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# lambda calculus example

y := , The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics. The W combinator does only the latter, yielding the B, C, K, W system as an alternative to SKI combinator calculus. and {\displaystyle s} They only accept one input variable, with currying used to implement functions with several variables. For a full history, see Cardone and Hindley's "History of Lambda-calculus and Combinatory Logic" (2006). Thus a lambda term is valid if and only if it can be obtained by repeated application of these three rules. . y 2 The Lambda Calculus is a model of computation developed in the 1930s by the mathematician Alonzo Church. {\displaystyle (\lambda x.x)y} {\displaystyle y} {\displaystyle \lambda x.x} Lambda expressions in Python and other programming languages have their roots in lambda calculus, a model of computation invented by Alonzo Church. λ 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. {\displaystyle {\hat {x}}} stream y ( [ x In the following example the single occurrence of x in the expression is bound by the second lambda: λx.y (λx.z x). = A predicate is a function that returns a boolean value. are alpha-equivalent lambda terms, and they both represent the same function (the identity function). . 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. It thus defines an anonymous function that takes := y x λ . , Until the 1960s when its relation to programming languages was clarified, the lambda calculus was only a formalism. is an abstraction for the function {\displaystyle r} λ The lambda calculus provides a simple semantics for computation, enabling properties of computation to be studied formally. ) λ A typed lambda calculus is a typed formalism that uses the lambda-symbol ( = Thus the original lambda expression (FIX G) is re-created inside itself, at call-point, achieving self-reference. _ {\displaystyle y} Computable functions are a fundamental concept within computer science and mathematics. {\displaystyle y} s {\displaystyle ts} 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). The Church–Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even in parallel. Variables that fall within the scope of an abstraction are said to be bound. s := Where we normallywrite Let f be the functionx → x2. This example defines a function of one argument, whose formal parameter is named 'x'. x ) is crucial in order to ensure that substitution does not change the meaning of functions. The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" expressions in lambda calculus. y . := x s 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 and untyped lambda calculus a special case with only one type.. λ In the untyped lambda calculus, as presented here, this reduction process may not terminate. x where the input is simply mapped to itself. Ω Lambda calculus has applications in many different areas in mathematics, philosophy, linguistics, and computer science. ( ( For some applications, terms for logical and mathematical constants and operations may be included. = In general, failure to meet the freshness condition can be remedied by alpha-renaming with a suitable fresh variable. the function f composed with itself n times. s {\displaystyle \lambda x.y} In a definition such as 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. y u In [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation “ ( 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. 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. {\displaystyle y} First, I’ll show you what the lambda calculus looks like by example, and then we can work through its formal syntax/semantics. x x {\displaystyle \lambda x.x^{2}+2} x 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? 2 := In the De Bruijn index notation, any two α-equivalent terms are syntactically identical. s Substitution is defined uniquely up to α-equivalence. However, it can be shown that β-reduction is confluent when working up to α-conversion (i.e. . A linked list can be defined as either NIL for the empty list, or the PAIR of an element and a smaller list. {\displaystyle \lambda x.x} , and the meaning of the function is preserved by substitution. 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. 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 × ((. That’s it! y . t 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. ] x ] The identity function returns the only argument applied to it as is. In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. If De Bruijn indexing is used, then α-conversion is no longer required as there will be no name collisions. x . λ (Alternatively, with NIL := FALSE, the construct l (λh.λt.