming Languages and Lambda Calculi

publisher =

date = 2006

location =

pages = 26

url = http://www.cs.utah.edu/plt/publications/pllc.pdf

doi =

id =

isbn =`MULT := λ "m" "n" "f" . "m" ("n" "f")`[*Citation*] Alternatively:

first = Peter

last = Selinger

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first2 =

last2 =

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title = Lecture Notes on the Lambda Calculus

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pages = 16

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publisher = Department of Mathematics and Statistics, University of Ottawa

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id =`MULT := λ "m" "n". "m" (PLUS "n") 0`,since multiplying`"m"`and`"n"`is the same as repeating the "add`"n"`" function`"m"`times and then applying it to zero.The predecessor function defined by`PRED "n" = "n" - 1`for a positive integer`"n"`and`PRED 0 = 0`is considerably more difficult. The formula:`PRED := λ "n" "f" "x". "n" (λ "g" "h". "h" ("g" "f")) (λ "u". "x") (λ "u". "u")`can be validated by showing inductively that if`T`denotes`(λ "g" "h". "h" ("g" "f"))`, then`T`for^{("n")}(λ "u". "x") = (λ "h". "h"("f"^{("n"-1)}("x")) )`"n" > 0`. Two other definitions of`PRED`are given below, one using conditionals and the other using pairs. With the predecessor function, subtraction is straightforward. Defining:`SUB := λ "m" "n". "n" PRED "m"`,`SUB "m" "n"`yields`"m" - "n"`when`"m" > "n"`and`0`otherwise.**Logic and predicates**By convention, the following two definitions (known as Church booleans) are used for the boolean values

`TRUE`and`FALSE`::`TRUE := λ "x" "y". "x"`:`FALSE := λ "x" "y". "y"`::(Note that`FALSE`is equivalent to the Church numeral zero defined above)Then, with these two λ-terms, we can define some logic operators (these are just possible formulations; other expressions are equally correct)::`AND := λ "p q". "p q p"`:`OR := λ "p q". "p p q"`:`NOT := λ "p a b". "p b a"`:`IFTHENELSE := λ "p". "p"`We are now able to compute some logic functions, for example::

