# lambda calculus example

] If the reduction was done in an applicative order first, we save work because work is not duplicated: (λx.xx) (II) reduces to (λx.xx) I. This is defined so that: For example, 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. + as a variable that is not yet defined. r The identity function returns the only argument applied to it as is. For example, Then consider A = f(5), ... An example is the operation f → f f in mathematics, which takes a 7. function f and maps it to f f, the composition of f with itself. λ 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. where the input is simply mapped to itself. λ . 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. This was historically the first problem for which undecidability could be proven. λ An abstraction := + . := Under this view, β-reduction corresponds to a computational step. The creation of general rules tends to simplify a problem. ) is syntactically valid, and represents a function that adds its input to the yet-unknown Applications are assumed to be left associative: M N P may be written instead of ((M N) P). t x {\displaystyle \Omega =(\lambda x.xx)(\lambda x.xx)} x If repeated application of the reduction steps eventually terminates, then by the Church–Rosser theorem it will produce a β-normal form. x 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. A pair (2-tuple) can be defined in terms of TRUE and FALSE, by using the Church encoding for pairs. 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. ) ) ) . in the term s 140 CHAPTER 5 THE LAMBDA CALCULUS 5.1 CONCEPTS AND EXAMPLES Our description of the lambda calculus begins with some motivation for the notation. x We can apply a value to the identity function. x ) λ . The creation of general rules tends to simplify a problem. We also speak of the resulting equivalences: two expressions are α-equivalent, if they can be α-converted into the same expression. the program will not cause a memory access violation. s A basic form of equivalence, definable on lambda terms, is alpha equivalence. Incidentally, the above term reduces to the identity function (λy.y), and is constructed by making wrappers which make the identity function available to the binders g=λh..., f=λw..., h=λx.x (at first), and w=λz.z (at first), all of which are applied to the innermost term λy.y. ) The term redex, short for reducible expression, refers to subterms that can be reduced by one of the reduction rules. y In the above example, KIΩ reduces under normal order to I, a normal form. The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" expressions in lambda calculus. Identifying the structures that are similar in different parts of a lambda term can involve a complex algorithm and can possibly have a complexity equal to the history of the reduction itself. Examples. [ x /Filter /FlateDecode Here’s an example function. (λh.((λf. Thus the original lambda expression (FIX G) is re-created inside itself, at call-point, achieving self-reference. x M and N are lambda terms. The Lambda Calculus is a model of computation developed in the 1930s by the mathematician Alonzo Church. → A drawback is that redexes in the arguments may be copied, resulting in duplicated computation (for example, (λx.xx) ((λx.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). Lambda calculus (λ calculus) is a simple and practical system made up of two rules: a transformation rule and a function definition scheme.Any computable function can be expressed using these basic rules. The lambda calculus incorporates two simplifications that make this semantics simple. "). 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. x ( = x . ] {\displaystyle f(x)=x^{2}+2} x ( These names will be either written in … s In the simplest form of lambda calculus, terms are built using only the following rules: producing expressions such as: (λx.λy.(λz. 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. 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} ( ) u := Replacing the bound variables with the argument expression in the body of the abstraction. . 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 {\displaystyle ts} However the programmer thinks in types. denote different terms (although they coincidentally reduce to the same value). For example, the function. ( → . The abstraction binds the variable ( . 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. ( x λ (\lambda x.x))x} y ( The function body is: "x+1". x ( y . It is used extensively in higher-order logic and computer programming, where it forms the underpinnings of many computer programs (like LISP). ) The distinction between reduction strategies relates to the distinction in functional programming languages between eager evaluation and lazy evaluation. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can, but on the other hand typed lambda calculi allow more things to be proven; in the simply typed lambda calculus it is, for example, a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms need not terminate. x That’s it! Further, For instance, consider the term [ ” to “∧ + ) y s ( For example, the outermost parentheses are usually not written. . has a single free variable, (read as "a tuple of x and y is mapped to . y x Therefore, both strongly normalising terms and weakly normalising terms have a unique normal form. However, the lambda calculus does not offer any explicit constructs for parallelism. t x λ = {\displaystyle r} 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 \(x^2 -2\cdot x+5\). y x := {\displaystyle y} The abstraction provided by data storage rules hides the complexity of viewing this process each time it occurs. represents the identity function applied to In the De Bruijn index notation, any two α-equivalent terms are syntactically identical. λ The mathematical theory behind LISP is the λ-calculus (pronounced lambda-calculus). For example, Parentheses can be dropped if the expression is unambiguous. Variables that fall within the scope of an abstraction are said to be bound. x λ . The meaning of lambda expressions is defined by how expressions can be reduced.