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In mathematical logic and computer science, lambda calculus, also λ-calculus, is a formal system designed to investigate function definition, function application, and recursion. It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s as part of a larger effort to base the foundation of mathematics upon functions rather than sets (in the hopes of avoiding obstacles like Russell's Paradox). The Kleene-Rosser paradox shows that the lambda calculus is unable to avoid set-theoretic paradoxes, but the lambda calculus emerged as a useful tool in the investigation of problems in computability or recursion theory, and forms the basis of a paradigm of computer programming called functional programming. ^[1] Image File history File links This is a lossless scalable vector image. ... Image File history File links Mergefrom. ... Church encoding is a means of embedding data and operators into the lambda calculus, the most familiar form being the church numerals, a representation of the natural numbers using lambda notation. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... This article is about the concept of recursion. ... ‹ The template below (Expand) is being considered for deletion. ... Stephen Cole Kleene (January 5, 1909 – January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... The 1930s (years from 1930–1939) were described as an abrupt shift to more radical and conservative lifestyles, as countries were struggling to find a solution to the Great Depression, also known as the World Depression. ... Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... In mathematics, the Kleene-Rosser paradox is a paradox that shows Churchs original lambda calculus is inconsistent. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ...

The lambda calculus can be thought of as an idealized, minimalistic programming language. It is a close cousin of the Turing machine, another minimalist abstraction capable of expressing any algorithm. The difference between the two is that the lambda calculus takes a functional view of algorithms, while the original Turing machine takes an imperative view. That is, a Turing machine maintains 'state' - a 'notebook' of symbols that can change from one instruction to the next. The imperative paradigm can be seen in programming languages like C or Basic. By contrast, the lambda calculus is stateless, it deals exclusively with functions which accept and return data (including other functions), but produce no side effects in 'state' and do not make alterations to incoming data (immutability.) The functional paradigm can be seen in modern languages like Lisp, Scheme and Haskell. For the test of artificial intelligence, see Turing test. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ... In computer science, imperative programming, as opposed to declarative programming, is a programming paradigm that describes computation in terms of a program state and statements that change the program state. ... In computer science, imperative programming, as opposed to declarative programming, is a programming paradigm that describes computation in terms of a program state and statements that change the program state. ... In computer science, an immutable object, as opposed to a mutable object, is a kind of object whose internal states cannot be modified after it is created. ... “LISP” redirects here. ... Scheme is a multi-paradigm programming language. ... Haskell is a standardized purely functional programming language with non-strict semantics, named after the logician Haskell Curry. ...

Church used the lambda calculus to give a negative answer to the Entscheidungsproblem and to the halting problem. The lambda calculus - and the paradigm of functional programming - is still influential, especially among the artificial intelligence community. The Entscheidungsproblem (German for decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ... In computability theory the halting problem is a decision problem which can be stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ...

This article deals with the "untyped lambda calculus" as originally conceived by Church. Since then, some typed lambda calculi have been developed. Typed versions of the lambda calculus extend the standard lambda calculus with types. ...

History

Originally, Church had tried to construct a complete formal system for the foundations of mathematics; when the system turned out to be susceptible to the analog of Russell's paradox, he separated out the lambda calculus and used it to study computability, culminating in his negative answer to the Entscheidungsproblem. Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Computation can be defined as finding a solution to a problem from given inputs by means of an algorithm. ... The Entscheidungsproblem (German for decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ...

Informal description

In lambda calculus, every expression stands for a function with a single input, called its argument; the argument of the function is in turn a function with a single argument, and the value of the function is another function with a single argument. A function is anonymously defined by a lambda expression which expresses the function's action on its argument. For instance, the "add-two" function f such that  f(x) = x + 2  would be expressed in lambda calculus as  λ x. x + 2  (or equivalently as  λ y. y + 2;  the name of the formal argument is immaterial) and the application of the function f(3) would be written as  (λ x. x + 2) 3.  Note that part of what makes this description "informal" is that the expression x + 2 is not part of lambda calculus; for an explanation of arithmetic in lambda calculus, see below. Function application is left associative:  f x y = (f x) y.  Consider the function which takes a function as an argument and applies it to the number 3: λ f. f 3.  This latter function could be applied to our earlier "add-two" function as follows:  (λ f. f 3) (λ x. x + 2).  The three expressions: In mathematics, associativity is a property that a binary operation can have. ...

