Inmathematical logic, thelambda calculus (also written asλ-calculus) is aformal system for expressingcomputation based on functionabstraction andapplication using variablebinding andsubstitution. Untyped lambda calculus, the topic of this article, is auniversal machine, i.e. amodel of computation that can be used to simulate anyTuring machine (and vice versa). It was introduced by the mathematicianAlonzo Church in the 1930s as part of his research into thefoundations of mathematics. In 1936, Church found a formulation which waslogically consistent, and documented it in 1940.
The lambda calculus consists of a language oflambda terms, which are defined by a formal syntax, and a set of transformation rules for manipulating those terms. InBNF, the syntax is where variablesx,y,z range over an infinite set of names. TermsM,N,t,s,e,f range over all lambda terms. This corresponds to the followinginductive definition:
A lambda term is syntactically valid if it can be obtained by repeated application of these three rules. For convenience, parentheses can often be omitted when writing a lambda term—seeLambda calculus definition § Notation for details.
Within lambda terms, any occurrence of a variable that is not a parameter of some enclosing λ is said to befree. Any free occurrence of in a term isbound in. Any free occurrence of any other variable within remains free in.
For example, in the term, both and occur free. In, is free, but in the body (i.e. after the dot) is not free, and is said to bebound (to the parameter). While is free in, it is bound in. There are two occurrences of in – one is bound, and the other is free.
FV(M) is the set of free variables ofM, i.e. such variables that occur free inM at least once. It can be defined inductively as follows:
The notation denotescapture-avoiding substitution: substitutingN for every free occurrence ofx inM, while avoiding variable capture.[a] This operation is defined inductively as follows:
There are several notions of "equivalence" and "reduction" that make it possible to reduce lambda terms to equivalent lambda terms.[3]
The termredex, short forreducible expression, refers to subterms that can be reduced by one of the reduction rules. For example, (λx.M)N is a β-redex in expressing the substitution ofN forx inM. The expression to which a redex reduces is called itsreduct; the reduct of (λx.M)N isM[x :=N].
Lambda calculus isTuring complete, that is, it is a universalmodel of computation that can be used to simulate anyTuring machine.[4] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denotebinding a variable in afunction.
Lambda calculus may beuntyped ortyped. 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 strictlyweaker than the untyped lambda calculus, which is the primary subject of this article, in the sense thattyped lambda calculi can express less than the untyped calculus can. On the other hand, more things can be proven with typed lambda calculi. For example, insimply typed lambda calculus, it is a theorem that every evaluation strategy terminates for every simply typed lambda-term,[5] whereas evaluation of untyped lambda-terms need not terminate (seebelow). 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 inmathematics,philosophy,[6]linguistics,[7][8] andcomputer science.[9][10] Lambda calculus has played an important role in the development of thetheory ofprogramming languages.Functional programming languages implement lambda calculus. Lambda calculus is also a current research topic incategory theory.[11][failed verification]
Lambda calculus was introduced by mathematicianAlonzo Church in the 1930s as part of an investigation into thefoundations of mathematics.[12][c] The original system was shown to belogically inconsistent in 1935 whenStephen Kleene andJ. B. Rosser developed theKleene–Rosser paradox.[13][14]
Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus.[15] In 1940, he also introduced a computationally weaker, but logically consistent system, known as thesimply typed lambda calculus.[16]
Until the 1960s when its relation to programming languages was clarified, the lambda calculus was only a formalism. Thanks toRichard Montague and other linguists' applications in the semantics of natural language, the lambda calculus has begun to enjoy a respectable place in both linguistics[17] and computer science.[18]
There is some uncertainty over the reason for Church's use of the Greek letterlambda (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 [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation "" used for class-abstraction byWhitehead and Russell, by first modifying "" to "" to distinguish function-abstraction from class-abstraction, and then changing "" to "λ" for ease of printing.
