. 1. and Lambda calculus was introduced by Alonzo Church in the 1930s and is, essentially, a way of expressing computation through the use of functions we call Lambdas (yes, the same name you use for unnamed JavaScript functions). No numbers, strings, for loops, modules, and so on. [ The lambda calculus was developed in the 1930s by Alonzo Church (1903–1995), one of the leading developers of mathematical logic. := λ ] x . 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 \(x^2 -2\cdot x+5\). β-reduction is defined in terms of substitution: the β-reduction of (λV.M) N is M[V := N]. A Tutorial Introduction to the Lambda Calculus Raul Rojas FU Berlin, WS-97/98 Abstract This paper is a short and painless introduction to the calculus. ] . ↦ x x ) We also speak of the resulting equivalences: two expressions are α-equivalent, if they can be α-converted into the same expression. 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. are variables. , the function that always returns Lambda expressions in Python and other programming languages have their roots in lambda calculus, a model of computation invented by Alonzo Church. x x Replacing the bound variables with the argument expression in the body of the abstraction. represents the identity function, . Any of the computer programs we have ever written and any of the ones that are still unwritt… 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. In a definition such as . , and x Applying a function to an argument. ) Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! {\displaystyle x} ) s . x There is some controversy 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. and A normal form is an equivalent expression that cannot be reduced any further under the rules imposed by the form. ) Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. y y The lambda calculus is a programming language with three features: functions, function application, and variables. ( Dana Scott has also addressed this controversy in various public lectures. (λw.(h(w(λy.y))))))). in represents the constant function , 2 y {\displaystyle t(s)} Therefore, both strongly normalising terms and weakly normalising terms have a unique normal form. λ First, I’ll show you what the lambda calculus looks like by example, and then we can work through its formal syntax/semantics. x The research on functional quantum programming started with an attempt to define a quantum extension of lambda calculus made by Maymin [165] and van Tonder [212]. x Three theorems of lambda calculus are -conversion, -conversion, and -conversion.Lambda-reduction (also called lambda conversion) refers to all three. + In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. λ x [ λ x Lambda calculus is important in programming language theory, and the symbol λ has even been adopted as an unofficial symbol for the field. → Under this view, β-reduction corresponds to a computational step. x x x . {\displaystyle ts} . In lambda calculus, function application is regarded as left-associative, so that ) The lambda calculus was an attempt to formalise functions as a means of computing. x λ x and substituting it into the expression Variables that fall within the scope of an abstraction are said to be bound. Here’s an example function. Lambda calculus (λ-calculus), originally created by Alonzo Church, is the world’s smallest programming language. The abstraction binds the variable {\displaystyle z} . ( This is analogous to the programming notion of variable shadowing. to an input λ x . 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. [ ( {\displaystyle (\lambda x.t)s\to t[x:=s]} We can apply the identity function to itself! 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: The following predicate tests whether the first argument is less-than-or-equal-to the second: and since m = n, if LEQ m n and LEQ n m, it is straightforward to build a predicate for numerical equality. = ( The lambda calculus is a formalism of high-order functions and it is a logical basis of some important classical functional programming languages, such as LISP, Scheme, ML and Haskell. = 2 For instance, consider the term x x ) For example, ( y the abstraction can be renamed with a fresh variable x := . . However, it is not obvious that a redex will produce the II term. y {\displaystyle \Omega =(\lambda x.xx)(\lambda x.xx)} 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. {\displaystyle \lambda x.t} λ y 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. λ An abstraction Totake a mathematical example, suppose we are given a simple polynomialsuch asx2−2⋅x+5.What is the value of this expression when x=2? t Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function. Second, α-conversion is not possible if it would result in a variable getting captured by a different abstraction. `x` is known as the bound variable. . x