• (λ f. f 3)(λ x. x + 2)
• (λ x. x + 2) 3
• 3 + 2

are equivalent. A function of two variables is expressed in lambda calculus as a function of one argument which returns a function of one argument (see currying). For instance, the function  f(x, y) = x - y  would be written as  λ x. λ y. x - y. A common convention is to abbreviate curried functions as, in this example,  λ x y. x - y. Not every lambda expression can be reduced to a definite value like the ones above; consider for instance This article is about the function transformation technique. ...

(λ x. x x) (λ x. x x)

or

(λ x. x x x) (λ x. x x x)

and try to visualize what happens when you start to apply the first function to its argument.  (λ x. x x)  is also known as the ω combinator;  ((λ x. x x) (λ x. x x))  is known as Ω,  ((λ x. x x x) (λ x. x x x))  as Ω₂, etc. This article is about a topic in theoretical computer science, and is not to be confused with combinatorial logic, a topic in electronics. ...

While the lambda calculus itself does not contain symbols for integers or addition, these can be defined as abbreviations within the calculus and arithmetic can be expressed as is shown below.

Lambda calculus expressions may contain free variables, i.e. variables not bound by any λ. For example, the variable  y  is free in the expression  (λ x. y) , representing a function which always produces the result y. Occasionally, this necessitates the renaming of formal arguments, for instance in order to reduce

(λ x y. y x) (λ x. y)

to

λ z. z (λ x. y).

In the unsimplified expression in this example, the first "y" defines the formal parameter, the second uses the formal parameter, and the third is a free variable.

If one only formalizes the notion of function application and does not allow lambda expressions, one obtains combinatory logic. Not to be confused with combinational logic, a topic in digital electronics. ...

Formal definition

Formally, a lambda expression is defined inductively as one of the following: Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

1. V, a variable, where V is any identifier. (The precise set of identifiers is arbitrary, but must be infinite.)
2. (λ V. E), an abstraction, where V is any identifier and E is any lambda expression. An abstraction corresponds to an anonymous function.
3. E E′, an application, where E and E′ are any lambda expressions. An application corresponds to calling a function (E) with an argument (E′).

To unclutter the notation, parentheses may be omitted if they are redundant. When there are parentheses missing, function application is left associative, and a lambda binds as much as possible after it. For example, the expression  ((λ x. (x x)) (λ y. y))  can be simply written as  (λ x. x x) λ y. y. Identifiers (IDs) are lexical tokens that name entities. ... An anonymous function is a function (or a subroutine) defined, and possibly called, without being bound to a name. ...

Free and bound variables

Each variable in a lambda expression is free or bound. For example, the x in  (x y)  is free, while the x in  λ x. (x y)  is bound. A bound variable has a specific lambda it is associated with, while a free variable does not. Precisely, the free variables of a lambda expression are defined inductively as follows: Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

1. In an expression of the form  V,  where V is a variable, this V is the single free occurrence.
2. In an expression of the form  λ V. E,  the free occurrences are the free occurrences in E except for V. In this case the occurrences of V in E are said to be bound by the λ before V.
3. In an expression of the form  (E E′),  the free occurrences are the free occurrences in E and E′.

α-conversion

Alpha conversion allows bound variable names to be changed. For example, an alpha conversion of  λx.x  would be  λy.y . Frequently in uses of lambda calculus, terms that differ only by alpha conversion are considered to be equivalent.

The precise rules for alpha conversion are not completely trivial. First, when alpha-converting an abstraction, the only variable occurrences that are renamed are those that are bound to the same abstraction. For example, an alpha conversion of  λx.λx.x  could result in  λy.λx.x , but it could not result in  λy.λx.y . The latter has a different meaning from the original.

Second, alpha conversion is not possible if it would result in a variable getting captured by a different abstraction. For example, if we replace x with y in λx.λy.x, we get λy.λy.y, which is not at all the same.