This origin was also reported in [Rosser, 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.[19]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 theiota operator, Hilbert had theepsilon 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".
Computable functions are a fundamental concept within computer science and mathematics. The lambda calculus provides simplesemantics 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
can be rewritten inanonymous form as
(which is read as "atuple ofx andy ismapped to").[d] Similarly, the function
can be rewritten in anonymous form as
where the input is simply mapped to itself.[d]
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 function, can be reworked into an equivalent function that accepts a single input, and as output returnsanother function, that in turn accepts a single input. For example,
can be reworked into
This method, known ascurrying, transforms a function that takes multiple arguments into a chain of functions each with a single argument.
Function application of the function to the arguments (5, 2), yields at once
whereas evaluation of the curried version requires one more step
to arrive at the same result.
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 represents theidentity function,. Further, represents theconstant function, the function that always returns, no matter the input. As an example of a function operating on functions, thefunction composition can be defined as.
It can be shown that β-reduction isconfluent 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). Ifrepeated application of the reduction steps eventually terminates, then by theChurch–Rosser theorem it will produce a uniqueβ-normal form. However, the untyped lambda calculus as arewriting rule under β-reduction is neitherstrongly normalising norweakly normalising; there are terms with no normal form such asΩ.
Considering individual terms, 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.
The basic lambda calculus may be used to modelarithmetic, Booleans, data structures, and recursion, as illustrated in the following sub-sectionsi,ii,iii, and§ iv.
There are several possible ways to define thenatural numbers in lambda calculus, but by far the most common are theChurch numerals, which can be defined as follows:
and so on. Or using an alternative syntax allowing multiple uncurried arguments to a function:
A Church numeral is ahigher-order function—it takes a single-argument functionf, and returns another single-argument function. The Church numeraln is a function that takes a functionf as argument and returns then-thcomposition off, i.e. the functionf composed with itselfn times. This is denotedf(n) and is in fact then-th power off (considered as an operator);f(0) is defined to be the identity function. Functional composition isassociative, and so, such repeated compositions of a single functionf obey twolaws of exponents,f(m)∘f(n) =f(m+n) and(f(n))(m) =f(m*n), 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 of0 impossible.)
One way of thinking about the Church numeraln, which is often useful when analyzing programs, is as an instruction 'repeatn times'. For example, using thePAIR andNIL functions defined below, one can define a function that constructs a (linked)list ofn elements all equal tox by repeating 'prepend anotherx element'n times, starting from an empty list. The lambda term
creates, given a Church numeraln and somex, a sequence ofn applications
By varying what is being repeated, and what argument(s) 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 numeraln and returns its successorn + 1 by performing one additional application of the functionf it is supplied with, where(n f x) means "n applications off starting fromx":
Because them-th composition off composed with then-th composition off gives them+n-th composition off,f(m)∘f(n) =f(m+n), addition can be defined as
PLUS 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 beta-equivalent lambda expressions. Since addingm to a number can be accomplished by repeating the successor operationm times, an alternative definition is:
Similarly, following(f(n))(m) =f(m*n), multiplication can be defined as
Thus multiplication of Church numerals is simply their composition as functions. Alternatively
since multiplyingm andn is the same as addingn repeatedly,m times, starting from zero.
Exponentiation, being the repeated multiplication of a number with itself, translates as a repeated composition of a Church numeral with itself, as a function. And repeated composition is what Church numeralsare:
Alternatively here as well,
Simplifying, it becomes
but that is just an eta-expanded version ofPOW we already have, above.
Thepredecessor function, specified by two equationsPRED (SUCCn) =n andPRED 0 = 0, is considerably more involved. The formula
can be validated by showing inductively that ifT denotes(λg.λh.h (gf)), thenT(n)(λu.x) = (λh.h(f(n−1)(x))) forn > 0. Two other definitions ofPRED are given below, one usingconditionals and the other usingpairs. With the predecessor function, subtraction is straightforward. Defining
SUBmn yieldsm −n whenm >n and0 otherwise.