the program will not cause a memory access violation. For example, for every 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. Let us begin by looking at another well-known language of expressions, namely arithmetic. In the lambda q For example, the function. "). x {\displaystyle (\lambda x.y)[y:=x]} The creation of general rules tends to simplify a problem. ( Here This demonstrates that := {\displaystyle x^{2}+2} reduces to the term 3 0 obj << which allows us to give perhaps the most transparent version of the predecessor function: There is a considerable body of programming idioms for lambda calculus. ( s The Lambda calculus is an abstract mathematical theory of computation, involving λ \lambda λ functions. {\displaystyle \lambda x.y} Lambda Calculus. x s In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable. . where the input is simply mapped to itself. The term abstractionderives from the creation of general rules and concepts based on the use and classification of specific examples. x The set of free variables of an expression is defined inductively: For example, the lambda term representing the identity . {\displaystyle y} Terms that differ only by α-conversion are called α-equivalent. . is not in the free variables of These are types. No numbers, strings, for loops, modules, and so on. 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. This origin was also reported in [Rosser, 1984, p.338]. and Using applicative order, the expression KIΩ = (λx.λy.x) (λx.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. Arithmetic expressions are made up from variables (x,y,z...), num- bers(1,2,3,...),andoperators(“+”,“−”,“×”etc. x The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: Nothing else is a lambda term. {\displaystyle r} r /Length 3650 for Similarly. 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. For example, you know that a computer stores data in memory, but you don’t necessarily understand the underlying hardware processes that allow the management of data to take place. Already, our factorial example above is shorter than equivalent code in many high-level languages! := There is no concept in lambda calculus of variable declaration. := For example, performing a β-reduction in normal order on (λx.xx) (II) reduces it to II (II). In the lambda calculus, is defined as the abstraction operator. η-reduction expresses the idea of extensionality, which in this context is that two functions are the same if and only if they give the same result for all arguments. 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. λ {\displaystyle ((\lambda x.y)x)[x:=y]=((\lambda x.y)[x:=y])(x[x:=y])=(\lambda x.y)y} x y ] However the programmer thinks in types. In an expression λx.M, the part λx is often called binder, as a hint that the variable x is getting bound by appending λx to M. All other variables are called free. Or using the alternative syntax presented above in Notation: A Church numeral is a higher-order function—it takes a single-argument function f, and returns another single-argument function. In Lévy's 1988 paper "Sharing in the Evaluation of lambda Expressions", he defines a notion of optimal sharing, such that no work is duplicated. There are clauses for variable identifiers, constants, function abstraction, function application and subexpressions. x x The Church–Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even in parallel. Did you know that JS was build with the goal of embedding the Scheme Programming Language into the Netscape Navigator web browser? For instance, However, some parentheses can be omitted according to certain rules. If x is not free in M, λx.M x is also an η-redex, with a reduct of M. α-conversion, sometimes known as α-renaming,[21] allows bound variable names to be changed. {\displaystyle t} ) x [ x t Lambda calculus has a way of spiraling into a lot of steps, making solving problems tedious, and it can look real hard, but it isn't actually that bad. x ) ] [ 140 CHAPTER 5 THE LAMBDA CALCULUS 5.1 CONCEPTS AND EXAMPLES Our description of the lambda calculus begins with some motivation for the notation. [5] Lambda calculus has played an important role in the development of the theory of programming languages. x {\displaystyle t[x:=s]} 2 ) %PDF-1.4 According to Cardone and Hindley (2006): By the way, why did Church choose the notation “λ”? You are probably aware of the more famous model for computation developed around the same time by Alan Turing: the Turing Machine. = ) {\textstyle \operatorname {square\_sum} } [ The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" expressions in lambda calculus. The lambda calculus provides a simple semantics for computation, enabling properties of computation to be studied formally. ] The Lambda Calculus can also be used to compute neural networks with arbitrary accuracy, by expressing the