Substitution

Substitution, written E[V := E′], corresponds to the replacement of a variable V by expression E′ every place it is free within E. The precise definition must be careful in order to avoid accidental variable capture (See also Hygienic macro). For example, it is not correct for (λ x.y)[y := x] to result in (λ x.x), because the substituted x was supposed to be free but ended up being bound. The correct substitution in this case is (λ z.x). The Scheme programming language is a functional programming language and a dialect of Lisp. ...

The precise rules are defined inductively as follows:

1. V[V := E] == E
2. W[V := E] == W, if W and V are different
3. (E₁ E₂)[V := E] == (E₁[v := E] E₂[v := E])
4. (λ V. E′)[V := E] == (λ V. E′)
5. (λ W. E′)[V := E] == (λ W. E′[V := E]), if V and W are different and W is not free in E.
6. (λ W. E′)[V := E] == (λ W′. E′[W := W′])[V := E], if V and W are different and if W′ is not free in E.

β-reduction

Beta reduction expresses the idea of function application. The beta reduction of  ((λ V. E) E′)  is simply  E[V := E′] .

η-conversion

Eta conversion 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. Eta-conversion converts between  λ x. f x  and  f  whenever x does not appear free in f. In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them. ... ↔ ⇔ ≡ logical symbols representing iff. ...

This conversion is not always equivalent when lambda expressions are interpreted as programs. Evaluation of  λ x. f x  can terminate even when evaluation of f does not.

Arithmetic in lambda calculus

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: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... A Church integer is a representation of natural numbers as functions, invented by Alonzo Church as part of his lambda calculus. ...

0 := λ f x. x

1 := λ f x. f x

2 := λ f x. f (f x)

3 := λ f x. f (f (f x))

and so on. A Church numeral is a higher-order function—it takes a single-argument function f, and returns another single-argument function. The Church numeral n is a function that takes a function f as argument and returns the n-th composition of f, i.e. the function f composed with itself n times. This is like raising f to the n-th power and such repeated compositions (of a single function f) obey the laws of exponents, which is why these numerals can be used for arithmetic. Note that 1 returns f itself, i.e. it is essentially the identity function, and 0 returns the identity function. (Also note that in Church's original lambda calculus, the formal parameter of a lambda expression was required to occur at least once in the function body, which made the above definition of 0 impossible.) In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... “Exponent” redirects here. ...

We can define a successor function, which takes a number n and returns n + 1 by adding an additional application of f:

SUCC := λ n f x. f (n f 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 := λ m n f x. n f (m f x)

PLUS can be thought of as a function taking two natural numbers as arguments and returning a natural number; it can be verified that

PLUS 2 3    and    5

are equivalent lambda expressions. Similarly, multiplication can be defined as

MULT := λ m n f. m (n f)

Alternatively

MULT := λ m n. m (PLUS n) 0,

the idea being that multiplying m and n is the same as repeating the "add n" function m times and then applying it to zero. The predecessor  PRED n = n - 1  of a positive integer n is more difficult:

PRED := λ n f x. n (λ g h. h (g f)) (λ u. x) (λ u. u) 

or alternatively

PRED := λ n. n (λ g k. (g 1) (λ u. PLUS (g k) 1) k) (λ v. 0) 0

Note the trick (g 1) (λ u. PLUS (g k) 1) k which evaluates to k if g(1) is zero and to g(k) + 1 otherwise.

Logic and predicates

By convention, the following two definitions (known as Church booleans) are used for the boolean values TRUE and FALSE:

TRUE := λ x y. x

FALSE := λ x y. y

(Note that FALSE is equivalent to the Church numeral zero defined above)

Then, with these two λ-terms, we can define some logic operators (these are just possible formulations; other expressions are equally correct):

AND := λ p q. p q FALSE

OR := λ p q. p TRUE q

NOT := λ p. p FALSE TRUE

IFTHENELSE := λ p x y. p x y

We are now able to compute some logic functions, as for example:

AND TRUE FALSE

≡ (λ p q. p q FALSE) TRUE FALSE →_β TRUE FALSE FALSE

≡ (λ x y. x) FALSE FALSE →_β FALSE

and we see that AND TRUE FALSE is equivalent to FALSE.