By convention, the following two definitions (known as Church Booleans) are used for the Boolean valuesTRUE andFALSE:
Then, with these two lambda terms, we can define some logic operators (these are just possible formulations; other expressions could be equally correct):[22]
We are now able to compute some logic functions, for example:
and we see thatAND TRUE FALSE is equivalent toFALSE.
Apredicate is a function that returns a Boolean value. The most fundamental predicate isISZERO, which returnsTRUE if its argument is the Church numeral0, butFALSE if its argument were any other Church numeral:
The following predicate tests whether the first argument is less-than-or-equal-to the second:
and sincem =n ifLEQmn andLEQnm, it is straightforward to build a predicate for numerical equality.
The availability of predicates and the above definition ofTRUE andFALSE 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 thatn (λg.λk.ISZERO (g 1)k (PLUS (gk) 1)) (λv.0) is the addn − 1 function forn > 0.
A pair (2-tuple) encapsulates two values, and is represented by an abstraction that expects a handler to which it will pass the two values.FIRST returns the first element of the pair, andSECOND returns the second.
A linked list can either be NIL, representing the empty list, or aPAIR of an element (so-calledhead) and a smaller list (tail). The predicateNULL returnsTRUE for the valueNIL, andFALSE for a non-empty list:
Alternatively, withNIL := FALSE, the construct(l (λh.λt.λz. ...h...t...) _on_nil_) obviates the need for an explicit NULL test:
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
which allows us to give perhaps the most transparent version of the predecessor function:
Substituting the definitions and simplifying the resulting expression leads to streamlined definitions
(where I := λx.x ), evidently leading back to the original.
There is a considerable body ofprogramming idioms for lambda calculus. Many of these were originally developed in the context of using lambda calculus as a foundation forprogramming language semantics, effectively using lambda calculus as alow-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.
In lambda calculus, alibrary 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 usef to meanN (some explicit lambda-term) inM (another lambda-term, the "main program"), one can say
Authors often introducesyntactic sugar, such aslet,[e] to permit writing the above in the more intuitive order
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 thislet is that the namef may not be referenced inN, forN is outside the scope of the abstraction bindingf, which isM; this means a recursive function definition cannot be written withlet. Theletrec[f] construction would allow writing recursive function definitions, where the scope of the abstraction bindingf includesN as well asM. Or self-application a-la that which leads toY combinator could be used.
Recursion is when a function invokes itself. What would a value be which were to represent such a function? It has to refer to itself somehow inside itself, just as the definition refers to itself inside itself. If this value were to contain itself by value, it would have to be of infinite size, which is impossible. Other notations, which support recursion natively, overcome this by referring to the functionby name inside its definition. Lambda calculus cannot express this, since in it there simply are no names for terms to begin with, only arguments' names, i.e. parameters in abstractions. Thus, a lambda expression can receive itself as its argument and refer to (a copy of) itself via the corresponding parameter's name. This will work fine in case it was indeed called with itself as an argument. For example,(λx.xx)E = (E E) will express recursion whenE is an abstraction which is applying its parameter to itself inside its body to express a recursive call. Since this parameter receivesE as its value, its self-application will be the same(E E) again.
As a concrete example, consider thefactorial functionF(n), recursively defined by
In the lambda expression which is to represent this function, aparameter (typically the first one) will be assumed to receive the lambda expression itself as its value, so that calling it with itself as its first argument will amount to the recursive call. Thus to achieve recursion, the intended-as-self-referencing argument (calleds here, reminiscent of "self", or "self-applying") must always be passed to itself within the function body at a recursive call point:
and we have
Heres s becomesthe same (E E) inside the result of the application (E E), and using the same function for a call is the definition of what recursion is. 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 there by the parameter names to be called via the self-applicationss, again and again as needed, each timere-creating the lambda-term F = E E.