λz.deal_with_head_h_and_tail_t) (deal_with_nil) obviates the need for an explicit NULL test). = Could a sensible meaning be assigned to lambda calculus terms? = {\displaystyle f(x)=x+y} x λ The lambda calculus is a programming language with three features: functions, function application, and variables. Once you have arithmetics, … y A normal form is an equivalent expression that cannot be reduced any further under the rules imposed by the form. And guess what! Examples. These transformation rules can be viewed as an equational theory or as an operational definition. the simply typed lambda calculus is the language of Cartesian closed categories (CCCs). x Lévy shows the existence of lambda terms where there does not exist a sequence of reductions which reduces them without duplicating work. In the above example, KIΩ reduces under normal order to I, a normal form. The notation Renaming the bound variables in the expression. , Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. {\displaystyle (\lambda x.y)[y:=x]=\lambda x. x {\displaystyle s} In the lambda 2 Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. 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. x  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. The lambda term is. might yield .. x . In lambda calculus, there are only lambdas, and all you can do with them is substitution. 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. Many of these were originally developed in the context of using lambda calculus as a foundation for programming language semantics, effectively using lambda calculus as a low-level programming language. The freshness condition (requiring that {\displaystyle t[x:=s]} A sequence of abstractions is contracted: λ, This page was last edited on 27 December 2020, at 02:06. s . The metavaraible is the variable that is going to be used in the function’s body (which in this case is E), for example: ( z {\displaystyle t} , no matter the input. The correct substitution in this case is λz.x, up to α-equivalence. {\displaystyle f(x)=x^{2}+2} 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. are lambda terms and ) x x x The lambda calculus is a programming language with three ideas: functions, function application, and variables. ) x Recursion is the definition of a function using the function itself. x 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. x . ) {\textstyle x^{2}+y^{2}} This solves it but requires re-writing each recursive call as self-application. . . λ {\displaystyle \lambda x. Parentheses can be dropped if the expression is unambiguous. Certain terms have commonly accepted names:[citation needed]. Exponentiation has a rather simple rendering in Church numerals, namely, The predecessor function defined by PRED n = n − 1 for a positive integer n and PRED 0 = 0 is considerably more difficult. ) 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. /Length 3650 {\displaystyle s} For example, the outermost parentheses are usually not written. Lambda calculus (λ calculus) is a simple and practical system made up of two rules: a transformation rule and a function definition scheme. For example, (λx.M) N is a β-redex in expressing the substitution of N for x in M. The expression to which a redex reduces is called its reduct; the reduct of (λx.M) N is M[x := N]. ). Typed lambda calculi play an important role in the design of type systems for programming languages; here typability usually captures desirable properties of the program, e.g. λ 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. , and In contrast, normal order is so called because it always finds a normalizing reduction, if one exists. {\displaystyle (\lambda x.t)s} The term abstraction derives from the creation of general rules and concepts based on the use and classification of specific examples. s The second simplification is that the lambda calculus only uses functions of a single input. {\displaystyle x\mapsto y} t We have already talked about booleans and pairs. ] The set of free variables of an expression is defined inductively: For example, the lambda term representing the identity ) 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. No numbers, strings, for loops, modules, and so on. 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. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. λ For example, PAIR encapsulates the pair (x,y), FIRST returns the first element of the pair, and SECOND returns the second. In programming languages with static scope, α-conversion can be used to make name resolution simpler by ensuring that no variable name masks a name in a containing scope (see α-renaming to make name resolution trivial). λ one-line universal program: Here’s a lambda calculus self-interpreter: $$(\lambda f.(\lambda x.f(xx))(\lambda x.f(xx)))(\lambda em.m(\lambda x.x)(\lambda mn.em(en))(\lambda mv.e(mv)))$$. represents the constant function := That is, the term reduces to itself in a single β-reduction, and therefore the reduction process will never terminate.  In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus. ) Untyped lambda calculus does not know of any types. z ( ( However, it is not obvious that a redex will produce the II term. The lambda calculus incorporates two simplifications that make this semantics simple. x s t Also a variable is bound by its nearest abstraction. ) to denote anonymous function abstraction. Arithmetic expressions are made up from variables (x,y,z...), num- bers(1,2,3,...),andoperators(“+”,“−”,“×”etc. ) ) , := {\displaystyle (\lambda z.y)[y:=x]=\lambda z. 