A predicate is a function which returns a boolean value. 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:

ISZERO := λ n. n (λ x. FALSE) TRUE

The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" statements in lambda calculus.

Pairs

A pair (2-tuple) datatype can be defined in terms of TRUE and FALSE.

CONS := λf.λs. λb. f b s

CAR := λp. p TRUE

CDR := λp. p FALSE

NIL := λx.TRUE

NULL := λp. p (λx y.FALSE)

A linked list datatype can be defined as either NIL for the empty list, or the CONS of an element and a smaller list. CONS, Connection-Oriented Network Service, is one of the two OSI stack network layer protocols, the other being CLNS (Connectionless Network Service). ... Introduced in the Lisp programming language, car (IPA [kar], just like the English word car) and cdr (IPA [kʌ dər] or [ku dər]) are primitive operations upon linked lists composed of cons cells. ... Introduced in the Lisp programming language, car (IPA [kar], just like the English word car) and cdr (IPA [kʌ dər] or [ku dər]) are primitive operations upon linked lists composed of cons cells. ...

Recursion

Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this. However, this impression is misleading. Consider for instance the factorial function f(n) recursively defined by This article is about the concept of recursion. ... For factorial rings in mathematics, see unique factorisation domain. ...

f(n) = 1, if n = 0; and n·f(n-1), if n>0.

In lambda calculus, one cannot define a function which includes itself. To get around this, one may start by defining a function, here called g, which takes a function f as an argument and returns another function that takes n as an argument:

g := λ f n. (1, if n = 0; and n·f(n-1), if n>0).

The function that g returns is either the constant 1, or n times the application of the function f to n-1. Using the ISZERO predicate, and boolean and algebraic definitions described above, the function g can be defined in lambda calculus.

However, g by itself is still not recursive; in order to use g to create the recursive factorial function, the function passed to g as f must have specific properties. Namely, the function passed as f must expand to the function g called with one argument -- and that argument must be the function that was passed as f again!

In other words, f must expand to g(f). This call to g will then expand to the above factorial function and calculate down to another level of recursion. In that expansion the function f will appear again, and will again expand to g(f) and continue the recursion. This kind of function, where f = g(f), is called a fixed-point of g, and it turns out that it can be implemented in the lambda calculus using what is known as the paradoxical operator or fixed-point operator and is represented as Y -- the Y combinator: A fixed point combinator (or fixed-point operator) is a higher-order function which computes a fixed point of other functions. ...

Y = λ g. (λ x. g (x x)) (λ x. g (x x))

In the lambda calculus, Y g is a fixed-point of g, as it expands to g (Y g). Now, to complete our recursive call to the factorial function, we would simply call  g (Y g) n,  where n is the number we are calculating the factorial of.

Given n = 5, for example, this expands to:

(λ n.(1, if n = 0; and n·((Y g)(n-1)), if n>0)) 5

1, if 5 = 0; and 5·(g(Y g)(5-1)), if 5>0

5·(g(Y g) 4)

5·(λ n. (1, if n = 0; and n·((Y g)(n-1)), if n>0) 4)

5·(1, if 4 = 0; and 4·(g(Y g)(4-1)), if 4>0)

5·(4·(g(Y g) 3))

5·(4·(λ n. (1, if n = 0; and n·((Y g)(n-1)), if n>0) 3))

5·(4·(1, if 3 = 0; and 3·(g(Y g)(3-1)), if 3>0))

5·(4·(3·(g(Y g) 2)))

...

And so on, evaluating the structure of the algorithm recursively. Every recursively defined function can be seen as a fixed point of some other suitable function, and therefore, using Y, every recursively defined function can be expressed as a lambda expression. In particular, we can now cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively.

Computable functions and lambda calculus

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 == y,  where x and y are the Church numerals corresponding to x and y, respectively. This is one of the many ways to define computability; see the Church-Turing thesis for a discussion of other approaches and their equivalence. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... ↔ ⇔ ≡ logical symbols representing iff. ... Computation can be defined as finding a solution to a problem from given inputs by means of an algorithm. ... In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ...