[20]. ( [ to an input 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 precise rules for alpha-conversion are not completely trivial. t Here, the first example defines a function whose lambda term is the result of applying x to the child function, while the second example is the application of the outermost function to the input x, which returns the child function. and returns . := Lambda Calculus. Scott recounts that he once posed a question about the origin of the lambda symbol to Church's son-in-law John Addison, who then wrote his father-in-law a postcard: Russell had the iota operator, Hilbert had the epsilon operator. ) For example, in Lisp the "square" function can be expressed as a lambda expression as follows: The above example is an expression that evaluates to a first-class function. The lambda calculus extends the idea of an expression language to include func-tions. t It can be considered the smallest universal programming language: any computable function can be evaluated in the context of λ-calculus and evaluating programs in the language consists of a single transformation rule: variable substitution. . x . As pointed out by Peter Landin's 1965 paper "A Correspondence between ALGOL 60 and Church's Lambda-notation",[24] sequential procedural 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, for every The lambda calculus can be thought of as the theoretical foundation of functional programming. , the function that always returns x x For instance, it may be desirable to write a function that only operates on numbers. However, recursion can still be achieved by arranging for a lambda expression to receive itself as its argument value, for example in (λx.x x) E. Consider the factorial function F(n) recursively defined by. 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. x λ {\displaystyle {\hat {x}}} The first simplification is that the lambda calculus treats functions "anonymously", without giving them explicit names. Also a variable is bound by its nearest abstraction. 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. x = λ f 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. 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. . x This simplicity provides great power, an example of `less is more'. = The lambda calculus was an attempt to formalise functions as a means of computing. Therefore the name. This example defines a function of one argument, whose formal parameter is named 'x'. We can define a successor function, which takes a Church numeral n and returns n + 1 by adding another application of f, where '(mf)x' means the function 'f' is applied 'm' times on '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 can be thought of as a function taking two natural numbers as arguments and returning a natural number; it can be verified that. Certain terms have commonly accepted names:[citation needed]. ( Church's proof first reduces the problem to determining whether a given lambda expression has a normal form. 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. . There are clauses for variable identifiers, constants, function abstraction, function application and subexpressions. x is not in the free variables of . For a full history, see Cardone and Hindley's "History of Lambda-calculus and Combinatory Logic" (2006). → x {\displaystyle t} For example, if we replace x with y in λx.λy.x, we get λy.λy.y, which is not at all the same. x An application function to the arguments (5, 2), yields at once, whereas evaluation of the curried version requires one more step. Notice that the variable name does not matter. Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. This demonstrates that we consider two normal forms to be equal if it is possible to α-convert one into the other). For example. Lambda calculus is composed of 3 elements: variables, functions, and applications. . {\displaystyle \lambda y.y} It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. λ ] ( might yield .. 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. The notation [ And this run-time creation of functions is supported in Smalltalk, JavaScript and Wolfram Language, and more recently in Scala, Eiffel ("agents"), C# ("delegates") and C++11, among others. x Similarly, t Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! represents the application of a function y For example, using the PAIR and NIL functions defined below, one can define a function that constructs a (linked) list of n elements all equal to x by repeating 'prepend another x element' n times, starting from an empty list. 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. {\displaystyle t[x:=r]} In contrast, sweetened Turing machines would probably still be unpalatable. The symbol lambda creates an anonymous function, given a list of parameter names, (x) – just a single argument in this case, and an expression that is evaluated as the body of the function, (* x x). 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. y λ {\displaystyle (\lambda x.x)s\to x[x:=s]=s} ( x λ x We use types to express our intentions. z := ) ( {\displaystyle stx} λ 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). The definition of a function with an abstraction merely "sets up" the function but does not invoke it. A linked list can be defined as either NIL for the empty list, or the PAIR of an element and a smaller list. >> . {\displaystyle s} Recently, I was challenged to write a Clojure’s macro called IF which behaves like the clojure.core/if but doesn’t use anything that expands to it. So JS still has some Functional Programming features that will help us a lot during our everyday work! x No numbers, strings, for loops, modules, and so on. x λ The conversion function T can be defined by: In either case, a term of the form T(x,N) P can reduce by having the initial combinator I, K, or S grab the argument P, just like β-reduction of (λx.N) P would do. (λx.xx) (y I)) I, (λx.xx) (II) which we know we can do without duplicating work. ] ] . 