`AND TRUE FALSE``::``≡ (λ "p q". "p q p") TRUE FALSE →`_{β}TRUE FALSE TRUE::`≡ (λ "x y". "x") FALSE TRUE →`and we see that_{β}FALSE`AND TRUE FALSE`is equivalent to`FALSE`.`A "predicate" is a function which returns a boolean value. The most fundamental predicate is``ISZERO`which returns`TRUE`if its argument is the Church numeral`0`, and`FALSE`if its argument is any other Church numeral::`ISZERO := λ "n". "n" (λ "x". FALSE) TRUE`The following predicate tests whether the first argument is less-than-or-equal-to the second::`LEQ := λ "m n". ISZERO (SUB "m n")`,and since`"m" = "n"`iff`LEQ "m n"`and`LEQ "n m"`, it is straightforward to build a predicate for numerical equality.`The availability of predicates and the above definition of``TRUE`and`FALSE`make it convenient to write "if-then-else" expressions in lambda calculus. For example, the predecessor function can be defined as' ':`PRED := λ "n". "n" (λ "g k". ISZERO ("g" 1) k (PLUS ("g k") 1) ) (λ "v". 0) 0`which can be verified by showing inductively that`"n" (λ "g k". ISZERO ("g" 1) k (PLUS ("g k") 1) ) (λ "v". 0)`is the "add`"n"`- 1" function for`"n"`> 0.**Pairs**`A pair (2-tuple) can be defined in terms of``TRUE`and`FALSE`, by using the Church encoding for pairs. For example,`PAIR`encapsulates the pair (`"x"`,`"y"`),`FIRST`returns the first element of the pair, and`SECOND`returns the second.`:``PAIR := λ "x" "y" "f". "f" "x" "y"`:`FIRST := λ "p". "p" TRUE`:`SECOND := λ "p". "p" FALSE`:`NIL := λ x. TRUE`:`NULL := λp. p (λx y.FALSE)``A linked list can be defined as either NIL for the empty list, or the``PAIR`of an element and a smaller list. The predicate`NULL`tests for the value`NIL`.`As an example of the use of pairs, the shift-and-increment function that maps``("m", "n")`to`("n", "n"+1)`can be defined as:`Φ := λ "x". PAIR (SECOND x) (SUCC (SECOND "x"))`which allows us to give perhaps the most transparent version of the predecessor function::`PRED := λ "n". FIRST ("n" Φ (PAIR 0 0))`**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 thefactorial function`"f"("n")`recursively defined by`:``"f"("n") = 1, if "n" = 0; and "n"·"f"("n"-1), if "n">0`.`In lambda calculus, one cannot define a function which includes itself. To get around this, one may start by defining a function, here called``"g"`, which takes a function`"f"`as an argument and returns another function that takes`"n"`as an argument:`:``"g" := λ "f" "n". (1, if "n" = 0; and "n"·"f"("n"-1), if "n">0)`.`The function that``"g"`returns is either the constant`1`, or "n" times the application of the function`"f"`to`"n"-1`. Using the`ISZERO`predicate, and boolean and algebraic definitions described above, the function`"g"`can be defined in lambda calculus.`However,``"g"`by itself is still not recursive; in order to use`"g"`to create the recursive factorial function, the function passed to`"g"`as`"f"`must have specific properties. Namely, the function passed as`"f"`must expand to the function`"g"`called with one argument -- and that argument must be the function that was passed as`"f"`again!`In other words,``"f"`must expand to`"g"("f")`. This call to`"g"`will then expand to the above factorial function and calculate down to another level of recursion. In that expansion the function`"f"`will appear again, and will again expand to`"g"("f")`and continue the recursion. This kind of function, where`"f" = "g"("f")`, is called a "fixed-point" of`"g"`, and it turns out that it can be implemented in the lambda calculus using what is known as the "paradoxical operator" or "fixed-point operator" and is represented as`"Y"`-- the Y combinator:`:``"Y" = λ "g". (λ "x". "g" ("x" "x")) (λ "x". "g" ("x" "x"))``In the lambda calculus,``"Y g"`is a fixed-point of`"g"`, as it expands to`"g" ("Y" "g")`. Now, to complete our recursive call to the factorial function, we would simply call`"g" ("Y" "g") "n"`, where "n" is the number we are calculating the factorial of.`Given "n" = 5, for example, this expands to:``:``(λ "n".(1, if "n" = 0; and "n"·(("Y g")("n"-1)), if "n">0)) 5`:`1, if 5 = 0; and 5·("g"("Y g")(5-1)), if 5>0`:`5·("g"("Y g") 4)`:`5·(λ "n". (1, if "n" = 0; and "n"·(("Y g")("n"-1)), if "n">0) 4)`:`5·(1, if 4 = 0; and 4·("g"("Y g")(4-1)), if 4>0)`:`5·(4·("g"("Y g") 3))`:`5·(4·(λ "n". (1, if "n" = 0; and "n"·(("Y g")("n"-1)), if "n">0) 3))`:`5·(4·(1, if 3 = 0; and 3·("g"("Y g")(3-1)), if 3>0))`:`5·(4·(3·("g"("Y g") 2)))`:`...``And so on, evaluating the structure of the algorithm recursively. Every recursively defined function can be seen as a fixed point of some other suitable function, and therefore, using``"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.**Computable functions and lambda calculus**`A function``"F":`of**N**→**N**natural number s is acomputable 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" "x" = "y"`, where`"x"`and`"y"`are the Church numerals corresponding to "x" and "y", respectively. This is one of the many ways to definecomputability ; see theChurch-Turing thesis for a discussion of other approaches and their equivalence.**Undecidability of equivalence**`There is no algorithm which takes as input two lambda expressions and outputs``TRUE`or`FALSE`depending on whether or not the two expressions are equivalent. This was historically the first problem for which undecidability could be proven. As is common for a proof of undecidability, the proof shows that nocomputable function can decide the equivalence.Church's thesis is then invoked to show that no algorithm can do so.`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 constructing a`Gödel numbering 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 ownGödel number , a contradiction results.**Lambda calculus and programming languages**`As pointed out by`Peter Landin 's 1965 classic [*http://portal.acm.org/citation.cfm?id=363749&coll=portal&dl=ACM A Correspondence between ALGOL 60 and Church's Lambda-notation*] , mostprogramming language s are rooted in the lambda calculus, which provides the basic mechanisms for procedural abstraction and procedure (subprogram) application.`Implementing the lambda calculus on a computer involves treating "functions" as`first-class object s, which raises implementation issues for stack-based programming languages. This is known as theFunarg problem .`The most prominent counterparts to lambda calculus in programming are`functional programming language s, which essentially implement the calculus augmented with someconstant s anddatatype s. Lisp uses a variant of lambda notation for defining functions, but only its purely functional subset ("Pure Lisp ") is really equivalent to lambda calculus.