Undecidability of equivalence

There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressions are equivalent. This was historically the first problem for which the unsolvability could be proven. Of course, in order to do so, the notion of algorithm has to be cleanly defined; Church used a definition via recursive functions, which is now known to be equivalent to all other reasonable definitions of the notion. In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ...

Church's proof first reduces the problem to determining whether a given lambda expression has a normal form. A normal form is an equivalent expression which cannot be reduced any further. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and constructing a Gödel numbering for lambda expressions, he constructs a lambda expression e which closely follows the proof of Gödel's first incompleteness theorem. If e is applied to its own Gödel number, a contradiction results. In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ...

Lambda calculus and programming languages

As pointed out by Peter Landin's 1965 classic A Correspondence between ALGOL 60 and Church's Lambda-notation, most programming languages are rooted in the lambda calculus, which provides the basic mechanisms for procedural abstraction and procedure (subprogram) application. Peter Landin is a British computer scientist. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...

Implementing the lambda calculus on a computer involves treating "functions" as first-class objects, which raises implementation issues for stack-based programming languages. This is known as the Funarg problem. In computing, a first-class object (also value, entity, and citizen), in the context of a particular programming language, is an entity which can be used in programs without restriction (when compared to other kinds of objects in the same language). ... Funarg is an abbreviation for functional argument; in computer science, the funarg problem relates to the difficulty of implementing functions as first-class objects in stack-based programming language implementations. ...

The most prominent counterparts to lambda calculus in programming are functional programming languages, which essentially implement the calculus augmented with some constants and datatypes. Lisp uses a variant of lambda notation for defining functions, but only its purely functional subset ("Pure Lisp") is really equivalent to lambda calculus. Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In computer science, a datatype or data type (often simply a type) is a name or label for a set of values and some operations which one can perform on that set of values. ... Lisp is a family of computer programming languages with a long history and a distinctive fully-parenthesized syntax. ... Lispkit Lisp is a strictly functional subset of Lisp developed as a testbed for functional programming concepts. ...

Functional languages are not the only ones to support functions as first-class objects. Numerous imperative languages, e.g. Pascal, have long supported passing subprograms as arguments to other subprograms. In C and the C-like subset of C++ the equivalent result is obtained by passing pointers to the code of functions (subprograms). Such mechanisms are limited to subprograms written explicitly in the code, and do not directly support higher-level functions. Some imperative object-oriented languages have notations that represent functions of any order; such mechanisms are available in C++, Smalltalk and more recently in Eiffel ("agents") and C# ("delegates"). As an example, the Eiffel "inline agent" expression In computing, a first-class object (also value, entity, and citizen), in the context of a particular programming language, is an entity which can be used in programs without restriction (when compared to other kinds of objects in the same language). ... In computer science, imperative programming, as opposed to declarative programming, is a programming paradigm that describes computation in terms of a program state and statements that change the program state. ... Pascal is a structured imperative computer programming language, developed in 1970 by Niklaus Wirth as a language particularly suitable for structured programming. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ... An object-oriented programming language is one that allows or encourages, to some degree, object-oriented programming methods. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ... For other uses, see Small talk. ... Eiffel is an ISO-standardized object-oriented programming language designed for extensibility, reusability, reliability and programmer productivity. ... The title given to this article is incorrect due to technical limitations. ...

 agent (x: REAL): REAL do Result := x * x end 

denotes an object corresponding to the lambda expression λ x . x*x (with call by value). It can be treated like any other expression, e.g. assigned to a variable or passed around to routines. If the value of square is the above agent expression, then the result of applying square to a value a (β-reduction) is expressed as square.item ([a]), where the argument is passed as a tuple. In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

A Python example of this uses the lambda form of functions: Python is a high-level programming language first released by Guido van Rossum in 1991. ...

 func = lambda x: x ** 2 

This creates a new anonymous function and names it func which can be passed to other functions, stored in variables, etc. Python can also treat any other function created with the standard def statement as first-class objects. In computing, a first-class object (also value, entity, and citizen), in the context of a particular programming language, is an entity which can be used in programs without restriction (when compared to other kinds of objects in the same language). ...