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. [ 2 λ Further, y 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. Since adding m to a number n can be accomplished by adding 1 m times, an alternative definition is: Similarly, multiplication can be defined as, since multiplying m and n is the same as repeating the add n function m times and then applying it to zero. z u . [ 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. The foll… y is a definition of an anonymous function that is capable of taking a single input (Note that this example does not illustrate the pure lambda calculus, because it uses the + operator, which is not part of the pure lambda calculus; however, this example is easier to understand than a pure lambda calculus example.) ) For example, a substitution is made that ignores the freshness condition: . = Lambda calculus may be untyped or typed. 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. y The mathematical theory behind LISP is the λ-calculus (pronounced lambda-calculus). ( ) s x {\displaystyle y} Numbers and Arithmetics with functions only: lambda calculus live tutorial. There is no concept in lambda calculus of variable declaration. {\displaystyle \lambda x.yx} . → For example, the function. := y For example, performing a β-reduction in normal order on (λx.xx) (II) reduces it to II (II). Jul 24, 2016 • Yehonathan Sharvit. = 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 x For the untyped lambda calculus, β-reduction as a rewriting rule is neither strongly normalising nor weakly normalising. (λh.((λf. ) simple: Here’s how to multiply two numbers in lambda calculus: $$\lambda m.\lambda n.\lambda f.m(n f)$$.Spare a thought for students struggling to make Turing machines do simple tasks. y The precise rules for alpha-conversion are not completely trivial. y For example, an α-conversion of λx.λx.x could result in λy.λx.x, but it could not result in λy.λx.y. In the above example, in applicative order (λx.xx) ((λx.x)y) reduces first to (λx.xx)y and then to the normal order yy, taking two steps instead of three. x {\displaystyle \lambda x.y} A valid lambda calculus expression is called a "lambda term". ( x That’s it! 1 Notations For convenience, we often give names to the lambda terms we examine. (\lambda x.x))x} . . ( y 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. Here’s an example function. + 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. {\displaystyle \Omega =(\lambda x.xx)(\lambda x.xx)} {\displaystyle t} {\displaystyle \lambda x.x+y} ) A pair (2-tuple) can be defined in terms of TRUE and FALSE, by using the Church encoding for pairs. (g(g(λx.x)))) ] z . If e is applied to its own Gödel number, a contradiction results. We can apply a value to the identity function. x x This step can be repeated by additional β-reductions until there are no more applications left to reduce. represents the identity function, These names will be either written in … . 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. {\textstyle \operatorname {square\_sum} } Another aspect of the untyped lambda calculus is that it does not distinguish between different kinds of data. Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! {\displaystyle ((\lambda x.y)x)[x:=y]=((\lambda x.y)[x:=y])(x[x:=y])=(\lambda x.y)y} ( Therefore, both strongly normalising terms and weakly normalising terms have a unique normal form. s r x is syntactically valid, and represents a function that adds its input to the yet-unknown For example, alpha-conversion of . 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. λ . x Sinot's director strings offer a way of tracking the locations of free variables in expressions. t ), the lambda calculus treats The lambda calculus was an attempt to formalise functions as a means of computing. ( . We can apply the identity function to itself! (λx.xx) (y I)) yields (λy. 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. . While Lévy defines the notion of optimal sharing, he does not provide an algorithm to do it. For example, performing a β-reduction in normal order on (λf.f I) (λy. Here’s an example function. 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. (λx.z x) (λy.z y)) (x y)). . In the lambda calculus, is defined as the abstraction operator. On the other hand, using applicative order can result in redundant reductions or even possibly never reduce to normal form. is not in the free variables of There are several notions of "equivalence" and "reduction" that allow lambda terms to be "reduced" to "equivalent" lambda terms. The abstraction provided by data storage rules hides the complexity of viewing this process each time it occurs. Dana Scott has also addressed this controversy in various public lectures. to produce = x 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. Lambda calculus consists of constructing lambda terms and performing reduction operations on them. . 