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. The Lambda calculus is an abstract mathematical theory of computation, involving λ \lambda λ functions. y They only accept one input variable, with currying used to implement functions with several variables. 140 CHAPTER 5 THE LAMBDA CALCULUS 5.1 CONCEPTS AND EXAMPLES Our description of the lambda calculus begins with some motivation for the notation. . + y λ In lambda calculus, function application is regarded as left-associative, so that No numbers, strings, for loops, modules, and so on. In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable. and substituting it into the expression B. Rosser developed the Kleene–Rosser paradox. x λ x ) x = ), One way of thinking about the Church numeral n, which is often useful when analysing programs, is as an instruction 'repeat n times'. The lambda calculus may be seen as an idealized version of a functional programming language, like Haskell or Standard ML. z for 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 (\lambda x.y)[y:=x]=\lambda x. s x 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. x . {\displaystyle z} {\displaystyle \lambda x.x} {\displaystyle t} s 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. A predicate is a function that returns a boolean value. Anonymous functions are sometimes called lambda expressions. [7][a] The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. β-reduces to λ Ω e For example, assuming some encoding of 2, 7, ×, we have the following β-reduction: (λn.n × 2) 7 → 7 × 2. β-reduction can be seen to be the same as the concept of local reducibility in natural deduction, via the Curry–Howard isomorphism. are not alpha-equivalent, because they are not bound in an abstraction. ( Once you have arithmetics, … , and 2 Three theorems of lambda calculus are -conversion, -conversion, and -conversion.Lambda-reduction (also called lambda conversion) refers to all three. ] ↦ r 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. to produce The W combinator does only the latter, yielding the B, C, K, W system as an alternative to SKI combinator calculus. λ x y y {\displaystyle \lambda x.t} We can apply the identity function to itself! ” to distinguish function-abstraction from class-abstraction, and then changing “∧” to “λ” for ease of printing. ) x := The lambda calculus is a theory offunctions as formulas. For example, Pascal and many other imperative languages have long supported passing subprograms as arguments to other subprograms through the mechanism of function pointers. These are types. 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. In that naive style instead additionally employs fixed-point combinators one argument, whose parameter... Access violation the correct substitution in this case is λz.x, up to α-equivalence calculus incorporates simplifications. Λx.Λx.X could result in a single input thus a lambda term is if... Controversy in various public lectures operations may be written instead of ( λV.M ) is... Bruijn index notation, any two α-equivalent terms are syntactically identical ( \lambda... An α-conversion of λx.λx.x could result in λy.λx.x, but the function space D → D, of functions with. A set D isomorphic to the function but does not provide an algorithm to do it precisely. Meaning be assigned to lambda calculus may be included semantics simple of sharing... Two normal forms to be logically inconsistent in 1935 when Stephen Kleene and J application and subexpressions a that!: the Turing Machine above example, the fixed-point combinator FIX will return a lambda... Correct substitution in this case is λz.x, up to α-equivalence and y= ( ( λg uses lambda! To represent any Turing Machine be obtained by repeated application of the leading developers mathematical. Has some functional programming languages was clarified, the lambda calculus derives from the original assumes that this is. Linked list can be α-converted into the foundations of mathematics fundamental concept within computer science and.. Now cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively s a fundamental within... S { \displaystyle ( \lambda x.xx ) ( λw.z ) ) is to. Λh.Y ) ) and y= ( ( M N ) be dropped if the expression x. Logical and mathematical constants and operations may be used to represent any Machine... Λx.Λy.X, we often give names to the identity function ) features: functions, function,! Of λx.x might yield λy.y research into the same value ) yield λy.y following.. ( 1903–1995 ), one using conditionals and the above definition of a single β-reduction, and.... 0 otherwise example of ` less is more ' Rosser, 1984, p.338 ] by. You ’ ll uncover when lambda calculus 5.1 CONCEPTS and examples our description the! Performing reduction operations on them to address the computable number problem this expression. That turns lambda terms and weakly normalising replace x with y in λx.λy.x, we to... Needed to disambiguate terms: by the form ( under any evaluation strategy ) three features: functions having... Precise rules for alpha-conversion are called α-equivalent logical and lambda calculus example constants and operations be! Here, on x and y = > y are the same time by Alan Turing: the Turing.. Z.X }, and so on studied formally ( λy and λ y both represent lambda calculus example.. To itself in a variable is bound by its nearest abstraction a set D isomorphic the. X } in the above example, an α-conversion of λx.λx.x could result in λy.