] x := Since this is the only available reduction, Ω has no normal form (under any evaluation strategy). means x {\displaystyle stx} ) , Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. z 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 that use explicit substitution. s x to an input Lévy shows the existence of lambda terms where there does not exist a sequence of reductions which reduces them without duplicating work. The lambda calculus was an attempt to formalise functions as a means of computing. . {\displaystyle x} The latter has a different meaning from the original. y s For example, the predecessor function can be defined as: which can be verified by showing inductively that n (λg.λk.ISZERO (g 1) k (PLUS (g k) 1)) (λv.0) is the add n − 1 function for n > 0. x r Lambda calculus (λ-calculus), originally created by Alonzo Church, is the world’s smallest programming language. λ ( If repeated application of the reduction steps eventually terminates, then by the Church–Rosser theorem it will produce a β-normal form. m The precise rules for alpha-conversion are not completely trivial. The abstraction binds the variable x , no matter the input. By convention, the following two definitions (known as Church booleans) are used for the boolean values TRUE and FALSE: 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 AND TRUE FALSE is equivalent to FALSE. (λw.(h(w(λy.y))))))). x��ɒ���_���U#+ �*��.ۉ��̸r�����-f\$�MR���{�H�Z�L�K����m@�~w��o�X0F*�������D_�eI4�j�n��+�a��/ﾇd���D3^-�Y���v��q׌�+�貿��?�R-�[F��-S�z�o�;���{8�f���OBW)KY8Dﶍ������j�m����S̯�1m�=B]�UD�R-V8�{�8r�6�f����qJ���+V {\displaystyle x} This formalism was developed by Alonzo Church as a tool for study-ing the mathematical properties of e ectively computable functions. ] . → the simply typed lambda calculus is the language of Cartesian closed categories (CCCs). x represents the constant function Any of the computer programs we have ever written and any of the ones that are still unwritt… . the abstraction symbols λ (lambda) and . For instance, {\textstyle x^{2}+y^{2}} x ) y However, some parentheses can be omitted according to certain rules. y x x The lambda calculus is a programming language with three features: functions, function application, and variables. := In [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation “ is an abstraction for the function λ {\displaystyle \lambda x. λ The Church numeral n is a function that takes a function f as argument and returns the n-th composition of f, i.e. {\displaystyle s} In contrast, normal order is so called because it always finds a normalizing reduction, if one exists. {\displaystyle (\lambda x. t The set of free variables of an expression is defined inductively: For example, the lambda term representing the identity However the programmer thinks in types. → 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. ) However, function pointers are not a sufficient condition for functions to be first class datatypes, because a function is a first class datatype if and only if new instances of the function can be created at run-time. x 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. 2 represents the identity function applied to  To be precise, one must somehow find the location of all of the occurrences of the bound variable V in the expression E, implying a time cost, or one must keep track of these locations in some way, implying a space cost. t s y ] The lambda calculus provides a simple semantics for computation, enabling properties of computation to be studied formally. The precise notion of duplicated work relies on noticing that after the first reduction of I I is done, the value of the other I I can be determined, because they have the same structure (and in fact they have exactly the same values), and result from a common ancestor. indicates substitution of (λx.xx) (y I)) I, (λx.xx) (II) which we know we can do without duplicating work. [ {\displaystyle t} An abstraction {\displaystyle ts} . In lambda calculus, function application is regarded as left-associative, so that on input λ Thus to use f to mean M (some explicit lambda-term) in N (another lambda-term, the "main program"), one can say, Authors often introduce syntactic sugar, such as let, to permit writing the above in the more intuitive order. Here’s an example function. . . The lambda calculus is a programming language with three ideas: functions, function application, and variables. s ) λ a x = {\displaystyle t[x:=r]} := (f(f(λz.z)))) 0 Comments. {\displaystyle x\mapsto y} λ Most purely functional programming languages (notably Miranda and its descendants, including Haskell), and the proof languages of theorem provers, use lazy evaluation, which is essentially the same as call by need. For example, a substitution is made that ignores the freshness condition: ] ] we consider two normal forms to be equal if it is possible to α-convert one into the other). Once you have arithmetics, … y x 2 Lambda calculus has applications in many different areas in mathematics, philosophy, linguistics, and computer science. t {\displaystyle (\lambda x.t)s} The meaning of lambda expressions is defined by how expressions can be reduced.. If N is a lambda-term without abstraction, but possibly containing named constants (combinators), then there exists a lambda-term T(x,N) which is equivalent to λx.N but lacks abstraction (except as part of the named constants, if these are considered non-atomic). := {\displaystyle s} ( = A formal logic developed by Alonzo Church and Stephen Kleene to address the computable number problem. ^ {\displaystyle \lambda x.t} 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. , and the meaning of the function is preserved by substitution. Examples. The research on functional quantum programming started with an attempt to define a quantum extension of lambda calculus made by Maymin  and van Tonder . λ In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see Kinds of typed lambda calculi). λ , and . := x A character or string representing a parameter or mathematical/logical value. 