The application is an additional step just as the name lookup would be. It has the same delaying effect. Instead of having F inside itself as a wholeup-front, delaying its re-creation until the next call makes its existence possible by having twofinite lambda-terms E inside it re-create it on the flylater as needed.
This self-applicational approach solves it, but requires re-writing each recursive call as a self-application. We would like to have a generic solution, without the need for any re-writes:
Given a lambda term with first argument representing recursive call (e.g.G here), thefixed-point combinatorFIX will return a self-replicating lambda expression representing the recursive function (here,F). 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(FIX G) is re-created inside itself, at call-point, achievingself-reference.
In fact, there are many possible definitions for thisFIX operator, the simplest of them being:
In the lambda calculus,Yg is a fixed-point ofg, as it expands to:
Now, to perform the recursive call to the factorial function for an argumentn, we would simply call(Y G)n. Givenn = 4, for example, this gives:
Every recursively defined function can be seen as a fixed point of some suitably defined higher order function (also known as functional) closing over the recursive call with an extra argument. Therefore, usingY, every recursive function can be expressed as a lambda expression. In particular, we can now cleanly define the subtraction, multiplication, and comparison predicates of natural numbers, using recursion.
WhenY combinator is coded directly in astrict programming language, the applicative order of evaluation used in such languages will cause an attempt to fully expand the internal self-application prematurely, causingstack overflow or, in case oftail call optimization, indefinite looping.[24] A delayed variant of Y, theZ combinator, can be used in such languages. It has the internal self-application hidden behind an extra abstraction througheta-expansion, as, thus preventing its premature expansion:[25]
Certain terms have commonly accepted names:[26][27][28]
I is the identity function.SK andBCKW form completecombinator calculus systems that can express any lambda term - seethe next section.Ω isUU, the smallest term that has no normal form.YI is another such term.Y is standard and definedabove, and can also be defined asY=BU(CBU), so thatYg=g(Yg).TRUE andFALSE definedabove are commonly abbreviated asT andF.
IfN is a lambda-term without abstraction, but possibly containing named constants (combinators), then there exists a lambda-termT(x,N) which is equivalent toλx.N but lacks abstraction (except as part of the named constants, if these are considered non-atomic). This can also be viewed as anonymising variables, asT(x,N) removes all occurrences ofx fromN, while still allowing argument values to be substituted into the positions whereN contains anx. The conversion functionT can be defined by:
In either case, a term of the formT(x,N)P is reduced by having the initial combinatorI,K, orS grab the argumentP, just like β-reduction of(λx.N)P would do.I returns that argument.KN throws the argument away, just like(λx.N) does whenx has no free occurrence inN.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, just like(λx.MN)P is the same as((λx.M)P)((λx.N)P).
The combinatorsB andC are similar toS, but pass the argument on to only one subterm of an application (B to the "argument" subterm andC to the "function" subterm), thus saving a subsequentK if there is no occurrence ofx in one subterm. In comparison toB andC, theS combinator actually conflates two functionalities: rearranging arguments, and duplicating an argument so that it may be used in two places. TheW combinator does only the latter, yielding theB, C, K, W system as an alternative toSKI combinator calculus.
Atyped lambda calculus is a typedformalism that uses the lambda-symbol () 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 (seeKinds 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 anduntyped lambda calculus a special case with only one type.[29]
Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typedimperative programming languages. Typed lambda calculi play an important role in the design oftype 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 tomathematical logic andproof theory via theCurry–Howard isomorphism and they can be considered as theinternal language of classes ofcategories, e.g., the simply typed lambda calculus is the language of aCartesian closed category (CCC).[citation needed]
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:[30][31][32]
Weak reduction strategies do not reduce under lambda abstractions:
Strategies with sharing reduce computations that are "the same" in parallel:
There is no algorithm that takes as input any two lambda expressions and outputsTRUE orFALSE depending on whether one expression reduces to the other.[15] More precisely, nocomputable function candecide the question. This was historically the first problem for which undecidability could be proven. As usual for such a proof,computable means computable by anymodel of computation that isTuring complete. In fact computability can itself be defined via the lambda calculus: a functionF:N →N of natural numbers is a computable function if and only if there exists a lambda expressionf such that for every pair ofx,y inN,F(x)=y if and only iffx =β y, wherex andy are theChurch numerals corresponding tox andy, respectively and =β meaning equivalence with β-reduction. See theChurch–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 anormal 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 aGödel numbering for lambda expressions, he constructs a lambda expressione that closely follows the proof ofGödel's first incompleteness theorem. Ife is applied to its own Gödel number, a contradiction results.