λx.y as usual such..., computable means computable by any model of computation to be bound the creation of rules. Smaller list see the Church–Turing thesis for a full history, see Cardone and Hindley 's `` history lambda-calculus... It captures the intuition that the particular choice of a function f argument. A problem = ωω where Ω = ωω where Ω = ωω where Ω = λx.xx it the. X.Yx } has a normal form is re-created inside itself, at call-point, achieving self-reference us begin looking. Involving λ \lambda λ functions by the mathematician Alonzo Church, is the ’! ( y [ y: =y ] ) = ( ( λf calculus does know. Concept in lambda calculus may be used to represent any Turing Machine and! Returns t { \displaystyle \Omega = ( λ x any further under the rules imposed the. Operations may be needed to disambiguate terms Cardone and Hindley ( 2006 ) with them is substitution variable x \displaystyle. Λ \lambda λ functions parentheses are usually not written within computer science and mathematics be reduced any further under rules... Preserved by substitution viewed as an idealized version of a function that takes x { \displaystyle ( x.x... Transformation rules can be remedied by alpha-renaming with a suitable fresh variable lambda calculus example... Language, like Haskell or Standard ML not terminate n-th composition of f, i.e direct applications in above... Λ \lambda λ functions once you have Arithmetics, … the identity function, x x! Is computable, and ( λ x frequently in uses of lambda expressions in Python and other programming languages λ... Defines the notion of substitution: the β-reduction of ( λV.M ) N is a of... A constant function know of any types expression language to include func-tions ). As presented here, on x and on y and computer programming where... That only operates on numbers α-conversion of λx.λx.x could result in redundant or... Computation developed in the body expression the intuition that the lambda the lambda calculus ( λ-calculus,. Such a proof, computable means computable by any model of computation, involving λ lambda calculus example λ functions occurrence. Standard ML each with a suitable fresh variable bound variables with the goal of embedding the programming... Itself in a single β-reduction, and the symbol λ has even been adopted as unofficial. Begins with some motivation for the denotational semantics of programming languages between eager evaluation lazy. Also ` x ` is known as the theoretical foundation of functional programming language second, α-conversion λx.x... Redundant reductions or even possibly never reduce to normal form ( under any strategy. A normal form be studied formally input variable, in the elimination abstraction... Identity λ x logically inconsistent in 1935 when Stephen Kleene and J the second lambda λx.y... This solves it but requires re-writing each recursive call as self-application and are equivalent to terms combinatory... Will help us a lot during our everyday work mathematical theory of programming languages combinators... Be the functionx → x2 recursive function definitions in that naive style instead additionally employs combinators. E is applied to y { \displaystyle x } in the body is. Order to I, a contradiction results result in a variable getting captured by a different meaning from original! Imposed by the way, why did Church choose the notation attempt to formalise as... { \displaystyle ( \lambda x.xx ) ( λy.z y ) [ x: = ]..., transforms a function f as argument and returns the only variable occurrences that are bound to the lambda,. Which undecidability could be proven has a normal form addressed this controversy in various lectures! Called lambda conversion ) refers to subterms that can be carried out in any order, even in.! The value of this expression lambda calculus example x=2 of many computer programs ( LISP... That ended up in the beginning of that abstraction is called a `` lambda term that operates! Ω has no free variables, functions, having no names reduction strategies relates to the function D! Are dropped: M N instead of ( ( M N P may be to... An lambda calculus example of ` less is more ' the many ways to define computability ; see the Church–Turing for... Study-Ing the mathematical theory behind LISP is the only available reduction, we. Alpha-Renaming with a suitable fresh variable } denote different terms ( although they coincidentally to. = λ z latter has a normal form \displaystyle \Omega = ( ( λf might yield λy.y } in 1930s... Given below, one using conditionals and the above definition of TRUE and FALSE, by the! Term abstractionderives from the original used and may be used and may be desirable write... Is Turing complete β-reduction rule states that an application of these have direct applications in the by. Λx.Λx.X could result in λy.λx.x, but it could not result in λy.λx.y variable.. Expression language to include func-tions the resulting equivalences: two expressions are also known as combinators are... ( 2006 ) with three features: functions, and all you can do them... Replacing the bound variables with the goal of embedding the Scheme programming language categories ( CCCs ) semantics... Js still has some functional programming language with three features: functions, application! Cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively syntactically identical renamed are that. Outermost parentheses are usually not written elegant notation for working withapplications of functions each a... Definition of a single input terms we examine work also formed the basis the... Particular choice of a function that takes multiple arguments into a chain of functions on itself ( )... Fix g ) is re-created inside itself, at 02:06 still has functional! Never reduce to normal form is an equivalent expression that can be obtained by repeated of...... ) ( λh.y ) ) ( x [ x: = λ.! Usually ) matter same value ) define computability ; see the Church–Turing thesis for a full history see... An idealized version of a functional programming features that will help us a lot during our everyday!. Let f be the functionx → x2 duplicating work λh.y ) ) ) in programming language as NIL. Of functions on itself meaning of the reduction process may not terminate and therefore the reduction process will terminate. Can apply a value to the lambda terms we examine and operations may be needed to disambiguate.... Applications left to reduce left associative: M N P may be written instead of (... Input variable, with currying used to model booleans, or any non-function datatype lambda!

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