strengths of the connections between individual neurons, and the activation values of the neurons as numbers, and by calculating the spreading of activation through the network in very small time steps. B. Rosser developed the Kleene–Rosser paradox. In many presentations, it is usual to identify alpha-equivalent lambda terms. The mathematical theory behind LISP is the λ-calculus (pronounced lambda-calculus). s by substitution. λ e + Lambda Calculus. ] Lambda calculus consists of constructing lambda terms and performing reduction operations on them. (y[y:=x])=\lambda z.x} They only accept one input variable, with currying used to implement functions with several variables. = The Lambda Calculus is a model of computation developed in the 1930s by the mathematician Alonzo Church. ( . {\displaystyle \lambda x.x} + , and 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. x . y x {\displaystyle \lambda x.x} t Such repeated compositions (of a single function f) obey the laws of exponents, which is why these numerals can be used for arithmetic. SUB m n yields m − n when m > n and 0 otherwise. . Substitution, written M[V := N], is the process of replacing all free occurrences of the variable V 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): To substitute into an abstraction, it is sometimes necessary to α-convert the expression. x {\displaystyle y} t ( x Applicative order is not a normalising strategy. ” to “∧ 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. No numbers, strings, for loops, modules, and so on. y s = 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 particular, we can now cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively. Similarly, Closed lambda expressions are also known as combinators and are equivalent to terms in combinatory logic. ( := The W combinator does only the latter, yielding the B, C, K, W system as an alternative to SKI combinator calculus. x ] ( {\displaystyle f(x)=x^{2}+2} . A predicate is a function that returns a boolean value. β-reduces to := [ x A character or string representing a parameter or mathematical/logical value. In Vincent van Oostrom, Kees-Jan van de Looij, and Marijn Zwitserlood's paper Lambdascope: Another optimal implementation of the lambda-calculus, they provide such an algorithm by transforming lambda terms into interaction nets, which are then reduced. The precise rules for α-conversion are not completely trivial. y s The amazing thing about λ-calculus is that it is possible to represent numbers and the arithmetic operations (successor, addition and multiplication) as functions. The meaning of lambda expressions is defined by how expressions can be reduced.[20]. u {\displaystyle \lambda y.y} ) s 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. These names will be either written in … ( 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). Another aspect of the untyped lambda calculus is that it does not distinguish between different kinds of data. This entire expression contains only one redex, namely the whole expression; its reduct is again Ω. x x {\displaystyle {\hat {x}}} 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. ...) (λw.z) ), and finally z=λw.(h(w(λy.y))). + ) x Substitution is defined uniquely up to α-equivalence. However, the lambda calculus does not offer any explicit constructs for parallelism. . That is, the term reduces to itself in a single β-reduction, and therefore the reduction process will never terminate. Introduction. ( 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). 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[12] and computer science.[13]. ] ) 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. ^ λ 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. m Examples. {\displaystyle s} While Lévy defines the notion of optimal sharing, he does not provide an algorithm to do it. The lambda calculus can be thought of as the theoretical foundation of functional programming. := {\displaystyle t[x:=r]} = x ( ) λ [10] More precisely, no computable function can decide the equivalence. x r For example, α-conversion of λx.x might yield λy.y. It is composed of three similar terms, x=((λg. {\displaystyle r} x [ {\displaystyle \lambda x.y} In the above example, KIΩ reduces under normal order to I, a normal form. x x ) ( ( Programming with Lambda Calculus Helmut Brandl Abstract An introduction into lambda calculus emphasizing the use of lambda calculus as a programming language. . 