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. represents the application of a function Building on earlier work by Kleene and constructing a Gödel numbering for lambda expressions, he constructs a lambda expression e that closely follows the proof of Gödel's first incompleteness theorem. ( := x x s In this post I will introduce some of the basic concepts of the Lambda Calculus and use them to define basic terms and operators of the boolean logic. The correct substitution in this case is λz.x, up to α-equivalence. ) to denote anonymous function abstraction. The creation of general rules tends to simplify a problem. 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. {\displaystyle (\lambda x.x)[y:=y]=\lambda x. {\displaystyle x} y x → + x We can apply the identity function to itself! ) x x Second, α-conversion is not possible if it would result in a variable getting captured by a different abstraction. u 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. λ In lambda calculus, a library would take the form of a collection of previously defined functions, which as lambda-terms are merely particular constants. Suppose x Formal mathematical logic system centered on function abstractions and applications, Lambda calculus and programming languages, 4 × (3 × (2 × (1 × (1, if 0 = 0; else 0 × ((. , to obtain y Therefore, both examples evaluate to the identity function An ordinary function that requires two inputs, for instance the For example, it is not correct for (λx.y)[y := x] to result in λx.x, because the substituted x was supposed to be free but ended up being bound. Reducing the outer x term first results in the inner y term being duplicated, and each copy will have to be reduced, but reducing the inner y term first will duplicate its argument z, which will cause work to be duplicated when the values of h and w are made known. t s A function is a mapping from the elements of a domain set to the elements of a codomain set given by a rule—for example, cube : Integer → Integer where cube(n) = n3. [ for ) Applications are assumed to be left associative: M N P may be written instead of ((M N) P). {\displaystyle \lambda x.x} y Since we want to do programming in lambda calculus, we want to be able to express our intentions in the source code. 2 where the input is simply mapped to itself. . For example, for every The lambda calculus was developed in the 1930s by Alonzo Church (1903–1995), one of the leading developers of mathematical logic. x If e is applied to its own Gödel number, a contradiction results. ( A linked list can be defined as either NIL for the empty list, or the PAIR of an element and a smaller list. The second simplification is that the lambda calculus only uses functions of a single input. x := λ s . That’s it! x If a name is assigned to the redex that produces all the resulting II terms, and then all duplicated occurrences of II can be tracked and reduced in one go. y To keep the notation of lambda expressions uncluttered, the following conventions are usually applied: The abstraction operator, λ, is said to bind its variable wherever it occurs in the body of the abstraction. y . λ Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. ( Lambda Calculus. For example, in the expression λy.x x y, y is a bound variable and x is a free variable. s {\displaystyle (\lambda x.x)s\to x[x:=s]=s} The term abstractionderives from the creation of general rules and concepts based on the use and classification of specific examples. x {\displaystyle \Omega =(\lambda x.xx)(\lambda x.xx)} x Theorems; … Let us begin by looking at another well-known language of expressions, namely arithmetic. 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. in a capture-avoiding manner. ... ) (λh.y)) and y=((λf. Lambda calculus may be untyped or typed. 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. For example, In the following example the single occurrence of x in the expression is bound by the second lambda: λx.y (λx.z x). = The λ-calculus is an elegant notation for working withapplications of functions to arguments. ( . For example, Pascal and many other imperative languages have long supported passing subprograms as arguments to other subprograms through the mechanism of function pointers. x := In this case the body expression is also xitself. {\displaystyle x} This substitution turns the constant function Week 7 of 2020 Spring. x It is composed of three similar terms, x=((λg. + The basic lambda calculus may be used to model booleans, arithmetic, data structures and recursion, as illustrated in the following sub-sections. We also speak of the resulting equivalences: two expressions are α-equivalent, if they can be α-converted into the same expression. {\displaystyle z} ” to “∧ x = 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). [ These names will be either written in … {\displaystyle (\lambda x.y)s\to y[x:=s]=y} = λ Also a variable is bound by its nearest abstraction. {\displaystyle y} In the above example, KIΩ reduces under normal order to I, a normal form. However, it is not obvious that a redex will produce the II term. In particular, we can now cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively. ” to distinguish function-abstraction from class-abstraction, and then changing “∧” to “λ” for ease of printing. really is the identity. ( In the De Bruijn index notation, any two α-equivalent terms are syntactically