`Functional languages are not the only ones to support functions as`first-class object s. Numerous imperative languages, e.g. Pascal, have long supported passing subprograms as arguments to other subprograms. In C and the C-like subset ofC++ the equivalent result is obtained by passing "pointers" to the code of functions (subprograms). Such mechanisms are limited to subprograms written explicitly in the code, and do not directly support higher-level functions. Some imperativeobject-oriented language s have notations that represent functions of any order; such mechanisms are available inC++ ,Smalltalk and more recently in Eiffel ("agents") and C# ("delegates"). As an example, the Eiffel "inline agent" expression`A Python example of this uses the [`*http://docs.python.org/ref/lambdas.html#lambda lambda*] form of functions:`The same holds for Smalltalk expression``A similar C++ example (using the Boost.Lambda library):``A simple C# delegate taking a variable and returning the square. This function variable can then be passed to other methods (or function delegates)`**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. The distinction between reduction strategies relates to the distinction in functional programming languages between`eager evaluation andlazy evaluation .`The following uses the term 'redex', short for 'reducible expression'. For example,``(λ x. M) N`is a beta-redex;`λ x. M x`is an eta-redex if`x`is not free in`M`. The expression to which a redex reduces is called its reduct; using the previous example, the reducts of these expressions are respectively`M [x:=N]`and`M`.`;Full beta reductions: 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".;Applicative order: The rightmost, 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.:Most programming languages (including Lisp,`ML and imperative languages like C and Java) are described as "strict", meaning that functions applied to non-normalising arguments are non-normalising. This is done essentially using applicative order, call by value reduction (see below), but usually called "eager evaluation".;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.;Call by name: As normal order, but no reductions are performed inside abstractions. For example`λ x.(λ x.x)x`is in normal form according to this strategy, although it contains the redex`(λ x.x)x`.;Call by value: Only the outermost redexes are reduced: a redex is reduced only when its right hand side has reduced to a value (variable or lambda abstraction).;Call by need: As normal order, but function applications that would duplicate terms instead name the argument, which is then reduced only "when it is needed". Called in practical contexts "lazy evaluation". In implementations this "name" takes the form of a pointer, with the redex represented by a thunk.`Applicative order is not a normalising strategy. The usual counterexample is as follows: define``Ω = ωω`where`ω = λ x. xx`. This entire expression contains only one redex, namely the whole expression; its reduct is again`Ω`. Since this is the only available reduction,`Ω`has no normal form (under any evaluation strategy). Using applicative order, the expression`KIΩ = (λ x y . x)(λ x.x)Ω`is reduced by first reducing`Ω`to normal form (since it is the rightmost redex), but since`Ω`has no normal form, applicative order fails to find a normal form for`KIΩ`.`In contrast, normal order is so called because it always finds a normalising reduction if one exists. In the above example,``KIΩ`reduces under normal order to`I`, a normal form. A drawback is that redexes in the arguments may be copied, resulting in duplicated computation (for example,`(λ x.xx)((λ x.x)y)`reduces to`((λx.x)y)((λx.x)y)`using this strategy; now there are two redexes, so full evaluation needs two more steps, but if the argument had been reduced first, there would now be none).`The positive tradeoff of using applicative order is that it does not cause unnecessary computation if all arguments are used, because it never substitutes arguments containing redexes and hence never needs to copy them (which would duplicate work). In the above example, in applicative order``(λ x.xx)((λ x.x)y)`reduces first to`(λ x.xx)y`and then to the normal order`yy`, taking two steps instead of three.`Most "purely" functional programming languages (notably`Miranda and its descendents, includingHaskell ), and the proof languages oftheorem prover s, use "lazy evaluation ", which is essentially the same as call by need. This is like normal order reduction, but call by need manages to avoid the duplication of work inherent in normal order reduction using "sharing". In the example given above,`(λ x.xx)((λ x.x)y)`reduces to`((λx.x)y)((λx.x)y)`, which has two redexes, but in call by need they are represented using the same object rather than copied, so when one is reduced the other is too.**A note about complexity**`While the idea of beta reduction seems simple enough, it is not an atomic step, in that it must have a non-trivial cost when estimating`computational complexity . [*R. Statman, "The typed λ-calculus is not elementary recursive." "Theoretical Computer Science", (1979)*] To be precise, one must somehow find the location of all of the occurrences of the bound variable**9**pp73-81.`"V"`in the expression`"E"`, implying a time cost, or one must keep track of these locations in some way, implying a space cost. A naïve search for the locations of`"V"`in`"E"`is "O"("n") in the length "n" of`"E"`. This has led to the study of systems which useexplicit substitution . Sinot'sdirector string s [*F.-R. Sinot. " [*] offer a way of tracking the locations of free variables in expressions.*http://www.lsv.ens-cachan.fr/~sinot/publis.php?onlykey=sinot-jlc05 Director Strings Revisited: A Generic Approach to the Efficient Representation of Free Variables in Higher-order Rewriting.*] " "Journal of Logic and Computation"**15**(2), pages 201-218, 2005.**Concurrency and parallelism**`The Church-Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in "any order", even concurrently. This means that various nondeterministic evaluation strategies are relevant. However, the lambda calculus does not offer any explicit constructs for parallelism. Various process calculi have been proposed as minimal languages for concurrency and distributed computation.`**emantics**`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.`In the 1970s,`Dana Scott showed that, if only continuous functions were considered, a set or domain "D" with the required property could be found, thus providing a model for the lambda calculus.`This work also formed the basis for the`denotational semantics of programming languages**ee also**