The notion ofcomputational complexity for the lambda calculus is a bit tricky, because the cost of a β-reduction may vary depending on how it is implemented.[33] To be precise, one must somehow find the location of all of the occurrences of the bound variableV in the expressionE, 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 ofV inE isO(n) in the lengthn ofE.Director strings were an early approach that traded this time cost for a quadratic space usage.[34] More generally this has led to the study of systems that useexplicit substitution.
In 2014, it was shown that the number of β-reduction steps taken by normal order reduction to reduce a term is areasonable time cost model, that is, the reduction can be simulated on a Turing machine in time polynomially proportional to the number of steps.[35] This was a long-standing open problem, due tosize explosion, the existence of lambda terms which grow exponentially in size for each β-reduction. 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.[36]
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 β-reduction 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 β-reduction per se but rather the handling of the duplication of redexes during β-reduction.[37] It is not known if optimal reduction implementations are reasonable when measured with respect to a reasonable cost model such as the number of leftmost-outermost 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 leftmost-outermost.[36] In addition the BOHM prototype implementation of optimal reduction outperformed bothCaml Light and Haskell on pure lambda terms.[37]
As pointed out byPeter Landin's 1965 paper "A Correspondence betweenALGOL 60 and Church's Lambda-notation",[38] sequentialprocedural programming languages can be understood in terms of the lambda calculus, which provides the basic mechanisms for procedural abstraction and procedure (subprogram) application.
For example, inPython the "square" function can be expressed as a lambda expression as follows:
(lambdax:x**2)
The above example is an expression that evaluates to a first-class function. The symbollambda creates an anonymous function, given a list of parameter names—just the single argumentx, in this case—and an expression that is evaluated as the body of the function,x**2. Anonymous functions are sometimes calledlambda expressions.
Pascal and many other imperative languages have long supported passingsubprograms asarguments to other subprograms through the mechanism offunction pointers. However, function pointers are an insufficient condition for functions to befirst class datatypes, because a function is a first class datatype if and only if new instances of the function can be created atruntime. Such runtime creation of functions is supported inSmalltalk,JavaScript,Wolfram Language, and more recently inScala,Eiffel (as agents),C# (as delegates) andC++11, among others.
TheChurch–Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out inany order, even in parallel. This means that variousnondeterministicevaluation strategies are relevant. However, the lambda calculus does not offer any explicit constructs forparallelism. One can add constructs such asfutures to the lambda calculus. Otherprocess calculi have been developed for describing communication and concurrency.
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 setD isomorphic to the function spaceD →D, of functions on itself. However, no nontrivial suchD can exist, bycardinality constraints because the set of all functions fromD toD has greater cardinality thanD, unlessD is asingleton set.
In the 1970s,Dana Scott showed that if onlycontinuous functions were considered, a set ordomainD with the required property could be found, thus providing amodel for the lambda calculus.[39]
This work also formed the basis for thedenotational semantics of programming languages.
These extensions are in thelambda cube:
Theseformal systems are extensions of lambda calculus that are not in the lambda cube:
These formal systems are variations of lambda calculus:
These formal systems are related to lambda calculus:
Some parts of this article are based on material fromFOLDOC, used withpermission.
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