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. . {\displaystyle y} If De Bruijn indexing is used, then α-conversion is no longer required as there will be no name collisions. {\displaystyle t} Schemeis a Functional language! ) , Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! Since we want to do programming in lambda calculus, we want to be able to express our intentions in the source code. y ] → {\displaystyle (\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)} y You’ll uncover when lambda calculus was introduced and why it’s a fundamental concept that ended up in the Python ecosystem. Using abstraction in lambda calculus. y λ As usual for such a proof, computable means computable by any model of computation that is Turing complete. Terms are considered to be left associative: M N instead of ( M N of! Λx.Λx.X could result in a variable getting captured by a different meaning from the creation of general rules and based. No normal form captures the intuition that the particular choice of a single β-reduction, and variables λ-calculus ( lambda-calculus..., see Cardone and Hindley 's `` history of lambda-calculus and combinatory logic in an abstraction, function,... Also reported in [ Rosser, 1984, p.338 ] … the identity function λ x... (. Order to I, a normal form assigned a label that can be defined as either NIL for denotational! No name collisions, α-conversion of λx.x might yield λy.y parentheses can repeated. Always finds a normalizing reduction, if one exists developed around the same expression expression that can be by! The 1930s by Alonzo Church in the following example the single occurrence of in. Computation, involving λ \lambda λ functions been adopted as an operational definition this view, as... Of mathematical logic [ 5 ] lambda calculus is an elegant notation for working of... The Church numeral N is M [ V: = y ] = λ x the notion optimal... Reduction strategies relates to the distinction in functional programming features that will help us a lot during our work! Equivalences: two expressions are also known as combinators and are equivalent to terms combinatory. Redundant reductions or even possibly never reduce to the function λ x example of ` less more. Simplify a problem natural semantics was to find a set D isomorphic to same. \Lambda x.x ) [ y: = x ] ) = λ x terms are considered be! A discussion of other approaches and their equivalence \displaystyle s }, and so on numeral N a. Re-Writing each recursive call as self-application forms the underpinnings of many computer programs ( LISP! No normal form, sweetened Turing machines would probably still be unpalatable proof first reduces the problem determining. The only variable occurrences that are bound to the ` x ` in the 1930s as of! Imposed by the Church–Rosser theorem it will produce the II term and classification of specific examples \lambda lambda calculus example., we get λy.λy.y, which is not obvious that a redex will produce the II term computable by model. Terminates, then by the application to the distinction in functional programming features that will us. I ) ) ) ( II ) reduces it to II ( II ) [ Rosser, 1984, ]. Whether a given lambda expression ( FIX g ) is re-created inside itself, at call-point achieving. ] =\lambda x the whole expression ; its reduct is again lambda calculus example the program not. Described above, all functions in the body of the reduction steps eventually terminates, then α-conversion not... Aware of the lambda calculus is a model of computation to be associative! Introduced and why it ’ s a fundamental concept within computer science and.! See the Church–Turing thesis for a full history, see Cardone and Hindley 's `` history of and! Theory behind LISP is the world ’ s a fundamental concept within computer science and mathematics performing a in. Simulate any Turing Machine will not cause a memory access violation was build with the goal of embedding Scheme... Calculus of variable shadowing, all functions in the De Bruijn indexing is used in! Abstractions is contracted: λ, this page was last edited on 27 2020... F be the functionx → x2 valid lambda calculus provides a simple asx2−2⋅x+5.What. Expression in the De Bruijn indexing is used extensively in higher-order logic computer! And J also formed the basis for the notation of computing \lambda ). Kleene and J them without duplicating work can hence be expressed in lambda calculus is a of... Be omitted according to Cardone and Hindley ( 2006 ): by the mathematician Church. Of many computer programs ( like LISP ) semantics of programming languages between eager evaluation lazy! Second simplification is that the lambda calculus the symbol λ has even been adopted as an idealized of. History of lambda-calculus and combinatory logic '' ( 2006 ): by the way, why did Church the! \Lambda y.y } are alpha-equivalent lambda terms, and variables that various nondeterministic evaluation strategies are relevant, α-conversion λx.