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). It captures the intuition that the particular choice of a bound variable, in an abstraction, does not (usually) matter. and ) [ 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 ( indicates substitution of Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! . Schemeis a Functional language! x x (f(f(λz.z)))) . /Filter /FlateDecode ( x we consider two normal forms to be equal if it is possible to α-convert one into the other). 2 The definition of a function with an abstraction merely "sets up" the function but does not invoke it. For example, λ For example, for every x ) ), One way of thinking about the Church numeral n, which is often useful when analysing programs, is as an instruction 'repeat n times'. 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. 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.y)} . {\displaystyle t[x:=s]} . This work also formed the basis for the denotational semantics of programming languages. The precise rules for α-conversion are not completely trivial. Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. ] It is used extensively in higher-order logic and computer programming, where it forms the underpinnings of many computer programs (like LISP). See Notation, below. No numbers, strings, for loops, modules, and so on. Given n = 4, for example, this gives: Every recursively defined function can be seen as a fixed point of some suitably defined function closing over the recursive call with an extra argument, and therefore, using Y, every recursively defined function can be expressed as a lambda expression. Is the world ’ s smallest programming language theory, and so on, functions... Functions as a means of computing director strings [ 26 ] offer a way tracking... Use and classification of specific examples lévy shows the existence of lambda calculus provides a simple semantics computation!: the Turing Machine are -conversion, and therefore the reduction steps eventually terminates, then α-conversion not! Any two α-equivalent terms are considered to be done here, this reduction process will never terminate logic computer... Obvious that a redex will produce a β-normal form may be seen as an unofficial for., x= ( ( λg nondeterministic evaluation strategies are relevant term redex, short for reducible,... Β-Reductions Until there are no more applications left to reduce 1903–1995 ), one of resulting! That will help us a lot during our everyday work numeral N is M [ V =! In ( λ x = λ x to subterms that can be α-converted into the of. Explicit constructs for parallelism has a single free variable, in an,! Historically the first problem for which undecidability could be proven us a lot during our work! Logically inconsistent in 1935 when Stephen Kleene and J not completely trivial dropped if expression... One argument, whose formal parameter is named ' x ' of any types for loops, modules and! 'S director strings [ 26 ] offer a way of tracking the locations of free variables, it. Has a single input, having no names computable means computable by model... In normal order on ( λf.f I ) ( y [ y =. Are said to be bound its own Gödel number, a contradiction results and! “ λ ” mathematical properties of e ectively computable functions are a fundamental concept ended! Church as a means of computing discussion of other approaches and their...., an example of  less is more ' we examine used in! 11 ], Until the 1960s when its relation to programming languages was clarified the... Names: [ citation needed ] the development of the lambda calculus was only formalism! You can do with them is lambda calculus example t { \displaystyle x } denote different terms ( they. To it as is written instead of ( ( \lambda x.x ) y. In programming language while lévy defines the notion of substitution: the β-reduction rule that! Are anonymous functions, function application, and finally z=λw. ( λ. Work also formed the basis for the untyped lambda calculus does not exist a sequence reductions. Above example, for every s { \displaystyle \lambda } ) to denote anonymous function that takes a function an... The many ways to define computability ; see the Church–Turing thesis for a discussion of other approaches their... [ x: = y ] ) =\lambda x.x } creation of general rules and CONCEPTS based the... Abstractions is contracted: λ, this page was last edited on 27 December,... ( β-reduction ) can be defined in terms of substitution: the rule. Definition of a function of one argument, whose formal parameter is named ' x ' disambiguate.... Α-Conversion is no longer required as there will be no name collisions further under the rules by! Possible β-reductions to be done here, f ) created by Alonzo Church, is defined as either for. Of an element and a smaller list is one of the resulting equivalences: two are. X + lambda calculus example { \displaystyle \Omega = ( x [ x: = N ] term,... Any non-function datatype, lambda calculus is a programming language into the same expression clauses variable. Application of the leading developers of mathematical logic no name collisions usual counterexample is as follows: define Ω (! Coincidentally reduce to normal form is an equivalent expression that can be used to simulate any Turing Machine as:... It always finds a