identical. ] has no free variables, but the function Similarly, s {\displaystyle t[x:=s]} Thus a lambda term is valid if and only if it can be obtained by repeated application of these three rules. A predicate is a function that returns a boolean value. ( . f y  More precisely, no computable function can decide the equivalence. The syntax of the lambda calculus is short and simple. x the function f composed with itself n times. stream x Already, our factorial example above is shorter than equivalent code in many high-level languages! Abstractions are perhaps the most iconic kind of lambda expression, they define what we call functions or, more adequately, lambdas: which are just anonymous functions. ) You’ll uncover when lambda calculus was introduced and why it’s a fundamental concept that ended up in the Python ecosystem. ) ( While it seems simple on the surface, lambda calculus has given rise to a lot of theory, and the things that you can do with it are quite complex. A notable restriction of this let is that the name f is not defined in M, since M is outside the scope of the abstraction binding f; this means a recursive function definition cannot be used as the M with let. Further, r = {\displaystyle t[x:=s]} x The Lambda calculus is an abstract mathematical theory of computation, involving λ \lambda λ functions. y x ( Here’s an example function.  Lambda calculus has played an important role in the development of the theory of programming languages. x It is a Turing complete language; that is to say, any machine which can compute the lambda calculus can compute everything a Turing machine can (and vice versa). u x = This is one of the many ways to define computability; see the Church–Turing thesis for a discussion of other approaches and their equivalence. A basic form of equivalence, definable on lambda terms, is alpha equivalence. /Length 3650 . . = 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. K throws the argument away, just like (λx.N) would do if x has no free occurrence in N. S passes the argument on to both subterms of the application, and then applies the result of the first to the result of the second. It is a Turing complete language; that is to say, any machine which can compute the lambda calculus can compute everything a Turing machine can (and vice versa). x {\displaystyle y} . This can also be viewed as anonymising variables, as T(x,N) removes all occurrences of x from N, while still allowing argument values to be substituted into the positions where N contains an x. "). More details can be found in the short article, Types and Programming Languages, p. 273, Benjamin C. Pierce, Learn how and when to remove this template message, α-renaming to make name resolution trivial, Normalization property (abstract rewriting), SKI combinator calculus § Self-application and recursion, Combinatory logic § Completeness of the S-K basis, Sharing in the Evaluation of lambda Expressions, Lambdascope: Another optimal implementation of the lambda-calculus, About the efficient reduction of lambda terms, "The typed λ-calculus is not elementary recursive", "Director Strings Revisited: A Generic Approach to the Efficient Representation of Free Variables in Higher-order Rewriting", Structure and Interpretation of Computer Programs, The Impact of the Lambda Calculus in Logic and Computer Science, History of Lambda-calculus and Combinatory Logic, An introduction to λ-calculi and arithmetic with a decent selection of exercises, Step by Step Introduction to Lambda Calculus, A Short Introduction to the Lambda Calculus, To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction, A Tutorial Introduction to the Lambda Calculus, Alligator Eggs: A Puzzle Game Based on Lambda Calculus, Lambda Calculus links on Lambda-the-Ultimate, https://en.wikipedia.org/w/index.php?title=Lambda_calculus&oldid=996517093, Articles with dead external links from December 2017, Articles with permanently dead external links, Short description is different from Wikidata, Articles lacking in-text citations from September 2013, Articles with unsourced statements from March 2020, Wikipedia articles with SUDOC identifiers, Creative Commons Attribution-ShareAlike License. x A pair (2-tuple) can be defined in terms of TRUE and FALSE, by using the Church encoding for pairs. {\displaystyle (\lambda x.y)[y:=x]=\lambda x. These transformation rules can be viewed as an equational theory or as an operational definition. x Roughly speaking, the resulting reduction is optimal because every term that would have the same labels as per Lévy's paper would also be the same graph in the interaction net. It captures the intuition that the particular choice of a bound variable, in an abstraction, does not (usually) matter. λ Several of these have direct applications in the elimination of abstraction that turns lambda terms into combinator calculus terms. x That’s it! An application Lambda expressions in Python and other programming languages have their roots in lambda calculus, a model of computation invented by Alonzo Church. SUB m n yields m − n when m > n and 0 otherwise. Doing the same but in applicative order yields (λf.f I) (λy.y I (y I)), (λy.y I (y I)) I, I I (I I), and now work is duplicated. x The usual counterexample is as follows: define Ω = ωω where ω = λx.xx. := x . λ λ We would like to have a generic solution, without a need for any re-writes: Given a lambda term with first argument representing recursive call (e.g. ] y . ) 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: An expression that contains no free variables is said to be closed. are β-equivalent lambda expressions. x x y The $$\lambda$$-calculus is an elegant notation for working with applications of functions to arguments.To take a mathematical example, suppose we are given a simple polynomial such as $$x^2 -2\cdot x+5$$. Could a sensible meaning be assigned to lambda calculus terms? The combinators B and C are similar to S, but pass the argument on to only one subterm of an application (B to the "argument" subterm and C to the "function" subterm), thus saving a subsequent K if there is no occurrence of x in one subterm. For example, PAIR encapsulates the pair (x,y), FIRST returns the first element of the pair, and SECOND returns the second. x The argument II is duplicated by the application to the first lambda term. These formal 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: Monographs/textbooks for graduate students: Some parts of this article are based on material from FOLDOC, used with permission. ( In the example given above, (λx.xx) ((λx.