*Anonymous recursion

*Combinatory logic

*Curry-Howard isomorphism

*Evaluation strategy

*Explicit substitution

*Knights of the Lambda Calculus

*Lambda cube

*Rewriting

*SKI combinator calculus

*System F

*Calculus of constructions

*Typed lambda calculus

*Unlambda

*Lambda-mu calculus

*Cartesian closed category

*Categorical abstract machine

*Applicative computing systems

*Domain theory **References****Further reading**`*Abelson, Harold & Gerald Jay Sussman.`Structure and Interpretation of Computer Programs .The MIT Press . ISBN 0-262-51087-1.

*Hendrik Pieter Barendregt [*http://citeseer.ist.psu.edu/barendregt94introduction.html "Introduction to Lambda Calculus"*] .

*Barendregt, Hendrik Pieter, "The Type Free Lambda Calculus" pp1091-1132 of "Handbook of Mathematical Logic",North-Holland (1977) ISBN 0-7204-2285-X

*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.

*Landin, Peter, "A Correspondence Between ALGOL 60 and Church's Lambda-Notation",Communications of the ACM , vol. 8, no. 2 (1965), pages 89-101. Available from the [*http://portal.acm.org/citation.cfm?id=363749&coll=portal&dl=ACM ACM site*] . A classic paper highlighting the importance of lambda-calculus as a basis for programming languages.

*Larson, Jim, [*http://www.jetcafe.org/~jim/lambda.html "An Introduction to Lambda Calculus and Scheme"*] . A gentle introduction for programmers.

*Schalk, A. and Simmons, H. (2005) " [*http://www.cs.man.ac.uk/~hsimmons/BOOKS/lcalculus.pdf An introduction to λ-calculi and arithmetic with a decent selection of exercises*] . Notes for a course in the Mathematical Logic MSc at Manchester University."Some parts of this article are based on material from FOLDOC, used with ."**External links**`* Henk Barendregt, Erik Barendsen [`*http://www.cs.ru.nl/E.Barendsen/onderwijs/sl2/materiaal/lambda.pdf "Introduction to Lambda Calculus"*] -(PDF)

*Achim Jung, " [*http://www.cs.bham.ac.uk/~axj/pub/papers/lambda-calculus.pdf A Short Introduction to the Lambda Calculus*] "-(PDF)

*David C. Keenan, " [*http://users.bigpond.net.au/d.keenan/Lambda/ To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction*] "

*Raúl Rojas, " [*http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf A Tutorial Introduction to the Lambda Calculus*] "-(PDF)

* Peter Selinger, [*http://www.mscs.dal.ca/~selinger/papers.html#lambdanotes "Lecture Notes on the Lambda Calculus"*] -(PDF)

*L. Allison, " [*http://www.allisons.org/ll/FP/Lambda/Examples/ Some executable λ-calculus examples*] "

*Chris Barker, " [*http://homepages.nyu.edu/~cb125/Lambda/ Some executable (Javascript) simple examples, and text.*] "

*Georg P. Loczewski, [*http://www.lambda-bound.com/book/lambdacalc/lcalconl.html "The Lambda Calculus and A++"*]

* Bret Victor, " [*http://worrydream.com/AlligatorEggs/ Alligator Eggs: A Puzzle Game Based on Lambda Calculus*] "

*" [*http://www.safalra.com/science/lambda-calculus/ Lambda Calculus*] " on [*http://www.safalra.com/ Safalra’s Website*]

*"planetmath reference|id=2788|title=Lambda Calculus"

* [*http://lci.sourceforge.net/ LCI Lambda Interpreter*] a simple yet powerful pure calculus interpreter

* [*http://lambda-the-ultimate.org/classic/lc.html Lambda Calculus links on Lambda-the-Ultimate*]

*Mike Thyer, [*http://thyer.name/lambda-animator/ Lambda Animator*] , a graphical Java applet demonstrating alternative reduction strategies.

* [*http://www.jetcafe.org/~jim/lambda.html An Introduction to Lambda Calculus and Scheme*] , by Jim Larson

*Wikimedia Foundation.
2010.*

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