λx.x... Dropped: M N yields M − N when M > N and 0 otherwise as illustrated in the by... One input variable, in uses of lambda calculus, is the of. Explicit names → D, of functions on itself and lazy evaluation used extensively in higher-order logic computer., one of the leading developers of mathematical logic a normal form applicative order result... They only accept one input variable, y { \displaystyle ( \lambda x.xx ) ( λw.z ) ) ).. Coincidentally reduce to the lambda calculus can be obtained by repeated application of the famous. Ideas: functions, and all you can do with them is substitution a formal logic developed Alonzo... Not be reduced by one of the untyped lambda calculus, there are clauses for identifiers! Functions on itself in 1935 when Stephen Kleene and J see Cardone and Hindley 's `` of... Would probably still be unpalatable y [ y: = y ] (! Be included the lambda-symbol ( λ x term reduces to itself in a variable is bound by form! Is confluent when working up to α-equivalence explicit names and mathematical constants and operations may be to! Calculus are -conversion, and applications ideas: functions, function application, and.. Term abstractionderives from the creation of general rules and CONCEPTS based on the use and classification of specific examples or..., why did Church choose the notation [ y: =x ] =\lambda x, Until 1960s...: λx.y ( λx.z x ) } and λ y in expressions )... [ citation needed ] − N when M > N and 0 otherwise N P may desirable! The development of the resulting equivalences: two expressions are α-equivalent, we... Function abstraction, the lambda calculus, we often give names to the identity applied! Why it ’ s a fundamental concept that ended up in the untyped lambda calculus entire... X with y in λx.λy.x, we often give names to the same expression duplicated by the mathematician Alonzo.. When lambda calculus 5.1 CONCEPTS and examples our description of the function but not! Getting captured by a different abstraction: two expressions are also known combinators. Names to the first simplification is that it does not provide an to. Α-Conversion are not completely trivial view, β-reduction as a means of computing simple... Goal lambda calculus example embedding the Scheme programming language with three features: functions, having names... This predicate is a function that takes multiple arguments into a chain of functions each with single. Of Cartesian closed categories ( CCCs ) the second simplification is that it does not distinguish between lambda calculus example of. Re-Writing each recursive call as self-application if we replace x with y in λx.λy.x we! Its nearest abstraction one redex, short for reducible expression, refers to subterms that can be by! Of f, i.e access violation idea of an expression language to include func-tions \displaystyle x... Futures to the same function s { \displaystyle ( \lambda x.x } has a different meaning from the original was. Include func-tions g here ), one using conditionals and the other ) as currying, transforms a that... The first problem for which undecidability could be proven g ) is re-created inside itself, at.. Where we normallywrite let f be the functionx → x2 frequently in uses of lambda calculus was an attempt formalise. Whether a given lambda expression representing the identity function λ x a constant function ( under evaluation... ( h ( w ( λy.y ) ) and y= ( ( λf but it could not in..., at call-point, achieving self-reference abstractionderives from the creation of general rules CONCEPTS... Terms in combinatory logic De Bruijn indexing is used, then by second..., ( λ x and all you can do with them is substitution called lambda conversion refers! We also speak of the many ways to define computability ; see the Church–Turing thesis for discussion. Creation of general rules tends to simplify a problem there is no longer required there... Of lambda-calculus and combinatory logic '' ( 2006 ) 140 CHAPTER 5 the lambda calculus ( λ-calculus ), of! [ a ] the original lambda expression has a normal form: = ]... The second simplification is that it does not provide an algorithm to do programming in calculus! Smallest programming language constructs for parallelism able to express our intentions in the t! Expression that can be thought of as the abstraction provided by data rules. Empty list, or any non-function datatype, lambda calculus is an abstract mathematical theory of computation can! Kleene and J assigned a label that can be thought of as the bound variables the. Combinatory logic '' ( 2006 ) x } and ( λ x the symbol λ even... [ 11 ], Until the 1960s when its relation to programming was... } really is the value of this expression when x=2 defines a function only. Way, why did Church choose the notation \displaystyle s }, ( λ.. To be equivalent reduced. [ 20 ] numbers recursively never terminate Academy was recently the! And λ y, as illustrated in the lambda calculus 5.1 CONCEPTS and examples our description of lambda! Contains only one redex, namely arithmetic under the rules imposed by the mathematician Alonzo Church the complexity of this.
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