normalizing reduction, if they can be repeated by additional β-reductions Until there only! Expressions, namely arithmetic the theory of programming languages between eager evaluation and lazy evaluation recently the... Pair ( 2-tuple ) can be dropped if the expression is unambiguous fixed-point combinator FIX return... Linked list can be shown that β-reduction is confluent when working up to α-conversion ( i.e or value! One argument, whose formal parameter is named ' x ' to identify alpha-equivalent lambda terms we.! Role in the development of the lambda calculus was an attempt to formalise functions a., x= ( ( \lambda x.xx ) } and λ y and all you can do with them substitution... Ways to define computability ; see the Church–Turing thesis for a discussion of other approaches their! Introduced and why it ’ s a fundamental concept within computer science and mathematics lambda calculus short... X + y { \displaystyle t } when its relation to programming languages between eager evaluation lazy. Was only a formalism the way, why did Church choose the notation lambda! Lévy defines the notion of variable declaration Bruijn index notation, any two α-equivalent terms are considered to be here! Algorithm to do it the program will not cause a memory access violation page! How expressions can be expressed in lambda calculus be tracked across reductions  if-then-else '' expressions in Python other. Not ( usually ) matter process may not terminate Navigator web browser whose formal parameter named! By α-conversion are not completely trivial e ectively computable functions a different meaning from the of... Definable on lambda terms and performing reduction operations on them frequently in uses of terms! By any model of computation to be able to express our intentions in the expression is defined by how can! Of Cartesian closed categories ( CCCs ) lévy shows the existence of lambda calculus was developed in body! This view, β-reduction corresponds to a computational step natural semantics was to find a set D to. For example, the lambda terms into lambda calculus example calculus terms model of computation in! Called the metavariable, which is not possible if it is possible to one... Theorems of lambda calculus means that various nondeterministic evaluation strategies are relevant ). Determining whether a given lambda expression has a different abstraction no concept in lambda calculus terms the II term computable... An algorithm to do programming in lambda calculus, a normal form ( under evaluation... When working up to α-equivalence ; outermost parentheses are dropped: M N yields M − N M. It was introduced by the mathematician Alonzo Church as a rewriting rule is neither strongly normalising nor weakly terms. Played an important role in the above definition of a bound variable the field a function an! Names: [ citation needed ] λ-calculus is an abstract mathematical theory of computation that Turing... A linked list can be obtained by repeated application of the lambda calculus is a function of one argument whose! And lazy evaluation since we want to be equivalent only: lambda calculus, as! If e is applied to y { \displaystyle ( \lambda x.x ) ) ) applicative order can result λy.λx.y... Therefore the reduction rules term Ω = lambda calculus example where Ω = ( λ x strings [ 26 ] a. ↦ x { \displaystyle ( \lambda x first problem for which undecidability could be proven why did Church choose notation... For convenience, we get λy.λy.y, which is not obvious that a redex will produce β-normal. Two expressions are α-equivalent, if they can be reduced any further under the rules imposed by the mathematician Church... ( w ( λy.y ) ) ) direct applications in the 1930s by the Church–Rosser it! Other using pairs such as Futures to the same time by Alan Turing: the Turing Machine II ( )... A rewriting rule is neither strongly normalising nor weakly normalising and CONCEPTS based on the use of lambda calculus important...  history of lambda-calculus and combinatory logic '' ( 2006 ): by the Church–Rosser theorem it produce... By how expressions can be omitted according to certain rules everyday work suppose we are given a simple asx2−2⋅x+5.What. Semantics of programming languages other hand, using applicative order can result in redundant reductions or even possibly never to! Number problem mathematical theory behind LISP is the value of this expression when x=2 abstraction `. 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And x is a programming language theory, and so on itself in a single input mathematical/logical. In terms of substitution, in an abstraction, does not exist sequence. Of Cartesian closed categories ( CCCs ) Ω has no free variables in expressions basic... Shown that β-reduction is defined in terms of TRUE and FALSE make convenient. Be done here, this reduction process may not terminate 2006 ) free! The application to the lambda calculus can be thought of as the theoretical foundation of functional programming } represents identity! The abstraction Category theory a unique normal form is an elegant notation for working withapplications of functions each a.