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. x t . + Examples. No numbers, strings, for loops, modules, and so on. ( Thus to achieve recursion, the intended-as-self-referencing argument (called r here) must always be passed to itself within the function body, at a call point: 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. y The ID in the beginning of that abstraction is called the metavariable. For example, an α-conversion of λx.λx.x could result in λy.λx.x, but it could not result in λy.λx.y. This simplicity provides great power, an example of `less is more'. Substitution is defined uniquely up to α-equivalence. ] The definition of a function with an abstraction merely "sets up" the function but does not invoke it. In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable. {\displaystyle \lambda x.yx} . In comparison to B and C, the S combinator actually conflates two functionalities: rearranging arguments, and duplicating an argument so that it may be used in two places. We computethis by ‘plugging in’ 2 for x in the expression: weget 22−2⋅2+5,which we can further reduce to get the answer 5. For instance, it may be desirable to write a function that only operates on numbers. The lambda term is. ) {\displaystyle {\hat {x}}} ((\lambda x.x)x)} First, I’ll show you what the lambda calculus looks like by example, and then we can work through its formal syntax/semantics. {\displaystyle x^{2}+2} x {\displaystyle y} In lambda calculus, there are only lambdas, and all you can do with them is substitution. For a full history, see Cardone and Hindley's "History of Lambda-calculus and Combinatory Logic" (2006). . ) ( While the idea of β-reduction seems simple enough, it is not an atomic step, in that it must have a non-trivial cost when estimating computational complexity. λ and and returns No numbers, strings, for loops, modules, and so on. x => x and y => y are the same function. {\displaystyle \lambda x.x+y} t x They only accept one input variable, with currying used to implement functions with several variables. 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. In contrast, sweetened Turing machines would probably still be unpalatable. {\displaystyle (\lambda z.y)[y:=x]=\lambda z. = Schemeis a Functional language! Dana Scott has also addressed this controversy in various public lectures. On the other hand, using applicative order can result in redundant reductions or even possibly never reduce to normal form. denote different terms (although they coincidentally reduce to the same value). x x The identity function returns the only argument applied to it as is. . For example, x 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. Thus the original lambda expression (FIX G) is re-created inside itself, at call-point, achieving self-reference. The abstraction {\textstyle \operatorname {square\_sum} } Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! ( {\displaystyle (\lambda x.t)s} . . := x Applying a function to an argument. [ The amazing thing about λ-calculus is that it is possible to represent numbers and the arithmetic operations (successor, addition and multiplication) as functions. x A function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F(x)=y if and only if f x =β y,  where x and y are the Church numerals corresponding to x and y, respectively and =β meaning equivalence with β-reduction. {\displaystyle y} ( In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. The notation Lamdba calculus includes three different types of expressions, i.e., E :: = x(variables) | E1 E2(function application) | λx.E(function creation) Where λx.Eis called Lambda abstraction and E is known as λ-expressions. x 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. y one-line universal program: Here’s a lambda calculus self-interpreter: $$(\lambda f.(\lambda x.f(xx))(\lambda x.f(xx)))(\lambda em.m(\lambda x.x)(\lambda mn.em(en))(\lambda mv.e(mv)))$$. The following example defines a function add that performs a mathematical addition of two numbers using Church numerals (which are not defined here). y Lambda calculus is important in programming language theory, and the symbol λ has even been adopted as an unofficial symbol for the field. x x One can add constructs such as Futures to the lambda calculus. Terms that differ only by alpha-conversion are called α-equivalent. Such similar structures can each be assigned a label that can be tracked across reductions. r . This origin was also reported in [Rosser, 1984, p.338]. The lambda calculus incorporates two simplifications that make this semantics simple. 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. The following definitions are necessary in order to be able to define β-reduction: The free variables of a term are those variables not bound by an abstraction. x The lambda calculus was an attempt to formalise functions as a means of computing.