The same holds for Smalltalk expression

 [ :x | x * x ] 

This is first-class object (block closure), which can be stored in variables, passed as arguments, etc.

A similar C++ example (using the Boost.Lambda library):

 std::for_each(c.begin(), c.end(), std::cout << _1 * _1 << std::endl); 

Here the standard library function for_each iterates over all members of container 'c', and prints the square of each element. The _1 notation is Boost.Lambda's convention (originally derived from Boost.Bind) for representing the first placeholder element (the first argument), represented as x elsewhere.

A simple C# delegate taking a variable and returning the square. This function variable can then be passed to other methods (or function delegates) A delegate is a form of type-safe function pointer used in the . ...

 //Declare a delegate signature delegate double MathDelegate(double i); //Create an delegate instance MathDelegate f = delegate(double i) { return Math.Pow(i, 2); }; 

 //Passing 'f' function variable to another method, executing, // and returning the result of the function double Execute(MathDelegate func) { return func(100); } 

In the .NET Framework 3.5, C# has lambda expressions in a form similar to python or lisp. The expression resolves to a delegate like in the previous example but the above can be simplified to below.

 //Create an delegate instance MathDelegate f = i => i * i; Execute(f); // or more simply put Execute(i => i * i); 

Concurrency and parallelism

The Church-Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even concurrently. (Indeed, the lambda calculus is referentially transparent.) While this means the lambda calculus can model the various nondeterministic evaluation strategies, it does not offer any richer notion of parallelism, nor can it express any concurrency issues. The Actor model and Process calculi such as CSP, the CCS, the π calculus and the ambient calculus have been designed for such purposes. Referential transparency is a property of parts of computer programs. ... In computer science, an evaluation strategy is a set of (usually deterministic) rules for determining the evaluation of expressions in a programming language. ... Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obtain results faster. ... Concurrent programming languages are programming languages that use language constructs for concurrency. ... In computer science, the Actor model is a mathematical model of concurrent computation that treats actors as the universal primitives of concurrent digital computation: in response to a message that it receives, an actor can make local decisions, create more actors, send more messages, and determine how to respond to... In the first half of the 20th century, various formalisms were proposed to capture the informal concept of computable function, μ-recursive functions, Turing Machines and the lambda calculus possibly being the most well-known examples today. ... In computer science, Communicating Sequential Processes (CSP) is a formal language for describing patterns of interaction in concurrent systems. ... The Calculus of Communicating Systems (or CCS) (one of the first process calculi) was developed by Robin Milner. ... In theoretical computer science, the π-calculus is a notation originally developed by Robin Milner, Joachim Parrow and David Walker to model concurrency (just as the λ-calculus is a simple model of sequential programming languages). ... Ambient calculus is a form of notation devised by Luca Cardelli and Andrew D. Gordon in 1998 and used to describe and theorise about mobile systems. ...

Although the nondeterministic lambda calculus is capable of expressing all partial recursive functions, it is not capable of expressing all computations. For example it is not capable of expressing unbounded nondeterminism (i.e. every nondeterministic lambda expression that is guaranteed to terminate; terminates in a finite number of expressions). However, there are concurrent programs guaranteed to halt that terminate in an infinite number of states [Clinger 1981, Hewitt 2006]. In computer science, unbounded nondeterminism (sometimes called unbounded indeterminacy) is a property of concurrency by which the amount of delay in servicing a request can become unbounded as a result of arbitration of contention for shared resources while still guaranteeing that the request will eventually be serviced. ...

See also

This article needs to be cleaned up to conform to a higher standard of quality. ... Not to be confused with combinational logic, a topic in digital electronics. ... The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ... In computer science, an evaluation strategy is a set of (usually deterministic) rules for determining the evaluation of expressions in a programming language. ... This article or section does not adequately cite its references or sources. ... In mathematical logic and type theory, the λ-cube is a framework for exploring the axes of refinement in Coquands Calculus of Constructions, starting from the simply typed lambda calculus as the vertex of a (3-D) cube placed at the origin, and the calculus of constructions (= higher order... Rewriting in mathematics, computer science and logic covers a wide range of non-deterministic methods of replacing subterms of a formula with other terms. ... SKI combinator calculus is a computational system that is a reduced, untyped version of Lambda calculus. ... System F is a typed lambda calculus. ... The calculus of constructions (CoC) is a higher-order typed lambda calculus where types are first-class values. ... Typed versions of the lambda calculus extend the standard lambda calculus with types. ... Unlambda is a minimal functional programming language based on combinatory logic, a version of the lambda calculus that omits the lambda operator. ... In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus, and was introduced by M. Parigot in [lambda-mu]calculus: an algorithmic interpetation of classical natural deduction, Springer LNAI no. ...

References

References

• Abelson, Harold & Gerald Jay Sussman. Structure and Interpretation of Computer Programs. The MIT Press. ISBN 0-262-51087-1.
• Barendregt, Hendrik Pieter, The Lambda Calculus, Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, Volume 103, North-Holland (1984), ISBN 0-444-87508-5 is the comprehensive reference on the (untyped) lambda calculus; see also the paper Introduction to Lambda Calculus.
• Barendregt, Hendrik Pieter, The Type Free Lambda Calculus pp1091-1132 of Handbook of Mathematical Logic, North-Holland (1977) ISBN 0-7204-2285-X
• Church, Alonzo, An unsolvable problem of elementary number theory, American Journal of Mathematics, 58 (1936), pp. 345–363. This paper contains the proof that the equivalence of lambda expressions is in general not decidable.
• Clinger, William, Foundations of Actor Semantics. MIT Mathematics Doctoral Dissertation, June 1981.
• Punit,Gupta, Amit & Ashutosh Agte, Untyped lambda-calculus, alpha-, beta- and eta- reductions and recursion
• Henz, Martin, The Lambda Calculus. Formally correct development of the Lambda calculus.
• Hewitt, Carl, What is Commitment? Physical, Organizational, and Social COIN@AAMAS. April 27, 2006.
• Kleene, Stephen, A theory of positive integers in formal logic, American Journal of Mathematics, 57 (1935), pp. 153–173 and 219–244. Contains the lambda calculus definitions of several familiar functions.
• Landin, Peter, A Correspondence Between ALGOL 60 and Church's Lambda-Notation, Communications of the ACM, vol. 8, no. 2 (1965), pages 89-101. Available from the ACM site. A classic paper highlighting the importance of lambda-calculus as a basis for programming languages.
• Larson, Jim, An Introduction to Lambda Calculus and Scheme. A gentle introduction for programmers.

Some parts of this article are based on material from FOLDOC, used with permission. Front cover Structure and Interpretation of Computer Programs (SICP) is a textbook published in 1985 about general computer programming concepts from MIT press written by Massachusetts Institute of Technology professors Harold Abelson and Gerald Jay Sussman, with Julie Sussman. ... MIT Press Books The MIT Press is a university publisher affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts. ... Hendrik Pieter (Henk) Barendregt (b. ... Capital Haarlem Queens Commissioner Mr. ... This article is about the year. ... Hendrik Pieter (Henk) Barendregt (b. ... Capital Haarlem Queens Commissioner Mr. ... Also: 1977 (album) by Ash. ... American Journal of Mathematics, April 2006 issue. ... American Journal of Mathematics, April 2006 issue. ... 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ... Peter Landin is a British computer scientist. ... Communications of the ACM (CACM) is the flagship monthly magazine of the Association for Computing Machinery. ... Year 1965 (MCMLXV) was a common year starting on Friday (link will display full calendar) of the 1965 Gregorian calendar. ... This article does not cite any references or sources. ...

External links

• L. Allison, Some executable λ-calculus examples
• Chris Barker, Some executable (Javascript) simple examples, and text.
• Georg P. Loczewski, The Lambda Calculus and A++
• Raùl Rojas, A Tutorial Introduction to the Lambda Calculus -(PDF)
• David C. Keenan, To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction
• Bret Victor, Alligator Eggs: A Puzzle Game Based on Lambda Calculus
• Lambda Calculus on PlanetMath
• Mike Thyer, Lambda Animator a graphical Java applet demonstrating alternative reduction strategies.
• LCI Lambda Interpreter a simple yet powerful pure calculus interpreter.